Test3 Confidence Interval And Hypothesis Testing

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3. Parameter of interest for a matched pair.

The parameter of interest for a matched pair refers to the specific characteristic or value that is being studied or measured in the context of a matched pair design. This design involves comparing two sets of measurements or observations that are paired or matched in some way, such as before and after measurements on the same subjects. In this case, the correct answer D likely represents a specific parameter that is relevant to the matched pair design, but without further context or information, it is not possible to determine the exact parameter being referred to.

Explanation

The parameter of interest for a matched pair refers to the specific characteristic or value that is being studied or measured in the context of a matched pair design. This design involves comparing two sets of measurements or observations that are paired or matched in some way, such as before and after measurements on the same subjects. In this case, the correct answer D likely represents a specific parameter that is relevant to the matched pair design, but without further context or information, it is not possible to determine the exact parameter being referred to.

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5. Standard deviation of the sampling distribution of x-bar.

The correct answer is C. The standard deviation of the sampling distribution of x-bar refers to the variability of the sample means that would be obtained if an infinite number of samples were taken from the same population. It is a measure of how spread out the sample means are from the true population mean. This standard deviation is also known as the standard error of the mean and is typically denoted as σ/√n, where σ is the population standard deviation and n is the sample size.

Explanation

The correct answer is C. The standard deviation of the sampling distribution of x-bar refers to the variability of the sample means that would be obtained if an infinite number of samples were taken from the same population. It is a measure of how spread out the sample means are from the true population mean. This standard deviation is also known as the standard error of the mean and is typically denoted as σ/√n, where σ is the population standard deviation and n is the sample size.

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7. True or False:
Appropriate collection of data is an important condition for all statistical inferential procedures.

T

F

Appropriate collection of data is indeed an important condition for all statistical inferential procedures. In order to make accurate inferences and draw meaningful conclusions, it is crucial to collect data that is relevant, representative, and reliable. The quality of the data directly impacts the validity and reliability of any statistical analysis or inference made from it. Therefore, ensuring appropriate data collection methods and techniques is essential for the validity and accuracy of statistical inferential procedures.

Explanation

Appropriate collection of data is indeed an important condition for all statistical inferential procedures. In order to make accurate inferences and draw meaningful conclusions, it is crucial to collect data that is relevant, representative, and reliable. The quality of the data directly impacts the validity and reliability of any statistical analysis or inference made from it. Therefore, ensuring appropriate data collection methods and techniques is essential for the validity and accuracy of statistical inferential procedures.

8. While performing a statistical test of hypotheses, we decide to fail to reject the null hypothesis. What can we say about the Type I and type II errors of our decision?

We did not make an error because the P-value is small

We made a type II error, but not a type I error

We made a type I error, but not a type II error

We made both a type I and a type II error

Every time we Reject Ho we can commit a Type I error.
Every time we fail to Reject Ho we can commit a Type II error.

Explanation

Every time we Reject Ho we can commit a Type I error.
Every time we fail to Reject Ho we can commit a Type II error.

9. When performing a one-sample t-test of Ho: µ=µ_o versus Ha: µ>µ_o, the observed effect is equal to the difference between x-bar and µ_o (i.e.,x-bar - µ_o). For a fixed sample size, if the observed effect were to decrease, what would happen to the P-value?

It would get smaller.

It would stay the same.

It would get bigger.

If the observed effect were to decrease, it means that the difference between x-bar and µo becomes smaller. This would result in a smaller t-value and a larger p-value. A larger p-value indicates that the observed effect is more likely to occur by chance, therefore the null hypothesis (Ho) is more likely to be true. Therefore, the correct answer is that the p-value would get bigger.

Explanation

If the observed effect were to decrease, it means that the difference between x-bar and µo becomes smaller. This would result in a smaller t-value and a larger p-value. A larger p-value indicates that the observed effect is more likely to occur by chance, therefore the null hypothesis (Ho) is more likely to be true. Therefore, the correct answer is that the p-value would get bigger.

10. Researchers have postulated that there is no differences in GPA of the BYU Salt Lake Center and BYU-Provo students. Suppose the mean GPA of BYU-Provo students is known to be 3.2. What hypothesis are being tested?

Ho: µ = 3.2 and Ha: µ > 3.2

Ho: µ = 3.2 and Ha: µ < 3.2

Ho: µ = 3.2 and Ha: µ ≠ 3.2

Ho: x ̅ =3.2 and Ho: x ̅ ≠ 3.2

The hypothesis being tested is Ho: µ = 3.2 and Ha: µ ≠ 3.2. This hypothesis suggests that there is no difference in the mean GPA of BYU Salt Lake Center and BYU-Provo students, with the null hypothesis stating that the mean GPA is equal to 3.2 and the alternative hypothesis stating that the mean GPA is not equal to 3.2.

Explanation

The hypothesis being tested is Ho: µ = 3.2 and Ha: µ ≠ 3.2. This hypothesis suggests that there is no difference in the mean GPA of BYU Salt Lake Center and BYU-Provo students, with the null hypothesis stating that the mean GPA is equal to 3.2 and the alternative hypothesis stating that the mean GPA is not equal to 3.2.

11. True or False:
The P-value is the probability, computed assuming Ho is true, that the observed outcome would take a value as extreme or more extreme than that actually observed.

T

F

The statement is true. The p-value is a statistical measure that represents the probability of obtaining a result as extreme or more extreme than the observed outcome, assuming that the null hypothesis (Ho) is true. It is used in hypothesis testing to determine the significance of the results and make conclusions about the population being studied. A smaller p-value indicates stronger evidence against the null hypothesis, while a larger p-value suggests that the observed outcome could occur by chance even if the null hypothesis is true.

Explanation

The statement is true. The p-value is a statistical measure that represents the probability of obtaining a result as extreme or more extreme than the observed outcome, assuming that the null hypothesis (Ho) is true. It is used in hypothesis testing to determine the significance of the results and make conclusions about the population being studied. A smaller p-value indicates stronger evidence against the null hypothesis, while a larger p-value suggests that the observed outcome could occur by chance even if the null hypothesis is true.

12. True or False:
We use t procedures for inference on means when the population standard deviation is unknown.

T

F

We use t procedures for inference on means when the population standard deviation is unknown because the t procedures are designed to account for the uncertainty introduced by not knowing the population standard deviation. The t-distribution is used instead of the normal distribution when the population standard deviation is unknown, and it provides more accurate confidence intervals and hypothesis tests in these cases.

Explanation

We use t procedures for inference on means when the population standard deviation is unknown because the t procedures are designed to account for the uncertainty introduced by not knowing the population standard deviation. The t-distribution is used instead of the normal distribution when the population standard deviation is unknown, and it provides more accurate confidence intervals and hypothesis tests in these cases.

13. Consider the following sampling distributions. The normal curve on the top represents the sampling distribution for x-bars assuming Ho: µ=70 is true. At a=.05, x-bar values that are less than 67 will lead to the rejection of Ho in favor of Ha: µ > 70 The normal curve on the bottom is the sampling distribution for x-bars assuming

µ =75.

Which area represent the probability of Type I error?

A

B

C

D

The area represented by option a represents the probability of Type I error. Type I error occurs when the null hypothesis (Ho) is rejected even though it is true. In this case, the null hypothesis is µ = 70, and if the x-bar values are less than 67, it would lead to the rejection of Ho. Therefore, the area to the left of 67 on the normal curve represents the probability of Type I error.

Explanation

The area represented by option a represents the probability of Type I error. Type I error occurs when the null hypothesis (Ho) is rejected even though it is true. In this case, the null hypothesis is µ = 70, and if the x-bar values are less than 67, it would lead to the rejection of Ho. Therefore, the area to the left of 67 on the normal curve represents the probability of Type I error.

14. Coach Sloan suspects that the supplier of the basketball uniforms are sending shirts that easily get torn. He plans to randomly select some shirts from the next batch and perform a test of significance. What can he do to ensure that the power of the test is high?

Take a small sample of shirts.

Take a large sample of shirts.

Wear each one of them.

Let the other team wear them.

To ensure that the power of the test is high, Coach Sloan should take a large sample of shirts. By increasing the sample size, he will have a better chance of detecting any significant differences in the quality of the shirts. A larger sample size reduces the likelihood of random variation and increases the precision of the test, making it more likely to detect any true differences in the quality of the shirts.

Explanation

To ensure that the power of the test is high, Coach Sloan should take a large sample of shirts. By increasing the sample size, he will have a better chance of detecting any significant differences in the quality of the shirts. A larger sample size reduces the likelihood of random variation and increases the precision of the test, making it more likely to detect any true differences in the quality of the shirts.

15. Consider the following sampling distributions. The normal curve on the top represents the sampling distribution for x-bars assuming Ho: µ=70 At a=.05, x-bar values that are less than 67 will lead to the rejection of Ho in favor of Ha: µ > 70 The normal curve on the bottom is the sampling distribution for x-bars assuming

µ =75.

Which area represent the probability of the power of the test?

A

B

C

D

The area represented by option c represents the probability of the power of the test. The power of a test is the probability of correctly rejecting the null hypothesis when it is false. In this case, the null hypothesis is that the population mean is 70, and the alternative hypothesis is that the population mean is greater than 70. Option c represents the area under the sampling distribution curve for x-bars assuming a true population mean of 75. This area represents the probability of obtaining a sample mean less than 67, which would lead to correctly rejecting the null hypothesis in favor of the alternative hypothesis.

Explanation

The area represented by option c represents the probability of the power of the test. The power of a test is the probability of correctly rejecting the null hypothesis when it is false. In this case, the null hypothesis is that the population mean is 70, and the alternative hypothesis is that the population mean is greater than 70. Option c represents the area under the sampling distribution curve for x-bars assuming a true population mean of 75. This area represents the probability of obtaining a sample mean less than 67, which would lead to correctly rejecting the null hypothesis in favor of the alternative hypothesis.

16.
In practice, if the condition of Normality of the population for t procedures in not met and n < 40, confidence levels and P-values are approximately correct provided:

α is set very low.

The sample standard deviation is not large.

There are no outliers nor strong skewness in the data.

The data are paired.

If the condition of Normality of the population for t procedures is not met and n

Explanation

If the condition of Normality of the population for t procedures is not met and n

17. The significance level is set at α=.01 and a hypothesis test results in a P-value of .02. Which one of the following is a correct conclusion based on the P-value?

The data are consistent with the null hypothesis. Therefore, we do not reject the null hypothesis.

The data are consistent with the null hypothesis. Therefore, we reject the null hypothesis.

The data are not consistent with the null hypothesis. Therefore, we reject the null hypothesis.

There is a 2% chance that the null hypothesis is true. Therefore, we reject the null hypothesis.

There is a 2% chance that the alternative hypothesis is true. Therefore, we accept the null hypothesis.

The correct conclusion based on the P-value of .02 is that the data are consistent with the null hypothesis. This means that there is not enough evidence to reject the null hypothesis at the significance level of α=.01.

Explanation

The correct conclusion based on the P-value of .02 is that the data are consistent with the null hypothesis. This means that there is not enough evidence to reject the null hypothesis at the significance level of α=.01.

18. Researchers have postulated that because of differences in teachers, students at the BYU Salt Lake Center should have a higher GPA than BYU-Provo students. Suppose the mean GPA of BYU-Provo students is known to be 3.2. What hypothesis are being tested?

Ho: µ = 3.2 and Ha: µ > 3.2

Ho: µ = 3.2 and Ha: µ < 3.2

Ho: µ = 3.2 and Ha: µ ≠ 3.2

Ho: x ̅ =3.2 and Ho: x ̅ >3.2

The hypothesis being tested is that the mean GPA of BYU-Provo students is equal to 3.2 (Ho: µ = 3.2) and the alternative hypothesis is that the mean GPA of BYU-Provo students is greater than 3.2 (Ha: µ > 3.2). This suggests that the researchers are interested in determining if there is a significant difference in GPA between BYU-Provo students and BYU Salt Lake Center students, with the expectation that the BYU Salt Lake Center students have a higher GPA.

Explanation

The hypothesis being tested is that the mean GPA of BYU-Provo students is equal to 3.2 (Ho: µ = 3.2) and the alternative hypothesis is that the mean GPA of BYU-Provo students is greater than 3.2 (Ha: µ > 3.2). This suggests that the researchers are interested in determining if there is a significant difference in GPA between BYU-Provo students and BYU Salt Lake Center students, with the expectation that the BYU Salt Lake Center students have a higher GPA.

19. The mean percentage free-throw of 12^th graders is 65%. A researcher suspects that varsity players have higher free-throw percentages than 12^th graders in general. He conduct a study using Ho: µ=65% vs. Ha: µ<65% and computes t-test statistic of 1.5. Which of the following graphs show the appropriate shaded area for the P-value of this test?

A

B

C

D

not-available-via-ai

Explanation

not-available-via-ai

20. Students were randomly assigned to each of the three Stats221 classes at BYU Salt Lake Center. Their final scores after the semester were recorded. To see if there are differences between the average Final scores among the three classes, what statistical procedure should be used for the data in this study?

A one sample t-test for means (not matched pairs).

A two sample t-test for means.

A matched pairs t-test for means.

Analysis of Variance (ANOVA)

A one sample t confidence interval estimate

In this study, the researcher wants to compare the average final scores among the three Stats221 classes. Since there are three independent groups involved, the appropriate statistical procedure to use is Analysis of Variance (ANOVA). ANOVA allows for the comparison of means between multiple groups and determines if there are significant differences among them. The other options, such as one sample t-test or two sample t-test, are not suitable because they are used for comparing means between two groups or a single group, respectively.

Explanation

In this study, the researcher wants to compare the average final scores among the three Stats221 classes. Since there are three independent groups involved, the appropriate statistical procedure to use is Analysis of Variance (ANOVA). ANOVA allows for the comparison of means between multiple groups and determines if there are significant differences among them. The other options, such as one sample t-test or two sample t-test, are not suitable because they are used for comparing means between two groups or a single group, respectively.

21. True or False:
Alpha is the probability of Type I error.

T

F

The statement "Alpha is the probability of Type I error" is true. In hypothesis testing, Type I error refers to rejecting a true null hypothesis. Alpha, also known as the significance level, is the probability of making a Type I error. It is typically set at a predetermined value (e.g., 0.05 or 0.01) and represents the maximum acceptable probability of rejecting a true null hypothesis. Therefore, the statement is correct.

Explanation

The statement "Alpha is the probability of Type I error" is true. In hypothesis testing, Type I error refers to rejecting a true null hypothesis. Alpha, also known as the significance level, is the probability of making a Type I error. It is typically set at a predetermined value (e.g., 0.05 or 0.01) and represents the maximum acceptable probability of rejecting a true null hypothesis. Therefore, the statement is correct.

22. True or False:
Practical significance is only an issue after the results are declared statistically significant.

T

F

Practical significance refers to the real-world importance or relevance of a research finding. It is concerned with whether the observed effect size is meaningful or impactful in practical terms. In the context of this question, the statement suggests that practical significance is only considered after the results have been deemed statistically significant. This implies that statistical significance is a prerequisite for assessing practical significance.

Explanation

Practical significance refers to the real-world importance or relevance of a research finding. It is concerned with whether the observed effect size is meaningful or impactful in practical terms. In the context of this question, the statement suggests that practical significance is only considered after the results have been deemed statistically significant. This implies that statistical significance is a prerequisite for assessing practical significance.

23. True or False:
If a p-value is small, then either the null hypothesis is false or we got a very unlikely sample.

T

F

If a p-value is small, it indicates that the observed data is unlikely to have occurred under the assumption that the null hypothesis is true. Therefore, we reject the null hypothesis and conclude that either the null hypothesis is false or we obtained a very unlikely sample.

Explanation

If a p-value is small, it indicates that the observed data is unlikely to have occurred under the assumption that the null hypothesis is true. Therefore, we reject the null hypothesis and conclude that either the null hypothesis is false or we obtained a very unlikely sample.

24. Consider the following sampling distributions. The normal curve on the top represents the sampling distribution for x-bars assuming Ho: µ=70 is true. At a=.05, x-bar values that are less than 67 will lead to the rejection of Ho in favor of Ha: µ < 70 The normal curve on the bottom is the sampling distribution for x-bars assuming

µ =65.

Which area represent the probability of Type I error?

A

B

C

D

The area represented by option a represents the probability of Type I error. Type I error occurs when the null hypothesis (Ho) is rejected, even though it is true. In this case, the null hypothesis is that µ = 70. If the x-bar values are less than 67, it would lead to the rejection of Ho, indicating a Type I error.

Explanation

The area represented by option a represents the probability of Type I error. Type I error occurs when the null hypothesis (Ho) is rejected, even though it is true. In this case, the null hypothesis is that µ = 70. If the x-bar values are less than 67, it would lead to the rejection of Ho, indicating a Type I error.

25. A sports scientist took an SRS of twenty five high school basketball players. The scientist then tested their free-throw percentages to estimate the mean free-throw percentage of these players. Here are the data: Stem-and-leaf of free-throw percentages n=25 Leaf unit = 1.0

On the basis of these data, would you recommend using a one-sample t confidence interval estimate for µ?

Yes, because an SRS of 25 players was selected.

Yes, because an SRS was taken and we can assume that free-throw percentages is Normally distributed.

No, because the t distribution is not robust in this case since there is an outlier.

No, because no standard deviation is given.

The correct answer is No, because the t distribution is not robust in this case since there is an outlier. In order to use a one-sample t confidence interval estimate for µ, it is important to assume that the data is normally distributed. However, the presence of an outlier can greatly affect the normality assumption and make the t distribution less robust. Therefore, it would not be recommended to use a one-sample t confidence interval estimate in this case.

Explanation

The correct answer is No, because the t distribution is not robust in this case since there is an outlier. In order to use a one-sample t confidence interval estimate for µ, it is important to assume that the data is normally distributed. However, the presence of an outlier can greatly affect the normality assumption and make the t distribution less robust. Therefore, it would not be recommended to use a one-sample t confidence interval estimate in this case.

26. A study was conducted using growing rats to examine the effect of jumping height on the strength of bones. Thirty rats were randomly allocated into groups. The first group of rats did low jumps of 30 cm. And the second group of rats did high jumps of 60 cm. After 8 weeks of 10 jumps per day, 5 days per week, the bone density of the rats was measured in mg/cm³. What is the response variable?

Growing rats

Height of jump

Bone density as measured in mg/cm3.

Age of rat

The response variable in this study is the bone density of the rats, which is measured in mg/cm3. The study aims to examine the effect of jumping height on the strength of bones, and bone density is a measure of bone strength. By comparing the bone density of rats that did low jumps of 30 cm with those that did high jumps of 60 cm, the researchers can determine if jumping height has an impact on bone density.

Explanation

The response variable in this study is the bone density of the rats, which is measured in mg/cm3. The study aims to examine the effect of jumping height on the strength of bones, and bone density is a measure of bone strength. By comparing the bone density of rats that did low jumps of 30 cm with those that did high jumps of 60 cm, the researchers can determine if jumping height has an impact on bone density.

27. The mean percentage free-throw of 12^th graders is 65%. A researcher suspects that varsity players have higher free-throw percentages than 12^th graders in general. He conduct a study using
Ho: µ=65% vs. Ha: µ ≠ 65% and computes t-test statistic of 1.5. Which of the following graphs show the appropriate shaded area for the P-value of this test?

A

B

C

D

not-available-via-ai

Explanation

not-available-via-ai

28. In order to compare free-throw shooting skills of Deacons versus Teachers, 12 Deacons and 16 Teachers were randomly selected to test the hypotheses: Ho: μ_D=μ_T versus Ha: μ_D<μ_T . The results of the free-throw shooting skills test are: Two Sample T-test results (without pooled variances): μ_D=mean of Deacons μ_T=mean of Teachers Ho: μ_D-μ_T = 0 Ha: μ_D-μ_T < 0

Difference	Sample Mean	Std. Err.	DF	t-stat	P-value
μ_D-μ_T	-7.31	4.2917	11	-1.5	0.081

On the basis of the P-value, what should we conclude at α=0.10?

The mean free-throw score for Teachers equals the mean for Deacons.

The mean free-throw score for Teachers is significantly less than the mean for Deacons.

The mean free-throw score for Deacons is significantly less than the mean for Teachers.

The mean free-throw score for Deacons is not significantly less than the mean for Teachers.

If the P-value If the P-value > α, Fail to Reject Ho or Not Significant.

Explanation

If the P-value If the P-value > α, Fail to Reject Ho or Not Significant.

29. While performing a statistical test of hypotheses, we decide to reject the null hypothesis .What can we say about the Type I and type II errors of our decision?

We did not make an error because the P-value is small

We made a type II error, but not a type I error

We made a type I error, but not a type II error

We made both a type I and a type II error

Every time we Reject Ho we can commit a Type I error.
Every time we fail to Reject Ho we can commit a Type II error.

Explanation

Every time we Reject Ho we can commit a Type I error.
Every time we fail to Reject Ho we can commit a Type II error.

30. Which one of the following situations will best allow a cause-and-effect conclusion about the relationship between smoking and lung cancer?

Randomly ask 100 lawyers whether smoking causes cancer.

The best situation that allows a cause-and-effect conclusion about the relationship between smoking and lung cancer is to take a random sample of 100 men and randomly assign them into two groups. One group should be asked to smoke for 10 years while the other group should not smoke. After 10 years, the incidence of lung cancer between the two groups should be measured using a two sample t-test. This controlled experiment allows for the comparison of lung cancer rates between smokers and non-smokers, providing evidence for a cause-and-effect relationship.

Explanation

The best situation that allows a cause-and-effect conclusion about the relationship between smoking and lung cancer is to take a random sample of 100 men and randomly assign them into two groups. One group should be asked to smoke for 10 years while the other group should not smoke. After 10 years, the incidence of lung cancer between the two groups should be measured using a two sample t-test. This controlled experiment allows for the comparison of lung cancer rates between smokers and non-smokers, providing evidence for a cause-and-effect relationship.

31. True or False:
Standard deviation quantifies the variability.

T

F

Standard deviation is a statistical measure that quantifies the amount of variability or dispersion in a dataset. It measures how spread out the values in a dataset are around the mean. A higher standard deviation indicates greater variability, while a lower standard deviation indicates less variability. Therefore, the statement "Standard deviation quantifies the variability" is true.

Explanation

Standard deviation is a statistical measure that quantifies the amount of variability or dispersion in a dataset. It measures how spread out the values in a dataset are around the mean. A higher standard deviation indicates greater variability, while a lower standard deviation indicates less variability. Therefore, the statement "Standard deviation quantifies the variability" is true.

32. The following hypotheses were tested:
Ho: µ=75 versus Ha: µ > 75
where
µ
is the true mean score for Stats221 finals. The test scores of a random sample of students who have taken Stats221 had a mean x-bar = 78. The hypothesis test produced a P-value of 0.0314. With alpha=0.05, do the data give sufficient evidence that the mean final score is greater than 75?0

No, because the p-value is not less than alpha

No, because the results are statistical significant.

No, because the p-value is less than alpha.

Yes, because the p-value is not less than alpha.

Yes, because the p-value is less than alpha.

The correct answer is "Yes, because the p-value is less than alpha." In hypothesis testing, the p-value represents the probability of obtaining a test statistic as extreme as the one observed, assuming the null hypothesis is true. In this case, the p-value is 0.0314, which is less than the significance level alpha of 0.05. This means that there is sufficient evidence to reject the null hypothesis and conclude that the mean final score is greater than 75.

Explanation

The correct answer is "Yes, because the p-value is less than alpha." In hypothesis testing, the p-value represents the probability of obtaining a test statistic as extreme as the one observed, assuming the null hypothesis is true. In this case, the p-value is 0.0314, which is less than the significance level alpha of 0.05. This means that there is sufficient evidence to reject the null hypothesis and conclude that the mean final score is greater than 75.

33. A SRS of 64 BYU students found that the average GPA was x-bar=2.7. Assuming the population standard deviation is known to be 0.3, a margin of error for a 95% confidence interval for the population average GPA is calculated to be 0.0735. Which action below would result in a smaller margin of error?

Using a confidence level of 99%

Using a sample of 50 students.

Using a sample of 100 students.

Taking a different sample of 64 students.

Using a sample of 100 students would result in a smaller margin of error because a larger sample size leads to a more accurate estimate of the population parameter. As the sample size increases, the variability decreases, resulting in a smaller margin of error.

Explanation

Using a sample of 100 students would result in a smaller margin of error because a larger sample size leads to a more accurate estimate of the population parameter. As the sample size increases, the variability decreases, resulting in a smaller margin of error.

34. Consider the following sampling distributions. The normal curve on the top represents the sampling distribution for x-bars assuming Ho: µ=70 is true. At a=.05, x-bar values that are less than 67 will lead to the rejection of Ho in favor of Ha: µ > 70 The normal curve on the bottom is the sampling distribution for x-bars assuming

µ =75.

Which area represent the probability of Type II error?

A

B

C

D

The area represented by option d represents the probability of Type II error. Type II error occurs when we fail to reject the null hypothesis (Ho) when it is actually false. In this case, Ho: µ=70 is false, but if we observe a sample mean (x-bar) that falls within the shaded area represented by option d, we would fail to reject Ho and incorrectly conclude that the population mean (µ) is 70. Therefore, option d represents the probability of making a Type II error.

Explanation

The area represented by option d represents the probability of Type II error. Type II error occurs when we fail to reject the null hypothesis (Ho) when it is actually false. In this case, Ho: µ=70 is false, but if we observe a sample mean (x-bar) that falls within the shaded area represented by option d, we would fail to reject Ho and incorrectly conclude that the population mean (µ) is 70. Therefore, option d represents the probability of making a Type II error.

35. Mangosteen is a fruit containing chemicals called xanthones that are believed to help the body’s cells to function correctly and optimally. In one study four groups of people were compared; the first group was a control group and the other three groups of people were fed either a low dose, a medium dose or a high dose of xanthones from mangosteen. The number of good cells were counted. The table below gives the Analysis of Variance (ANOVA) of these data.
One of the requirements for Analysis of Variances must be equal. On the basis of the output given below, why is that requirement met?Assume that the conditions are met for performing this analysis.

The P-value for the F test statistic is less than α=0.05.

The largest standard deviation divided by the smallest standard deviation is less than 2.

The pooled standard deviation equals 0.4331 which is greater than α=0.05.

There is no information given in the ANOVA output to determine whether the variances are equal.

The requirement for equal variances in Analysis of Variance (ANOVA) is met because the largest standard deviation divided by the smallest standard deviation is less than 2. This indicates that the variability among the different groups is relatively similar and there is no significant difference in the spread of the data.

Explanation

The requirement for equal variances in Analysis of Variance (ANOVA) is met because the largest standard deviation divided by the smallest standard deviation is less than 2. This indicates that the variability among the different groups is relatively similar and there is no significant difference in the spread of the data.

36. In addition to having an SRS, what should be checked in order to validly use the formula

when n=15?

No other checks necessary.

Whether the plot of the population is approximately Normal.

No pattern in the residual plot.

No outliers or strong skewness in a plot of the data.

To validly use the formula when n=15, it is important to check for outliers or strong skewness in a plot of the data. This is because outliers or strong skewness can significantly affect the validity of the formula and the accuracy of the results. Therefore, it is necessary to ensure that the data does not contain any extreme values or unusual distributions that could impact the analysis.

Explanation

To validly use the formula when n=15, it is important to check for outliers or strong skewness in a plot of the data. This is because outliers or strong skewness can significantly affect the validity of the formula and the accuracy of the results. Therefore, it is necessary to ensure that the data does not contain any extreme values or unusual distributions that could impact the analysis.

37. The mean percentage free-throw of 12^th graders is 65%. A researcher suspects that varsity players have higher free-throw percentages than 12^th graders in general. He conduct a study using Ho: µ=65% vs. Ha: µ>65% and computes t-test statistic of 1.5. Which of the following graphs show the appropriate shaded area for the P-value of this test?

A

B

C

D

The t-test statistic of 1.5 indicates that the sample mean is 1.5 standard deviations above the population mean. The alternative hypothesis (Ha: µ>65%) suggests that the researcher is looking for a right-tailed test. Therefore, the appropriate shaded area for the p-value would be on the right side of the distribution curve. Graph b correctly shows the shaded area on the right side, indicating the p-value for the test.

Explanation

The t-test statistic of 1.5 indicates that the sample mean is 1.5 standard deviations above the population mean. The alternative hypothesis (Ha: µ>65%) suggests that the researcher is looking for a right-tailed test. Therefore, the appropriate shaded area for the p-value would be on the right side of the distribution curve. Graph b correctly shows the shaded area on the right side, indicating the p-value for the test.

38. True or False:
Even if the P-value is large, the null hypothesis could be false.

T

F

Even if the P-value is large, it means that there is a high probability of obtaining the observed data if the null hypothesis is true. However, this does not necessarily mean that the null hypothesis is true. There could be other factors or variables that are influencing the results, leading to a large P-value. Therefore, it is possible for the null hypothesis to be false even if the P-value is large.

Explanation

Even if the P-value is large, it means that there is a high probability of obtaining the observed data if the null hypothesis is true. However, this does not necessarily mean that the null hypothesis is true. There could be other factors or variables that are influencing the results, leading to a large P-value. Therefore, it is possible for the null hypothesis to be false even if the P-value is large.

39. Consider the following sampling distributions. The normal curve on the top represents the sampling distribution for x-bars assuming Ho: µ=70 is true. At a=.05, x-bar values that are less than 67 will lead to the rejection of Ho in favor of Ha: µ < 70 The normal curve on the bottom is the sampling distribution for x-bars assuming

µ =65.

Which area represent the probability of the power of the test?

A

B

C

D

The area represented by option c represents the probability of the power of the test.

Explanation

The area represented by option c represents the probability of the power of the test.

40.
The Provo Recreational Office conducted a research of the free-throw percentage of Jr Jazz kids. A percentage of 60% is a “basic” shooting ability and a percentage of 90% is “proficient”. Percentages for a random sample of 1500 Jr Jazz kids from Provo had a mean of 55% with a standard deviation of 20%. What is the value of the standard error of the mean?

0.0136

0.1876

0.5164

1.8754

4.5164

20/sqrt(1500)=.5164

Explanation

20/sqrt(1500)=.5164

41. When is a statistical procedure robust?

When sample is at least 20% of the population?

When it is used even though the sample is not SRS.

When confidence level or the P-value does not change very much even when the conditions are not fully met.

When confidence level or P-value does not change very much even when the conditions are not met. For example, confidence level or P-value using t-distribution is robust because these values do not change very much even if the distribution is not Normal as long as there is no extreme outlier or extreme skewness of the data.

Explanation

When confidence level or P-value does not change very much even when the conditions are not met. For example, confidence level or P-value using t-distribution is robust because these values do not change very much even if the distribution is not Normal as long as there is no extreme outlier or extreme skewness of the data.

42. Consider an SRS of size 20 from a Normally distributed population, If x-bar=45 and s=15, what is the appropriate formula for a 95% confidence interval for µ?

X ̅ ± z*σ / √n

X ̅ ± t*σ ⁄ √n

X ̅ ± z*s ⁄ √n

X ̅± t*s ⁄ √n

When σ is unknown and the population is Normal and n x ̅±t*s⁄√n

Explanation

When σ is unknown and the population is Normal and n x ̅±t*s⁄√n

43.
The life in hours of a particular brand of plasma TV is advertised to have a mean of 30,000 hours. A nationwide electronics chain wants to determine whether to purchase this particular brand. They decide to test a sample of the plasma tvs and purchase these plasma tv unless the test of significance shows evidence that the mean is less 30,000 hours. In other words, they will test the hypotheses ho: µ =30,000 versus Ha: µ < 30,000 and purchase the plasma tv if they fail to reject the null hypotheses. If they reject the null hypothesis, they will not purchase this particular brand of plasma tv. What is the type I error of this test?

Decide to purchase the plasma tv when the mean life in hours really is 30,000 hours.

Decide to purchase the plasma tv when the mean life in hours reall is less than 30,000 hours.

Decide NOT to purchase the plasma tv when the mean life in hours really is 30,000 hours.

Decide NOT to purchase the plasma tv when the mean life in hours really is less than 30,000 hours.

The type I error of this test is deciding NOT to purchase the plasma TV when the mean life in hours really is 30,000 hours. This means that the electronics chain incorrectly rejects the null hypothesis and concludes that the mean life of the plasma TV is less than 30,000 hours, leading to the decision not to purchase the brand. However, in reality, the mean life is actually 30,000 hours.

Explanation

The type I error of this test is deciding NOT to purchase the plasma TV when the mean life in hours really is 30,000 hours. This means that the electronics chain incorrectly rejects the null hypothesis and concludes that the mean life of the plasma TV is less than 30,000 hours, leading to the decision not to purchase the brand. However, in reality, the mean life is actually 30,000 hours.

44. The significance level is set at α=.05 and a hypothesis test results in a P-value of .02. Which one of the following is a correct conclusion based on the P-value?

The data are consistent with the null hypothesis. Therefore, we do not reject the null hypothesis.

The data are consistent with the null hypothesis. Therefore, we reject the null hypothesis.

The data are not consistent with the null hypothesis. Therefore, we do not reject the null hypothesis.

The data are not consistent with the null hypothesis. Therefore, we reject the null hypothesis.

There is a 2% chance that the null hypothesis is true. Therefore, we reject the null hypothesis.

The correct conclusion based on the P-value of .02 is that the data are not consistent with the null hypothesis. Therefore, we reject the null hypothesis. This is because the P-value is less than the significance level of α=.05, indicating that the probability of obtaining the observed data under the assumption that the null hypothesis is true is very low. Thus, we have evidence to reject the null hypothesis and support an alternative hypothesis.

Explanation

The correct conclusion based on the P-value of .02 is that the data are not consistent with the null hypothesis. Therefore, we reject the null hypothesis. This is because the P-value is less than the significance level of α=.05, indicating that the probability of obtaining the observed data under the assumption that the null hypothesis is true is very low. Thus, we have evidence to reject the null hypothesis and support an alternative hypothesis.

45. A study was conducted to determine the average GPA of students enrolled at the BYU Salt Lake Center. A random sample of 50 students was selected, the mean GPA computed and a 90% confidence interval obtained. The resulting confidence interval is (2.35, 3.87). This interval gives us

The range of reasonable values for the true mean GPA of students enrolled at the BYU Salt Lake Center.

The range of reasonable values for the sample mean GPA of students enrolled at the BYU Salt Lake Center.

The range for 90% of GPA’s of all students enrolled at the BYU Salt Lake Center.

The confidence interval (2.35, 3.87) provides a range of reasonable values for the true mean GPA of students enrolled at the BYU Salt Lake Center. This means that we can be 90% confident that the true mean GPA falls within this interval.

Explanation

The confidence interval (2.35, 3.87) provides a range of reasonable values for the true mean GPA of students enrolled at the BYU Salt Lake Center. This means that we can be 90% confident that the true mean GPA falls within this interval.

46. Which of the following questions does a test of significance answer?

Is the sample or experiment properly designed?

Is the observed effect too large to be due to chance alone?

How much confidence can be placed in the observed effect?

What is the probability that the parameter is different from the statistic?

A test of significance answers the question of whether the observed effect is too large to be due to chance alone. It determines the probability of obtaining the observed result if there is no real effect or difference. If the probability is very low (typically below a predetermined significance level), it suggests that the observed effect is unlikely to be due to chance and is therefore considered statistically significant.

Explanation

A test of significance answers the question of whether the observed effect is too large to be due to chance alone. It determines the probability of obtaining the observed result if there is no real effect or difference. If the probability is very low (typically below a predetermined significance level), it suggests that the observed effect is unlikely to be due to chance and is therefore considered statistically significant.

47. True or False:
A small P-value means the result have both practical and statistical significance.

T

F

A small P-value indicates that there is strong evidence against the null hypothesis, suggesting that the observed results are unlikely to have occurred by chance. However, it does not necessarily imply practical significance. Practical significance refers to the real-world importance or relevance of the findings, which may not always align with statistical significance. Therefore, the statement that a small P-value means the result has both practical and statistical significance is false.

Explanation

A small P-value indicates that there is strong evidence against the null hypothesis, suggesting that the observed results are unlikely to have occurred by chance. However, it does not necessarily imply practical significance. Practical significance refers to the real-world importance or relevance of the findings, which may not always align with statistical significance. Therefore, the statement that a small P-value means the result has both practical and statistical significance is false.

48. You want to compare the daily items sold for two game consoles: Playstation3(PS3) and NintendoWII(WII). Over the next 80 days, 40 days are randomly assigned to PS3 and 40 days to WII. At the end, you compute a 95% confidence interval for the difference in mean daily items sold for the two game consoles to be (-20, -10). On the basis of this confidence interval, can you conclude that there is a significant difference between the mean daily items sold for the two game consoles at α=0.05? (i.e., can you reject

No because the mean daily items sold cannot be negative.

No, because the confidence interval contains zero.

Yes, because the confidence interval does not contains zero.

Using confidence interval to accept or reject Ho:
When the hypothesized value is in the interval then we fail to reject Ho. When it is NOT in the interval, we Reject Ho.
When comparing two means, the hypothesized value is zero.

Explanation

Using confidence interval to accept or reject Ho:
When the hypothesized value is in the interval then we fail to reject Ho. When it is NOT in the interval, we Reject Ho.
When comparing two means, the hypothesized value is zero.

49. Coach Rose claims that by using his method of shooting, basketball players can increase their scores by an average of 15 points. Wesley, a former basketball player is skeptical of this claim and wants to test the hypotheses: Ho: µ =15 versus Ha: µ >15 where
µ
represents the mean increase in scores of the population of all basketball players who have used the Rose method. Wesley collects data from an SRS of 25 players who use the Rose method. He finds that the sample mean increase in scores of these 25 players is 13 points with s=7. Assuming that the distribution of their scores is approximately normal, what is the p-value for this test?

0.0830

0.0228

0.05 < p-value

0.10 < p-value < 0.20

The p-value for this test is 0.05

Explanation

The p-value for this test is 0.05

50. True or False:
alpha denotes the significance level.

T

F

The statement is true because in statistical hypothesis testing, the significance level, denoted by alpha, is the predetermined threshold at which the null hypothesis is rejected. It represents the probability of rejecting the null hypothesis when it is actually true. By convention, alpha is typically set at 0.05 or 0.01, indicating a 5% or 1% chance of making a Type I error (rejecting the null hypothesis when it is true). Therefore, alpha does indeed denote the significance level.

Explanation

The statement is true because in statistical hypothesis testing, the significance level, denoted by alpha, is the predetermined threshold at which the null hypothesis is rejected. It represents the probability of rejecting the null hypothesis when it is actually true. By convention, alpha is typically set at 0.05 or 0.01, indicating a 5% or 1% chance of making a Type I error (rejecting the null hypothesis when it is true). Therefore, alpha does indeed denote the significance level.

51. Mangosteen is a fruit containing chemicals called xanthones that are believed to help the body’s cells to function correctly and optimally. In one study four groups of people were compared; the first group was a control group and the other three groups of people were fed either a low dose, a medium dose or a high dose of xanthones from mangosteen. The number of good cells were counted. The following gives the Analysis of Variance (ANOVA) of these data. What can you conclude about the means of the four groups at α=0.05? Assume that the conditions are met for performing this analysis.

There is no significance difference between the mean count of good cells of the four groups.

The mean count of good cells is significantly different for all four groups.

Based on the given information, the ANOVA analysis suggests that there is a significant difference in the mean count of good cells between the high dosage group and the control and low dosage groups. However, there is no information provided in the ANOVA output to determine if the mean count of good cells is significantly different for all four groups or if there is no significant difference between the mean count of good cells of the four groups. Therefore, the correct answer is that the mean count of good cells of the high dosage group is significantly greater than the mean count of good cells of the control and low dosage groups.

Explanation

Based on the given information, the ANOVA analysis suggests that there is a significant difference in the mean count of good cells between the high dosage group and the control and low dosage groups. However, there is no information provided in the ANOVA output to determine if the mean count of good cells is significantly different for all four groups or if there is no significant difference between the mean count of good cells of the four groups. Therefore, the correct answer is that the mean count of good cells of the high dosage group is significantly greater than the mean count of good cells of the control and low dosage groups.

52. The average hours spent per week doing the Stats221 homework for BYU students has been 10 hours with a standard deviation of 4 hours. The Statistics Department wanted to test the hypotheses
Ho: µ=10 versus Ha: µ<10. They selected an α=0.05 and took a random sample of 100 students who had taken the class. The sample average obtained was 9.75 hours. This result was statistically significant with a P-value <0.01. Are these results also practically significant?

No, because a 15-minute difference is probably too small to matter.

No, because the sample is not large enough.

Yes, because the P-value is less than α

Yes, because the observed mean is less than 10.

Yes, because results that are statistically significant are also practically significant.

The answer "No, because a 15-minute difference is probably too small to matter" is correct because the given information states that the sample average obtained was 9.75 hours, which is only a 15-minute difference from the hypothesized mean of 10 hours. This small difference suggests that there is not a significant practical difference in the amount of time spent on Stats221 homework for BYU students.

Explanation

The answer "No, because a 15-minute difference is probably too small to matter" is correct because the given information states that the sample average obtained was 9.75 hours, which is only a 15-minute difference from the hypothesized mean of 10 hours. This small difference suggests that there is not a significant practical difference in the amount of time spent on Stats221 homework for BYU students.

53. True or False:
P-value is the probability that the Null hypothesis is false.

T

F

The given answer is false because the p-value is the probability of obtaining a test statistic as extreme as the one observed, assuming that the null hypothesis is true. It is not the probability that the null hypothesis is false.

Explanation

The given answer is false because the p-value is the probability of obtaining a test statistic as extreme as the one observed, assuming that the null hypothesis is true. It is not the probability that the null hypothesis is false.

54. An SRS of 500 students in BYU-Provo (population 25,000) responded to a survey which asked their GPA. A 95% confidence interval for the mean GPA was obtained. This survey was also given to a separate SRS of 500 students at the BYU-Salt Lake Center (population 7,000) who also answered the GPA question. A separate 95% confidence interval statement about the mean GPA of all students in the BYU-Slat Lake Center was also constructed. Assume the standard deviation, s, is known and is the same for both groups. The margin of error for the BYU-Salt Lake Center is

The same as in BYU-Provo, because the two sample sizes are the same.

Smaller than in BYU-Provo, because the population is much larger in BYU-Provo than BYU-Salt Lake Center.

Larger than BYU-Provo, because there are fewer students in BYU-Salt Lake

May be either smaller or larger than BYU-Provo.

The margin of error for the BYU-Salt Lake Center is the same as in BYU-Provo because the two sample sizes are the same. The margin of error is influenced by the sample size, with larger sample sizes resulting in smaller margins of error. Since both surveys had a sample size of 500 students, the margin of error would be the same for both groups.

Explanation

The margin of error for the BYU-Salt Lake Center is the same as in BYU-Provo because the two sample sizes are the same. The margin of error is influenced by the sample size, with larger sample sizes resulting in smaller margins of error. Since both surveys had a sample size of 500 students, the margin of error would be the same for both groups.

55. The P-value for a significance test is defined as

The probability that the null hypothesis is true.

The probability that the alternative hypothesis is wrong.

The probability of rejecting the null hypothesis when in fact the null hypothesis is true.

The probability that the alternative hypothesis is false, assuming the null hypothesis is true.

The P-value for a significance test is the probability of obtaining a test statistic that has a value at least as extreme as that actually observed, assuming the null hypothesis is true. This means that if the P-value is small, it suggests that the observed data is unlikely to have occurred by chance alone, and we have evidence to reject the null hypothesis. Conversely, if the P-value is large, it suggests that the observed data is likely to have occurred by chance alone, and we do not have enough evidence to reject the null hypothesis.

Explanation

The P-value for a significance test is the probability of obtaining a test statistic that has a value at least as extreme as that actually observed, assuming the null hypothesis is true. This means that if the P-value is small, it suggests that the observed data is unlikely to have occurred by chance alone, and we have evidence to reject the null hypothesis. Conversely, if the P-value is large, it suggests that the observed data is likely to have occurred by chance alone, and we do not have enough evidence to reject the null hypothesis.

56. Consider an SRS of size 16 from a Normally distributed population with σ=16, If x-bar 12x">=45 and s=12, what is the appropriate formula for a 95% confidence interval for µ?

X ̅ ± z* σ ⁄ √n

X ̅ ± z* s ⁄ √n

X ̅ ± t* σ ⁄ √n

X ̅ ± t* s ⁄ √n

Use z* and σ.

Explanation

Use z* and σ.

57. Consider the following sampling distributions. The normal curve on the top represents the sampling distribution for x-bars assuming Ho: µ=70 is true. At a=.05, x-bar values that are less than 67 will lead to the rejection of Ho in favor of Ha: µ < 70 The normal curve on the bottom is the sampling distribution for x-bars assuming

µ =65.

Which area represent the probability of Type II error?

A

B

C

D

The area represented by choice d represents the probability of Type II error. Type II error occurs when we fail to reject the null hypothesis (Ho) when it is actually false. In this case, the null hypothesis is that the population mean (µ) is equal to 70. The alternative hypothesis (Ha) is that the population mean is less than 70. Therefore, if the true population mean is actually 65 (as represented by the normal curve on the bottom), any x-bar values that fall within the shaded area (choice d) would lead to a Type II error.

Explanation

The area represented by choice d represents the probability of Type II error. Type II error occurs when we fail to reject the null hypothesis (Ho) when it is actually false. In this case, the null hypothesis is that the population mean (µ) is equal to 70. The alternative hypothesis (Ha) is that the population mean is less than 70. Therefore, if the true population mean is actually 65 (as represented by the normal curve on the bottom), any x-bar values that fall within the shaded area (choice d) would lead to a Type II error.

58. Suppose that a researcher would like to predict the results of mayoral election on 20 cities in Utah. We randomly poll 1501 voters from all these cities. For each city he either “called” a winner or declared the election “too close to call.” The researcher correctly predicted the outcome of 19 elections. However, for one of the mayoral election, the researcher “called Anderson the winner when Sotomayor actually won the election. What is the most likely explanation for this mistake?

The researcher is biased against minorities.

The researcher does not like Sotomayor.

Multiple analyses increase the chances of making a Type I error.

Multiple analyses increase the chances of making a Type II error.

The most likely explanation for the researcher's mistake is that multiple analyses increase the chances of making a Type I error. This means that with multiple predictions, there is a higher probability of incorrectly predicting the outcome of an election. In this case, the researcher correctly predicted the outcome of 19 elections, but made an error in one. This suggests that the mistake was likely due to the increased likelihood of making a Type I error when analyzing multiple elections.

Explanation

The most likely explanation for the researcher's mistake is that multiple analyses increase the chances of making a Type I error. This means that with multiple predictions, there is a higher probability of incorrectly predicting the outcome of an election. In this case, the researcher correctly predicted the outcome of 19 elections, but made an error in one. This suggests that the mistake was likely due to the increased likelihood of making a Type I error when analyzing multiple elections.

59. When performing a one-sample t-test of Ho: µ=µ_o versus Ha: µ>µ_o, the observed effect is equal to the difference between x-bar and µ_o (i.e.,x-bar - µ_o). For a fixed sample size, if the observed effect were to increase, what would happen to the P-value?

It would get smaller.

It would stay the same.

It would get bigger.

When performing a one-sample t-test, the P-value represents the probability of obtaining a test statistic as extreme as the observed effect, assuming the null hypothesis is true. As the observed effect increases, the test statistic becomes more extreme, resulting in a smaller P-value. Therefore, if the observed effect were to increase, the P-value would get smaller.

Explanation

When performing a one-sample t-test, the P-value represents the probability of obtaining a test statistic as extreme as the observed effect, assuming the null hypothesis is true. As the observed effect increases, the test statistic becomes more extreme, resulting in a smaller P-value. Therefore, if the observed effect were to increase, the P-value would get smaller.

60. We wish to compare two game consoles on the market which were deemed preferred by video game players. Initial testing leads us to believe that Nintendo WII will be more preferred by video game players than Playstation 3. Eighty players are randomly assigned to the two game consoles so that 40 players get WII and 40 players get PS3. The researcher determines in each case whether or not the game console is preferred by video game players. Which statistical procedure should the researcher use for an appropriate test of significance?

A one sample t-test for means (not matched pairs).

A two sample t-test for means.

A matched pairs t-test for means.

Analysis of Variance (ANOVA)

A one sample t confidence interval estimate

The researcher wants to compare two game consoles to determine if there is a significant difference in preference among video game players. Since there are two independent groups (WII and PS3) and the researcher wants to compare means (preference), a two sample t-test for means is the appropriate statistical procedure. This test will allow the researcher to determine if there is a significant difference in preference between the two game consoles.

Explanation

The researcher wants to compare two game consoles to determine if there is a significant difference in preference among video game players. Since there are two independent groups (WII and PS3) and the researcher wants to compare means (preference), a two sample t-test for means is the appropriate statistical procedure. This test will allow the researcher to determine if there is a significant difference in preference between the two game consoles.

61.
A study conducted by researchers at BYU investigated the number of months Returned Missionaries get married after coming back from their missions. A random sample of 25 married RM were selected. The average number of months from returning to getting married for these RM’s was 16 months. When testing Ho: µ = 12 months versus Ha: µ > 12 months, the P-value was found to be 0.04. Which of the following is a correct interpretation of this P-value?

The correct interpretation of the P-value is that if the average number of months RM's get married after their mission was indeed 12 months, the probability that RM's get married after 16 months or greater is 0.04.

Explanation

The correct interpretation of the P-value is that if the average number of months RM's get married after their mission was indeed 12 months, the probability that RM's get married after 16 months or greater is 0.04.

62. A SRS of 64 BYU students found that the average GPA was x-bar=2.7. Assuming the population standard deviation is known to be 0.3, a margin of error for a 95% confidence interval for the population average GPA is calculated to be 0.0735. Which action below would result in a larger margin of error?

Using a confidence level of 99%.

Using a confidence level of 90%

Using a sample of 100 students

Taking a different sample of 64 students.

Using a confidence level of 99% would result in a larger margin of error because a higher confidence level requires a wider interval to capture a larger range of possible values. This means that there is a higher level of certainty in the estimate, but it also leads to a larger margin of error.

Explanation

Using a confidence level of 99% would result in a larger margin of error because a higher confidence level requires a wider interval to capture a larger range of possible values. This means that there is a higher level of certainty in the estimate, but it also leads to a larger margin of error.

63. Suppose we want a 95% confidence interval for the average amount spend on dates at BYU. The amount spend on dates follow a normal distribution with a standard deviation σ= $20. How large should the sample be so that 95% confidence interval has a margin of error of $3?

170

170.7

171

100

To calculate the sample size needed for a 95% confidence interval with a margin of error of $3, we need to use the formula for sample size calculation. The formula is:

n = (Z * σ / E)^2

Where:
n = sample size
Z = Z-score for the desired confidence level (for 95% confidence level, Z = 1.96)
σ = standard deviation
E = margin of error

Plugging in the values, we have:
n = (1.96 * 20 / 3)^2
n = (39.2 / 3)^2
n = 13.07^2
n ≈ 170.7

Therefore, the sample size needed for a 95% confidence interval with a margin of error of $3 is approximately 171.

Explanation

To calculate the sample size needed for a 95% confidence interval with a margin of error of $3, we need to use the formula for sample size calculation. The formula is:

n = (Z * σ / E)^2

Where:
n = sample size
Z = Z-score for the desired confidence level (for 95% confidence level, Z = 1.96)
σ = standard deviation
E = margin of error

Plugging in the values, we have:
n = (1.96 * 20 / 3)^2
n = (39.2 / 3)^2
n = 13.07^2
n ≈ 170.7

Therefore, the sample size needed for a 95% confidence interval with a margin of error of $3 is approximately 171.

64. Coach Rose claims that by using his method of shooting, basketball players can increase their scores by an average of 15 points. Wesley, a former basketball player is skeptical of this claim and wants to test the hypotheses: Ho: µ = 15 versus Ha:this claim and wants to test the hypothesis: Ha: µ NE 15 where mu represents the mean increase in scores of the population of basketball players who have used the Rose method. Wesley collects data from an SRS of 25 players who use the Rose method. He finds the sample mean increase in scores of these 25 players is 13 points with s=7. The test statistic was computed to be -1.43, what is the p-value for this test?

0.0830

0.0228

0.05 < p-value

0.10 < p-value < 0.20

The p-value for this test is between 0.10 and 0.20. This means that there is moderate evidence against the null hypothesis and suggests that the mean increase in scores using the Rose method may be different from 15 points. However, the evidence is not strong enough to conclude that the mean increase is definitely not 15 points.

Explanation

The p-value for this test is between 0.10 and 0.20. This means that there is moderate evidence against the null hypothesis and suggests that the mean increase in scores using the Rose method may be different from 15 points. However, the evidence is not strong enough to conclude that the mean increase is definitely not 15 points.

65. True or False:
The t-distribution with df=8 has a smaller spread than the standard normal distribution.

T

F

The t-distribution with df=8 does not have a smaller spread than the standard normal distribution. The t-distribution has fatter tails than the standard normal distribution, which means it has a larger spread. The degrees of freedom (df) in the t-distribution affect the shape and spread of the distribution, but a t-distribution with df=8 will still have a larger spread than the standard normal distribution.

Explanation

The t-distribution with df=8 does not have a smaller spread than the standard normal distribution. The t-distribution has fatter tails than the standard normal distribution, which means it has a larger spread. The degrees of freedom (df) in the t-distribution affect the shape and spread of the distribution, but a t-distribution with df=8 will still have a larger spread than the standard normal distribution.

66. The following hypotheses were tested:
Ho: µ=75 versus
Ha: µ > 75
where
µ
is the true mean score for Stats221 finals. The test scores of a random sample of students who have taken Stats221 had a mean x-bar = 78. The hypothesis test produced a P-value of 0.314. With alpha=0.05, do the data give sufficient evidence that the mean final score is greater than 75?

No, because the p-value is not less than alpha

No, because the results are statistical significant.

No, because the p-value is less than alpha.

Yes, because the p-value is not less than alpha.

Yes, because the p-value is less than alpha.

The correct answer is "No, because the p-value is not less than alpha." This means that the data does not provide sufficient evidence to reject the null hypothesis. The p-value is the probability of obtaining a test statistic as extreme as the one observed, assuming the null hypothesis is true. In this case, the p-value is greater than the significance level (alpha=0.05), indicating that the observed mean score of 78 is not significantly different from the hypothesized mean score of 75.

Explanation

The correct answer is "No, because the p-value is not less than alpha." This means that the data does not provide sufficient evidence to reject the null hypothesis. The p-value is the probability of obtaining a test statistic as extreme as the one observed, assuming the null hypothesis is true. In this case, the p-value is greater than the significance level (alpha=0.05), indicating that the observed mean score of 78 is not significantly different from the hypothesized mean score of 75.

67. You want to compare the daily items sold for two game consoles: Playstation3(PS3) and NintendoWII(WII). Over the next 80 days, 40 days are randomly assigned to PS3 and 40 days to WII. At the end, you compute a 95% confidence interval for the difference in mean daily items sold for the two game consoles to be (-20, 10). On the basis of this confidence interval, can you conclude that there is a significant difference between the mean daily items sold for the two game consoles at α=0.05? (i.e., can you reject

No because the mean daily items sold cannot be negative.

No, because the confidence interval contains zero.

Yes, because the confidence interval contains zero.

Using confidence interval to accept or reject Ho:
When the hypothesized value is in the interval then we fail to reject Ho. When it is NOT in the interval, we Reject Ho.
When comparing two means, the hypothesized value is zero.

Explanation

Using confidence interval to accept or reject Ho:
When the hypothesized value is in the interval then we fail to reject Ho. When it is NOT in the interval, we Reject Ho.
When comparing two means, the hypothesized value is zero.

68. For a one sided test on µ with σ known, the P-value is represented as the area in the tail of a Normal curve. What does this Normal curve represent?

Values of the measurements in the sample data set.

All possible measurements for the response variable and their frequencies.

The population of measurements about which we wish to make an inference.

All possible values of x-bars and how often they occur if Ho were true.

The Normal curve represents all possible values of x-bars and how often they occur if the null hypothesis (Ho) were true. This means that the curve represents the distribution of sample means that would be obtained if the null hypothesis were true. The area in the tail of the Normal curve represents the probability of observing a sample mean as extreme as the one obtained in the actual data, assuming that the null hypothesis is true.

Explanation

The Normal curve represents all possible values of x-bars and how often they occur if the null hypothesis (Ho) were true. This means that the curve represents the distribution of sample means that would be obtained if the null hypothesis were true. The area in the tail of the Normal curve represents the probability of observing a sample mean as extreme as the one obtained in the actual data, assuming that the null hypothesis is true.

69.
In order to estimate the mean GPA for BYU students, a researcher takes a SRS of GPAs for 81 students. A 96% confidence interval for the mean GPA was computed to be (2.84, 3.06) using x-bar= 2.95 and s=0.5. On the basis of this confidence interval, can we conclude at alpha= 0.04 that the mean GPA for BYU students differs from 3.1?

No, because 3.1 is within one standard deviation of the sample mean

No, because the confidence interval does not include the value 3.1.

Yes, because the BYU students are smarter.

Yes, because the sample mean of 2.95 is less than 3.1.

Yes, because the confidence interval does not include the value 3.1.

Based on the given information, a 96% confidence interval for the mean GPA was computed to be (2.84, 3.06) using x-bar= 2.95 and s=0.5. The confidence interval does not include the value 3.1, which means that there is evidence to suggest that the mean GPA for BYU students is different from 3.1. Therefore, the correct answer is "Yes, because the confidence interval does not include the value 3.1."

Explanation

Based on the given information, a 96% confidence interval for the mean GPA was computed to be (2.84, 3.06) using x-bar= 2.95 and s=0.5. The confidence interval does not include the value 3.1, which means that there is evidence to suggest that the mean GPA for BYU students is different from 3.1. Therefore, the correct answer is "Yes, because the confidence interval does not include the value 3.1."

70. Based on a random sample of 50 students and a known population and sigma, a 90% confidence interval for the mean GPA of all students was calculated as (2.35, 3.87). Which of the following is a correct statement regarding this confidence interval?

90% of the GPA of students are between (2.35, 3.87).

90% of the time, the mean GPA of students will be in the interval from 2.35 to 3.87.

We are 90% confident that the mean GPA of students is between 2.35 and 3.87.

We are 90% confident that the interval (2.35, 3.87) contains the mean GPA of this sample of 50 students.

The correct answer is "We are 90% confident that the mean GPA of students is between 2.35 and 3.87." This statement accurately reflects the interpretation of a confidence interval. A confidence interval provides a range of values within which the true population parameter is likely to fall, with a specified level of confidence. In this case, the confidence interval suggests that we can be 90% confident that the mean GPA of all students falls between 2.35 and 3.87.

Explanation

The correct answer is "We are 90% confident that the mean GPA of students is between 2.35 and 3.87." This statement accurately reflects the interpretation of a confidence interval. A confidence interval provides a range of values within which the true population parameter is likely to fall, with a specified level of confidence. In this case, the confidence interval suggests that we can be 90% confident that the mean GPA of all students falls between 2.35 and 3.87.

71. A random sample of 70 measurements of the free-throw percentage of Jr. Jazz players gave a mean of .60 and standard deviation of .10. Which statistical procedure should be used if we want to estimate the true mean free-throw percentage of the JR Jazz players with 95% confidence?

A one sample t-test for means (not matched pairs).

A two sample t-test for means.

A matched pairs t-test for means.

Analysis of Variance (ANOVA)

A one sample t confidence interval estimate

A one sample t confidence interval estimate should be used to estimate the true mean free-throw percentage of the Jr. Jazz players with 95% confidence. This procedure is appropriate when we have a single sample and want to estimate the population mean. The t-test takes into account the sample mean, sample size, and sample standard deviation to calculate a confidence interval that represents the range within which the true population mean is likely to fall. In this case, the mean of .60 and standard deviation of .10 from the random sample of 70 measurements can be used to calculate the confidence interval.

Explanation

A one sample t confidence interval estimate should be used to estimate the true mean free-throw percentage of the Jr. Jazz players with 95% confidence. This procedure is appropriate when we have a single sample and want to estimate the population mean. The t-test takes into account the sample mean, sample size, and sample standard deviation to calculate a confidence interval that represents the range within which the true population mean is likely to fall. In this case, the mean of .60 and standard deviation of .10 from the random sample of 70 measurements can be used to calculate the confidence interval.

72. Which one of the following statements best describes the logic of tests of significance?

An outcome that is quite likely to happen if Ho were true is good evidence that Ho is true.

If the probability of Ho being true is very small, then Ho cannot be true.

An outcome that would rarely happen if Ho were true is good evidence that Ho is not true.

The correct answer suggests that if an outcome is unlikely to occur if the null hypothesis (Ho) is true, then it is strong evidence that the null hypothesis is not true. This is because if the null hypothesis were true, the outcome would be expected to happen more frequently. Therefore, the occurrence of a rare outcome contradicts the null hypothesis and supports an alternative hypothesis.

Explanation

The correct answer suggests that if an outcome is unlikely to occur if the null hypothesis (Ho) is true, then it is strong evidence that the null hypothesis is not true. This is because if the null hypothesis were true, the outcome would be expected to happen more frequently. Therefore, the occurrence of a rare outcome contradicts the null hypothesis and supports an alternative hypothesis.

73. True or False:
Alpha should be large if the consequences of a type I error are very serious.

T

F

If the consequences of a type I error are very serious, alpha should be small, not large. Alpha is the level of significance or the probability of committing a type I error. A type I error occurs when a null hypothesis is rejected incorrectly. By setting a small alpha value, the researcher is being more cautious and reducing the chances of falsely rejecting the null hypothesis. Therefore, the correct answer is false.

Explanation

If the consequences of a type I error are very serious, alpha should be small, not large. Alpha is the level of significance or the probability of committing a type I error. A type I error occurs when a null hypothesis is rejected incorrectly. By setting a small alpha value, the researcher is being more cautious and reducing the chances of falsely rejecting the null hypothesis. Therefore, the correct answer is false.

74. True or False:
Margin of error accounts for sampling variability as well as variability due to non-response and measurement error.

T

F

The correct answer is False. Margin of error only accounts for sampling variability and does not take into consideration variability due to non-response and measurement error.

Explanation

The correct answer is False. Margin of error only accounts for sampling variability and does not take into consideration variability due to non-response and measurement error.

75. Researchers want to compare the mean levels of the good cholesterol and in order to do this, they should perform

A one sample t-test for means (not matched pairs).

A two sample t-test for means.

A matched pairs t-test for means.

Analysis of Variance (ANOVA)

A one sample t confidence interval estimate

In order to compare the mean levels of the good cholesterol, researchers should perform a matched pairs t-test for means. This test is appropriate when the same individuals are measured twice, such as before and after an intervention or treatment. It allows for the comparison of the means between the two time points for the same group of individuals, taking into account the paired nature of the data. This test is suitable for determining if there is a significant difference in the mean levels of the good cholesterol before and after the intervention.

Explanation

In order to compare the mean levels of the good cholesterol, researchers should perform a matched pairs t-test for means. This test is appropriate when the same individuals are measured twice, such as before and after an intervention or treatment. It allows for the comparison of the means between the two time points for the same group of individuals, taking into account the paired nature of the data. This test is suitable for determining if there is a significant difference in the mean levels of the good cholesterol before and after the intervention.

76. GPA of students of a specified population are Normally distributed with known standard deviation of 1.1 points. A 95% confidence interval, (2.35, 3.87), was calculated from a simple random sample of twenty-five students. Which of the following is a correct interpretation of “95% confidence”?

95% of the students’ GPA in the population are contained in the confidence interval.

The interval contains 95% of the sample GPAs.

The correct interpretation of "95% confidence" is that if this study were to be repeated a large number of times, approximately 95% of the confidence intervals calculated would include the true mean GPA of the population. This means that there is a high level of confidence that the true mean falls within the given interval.

Explanation

The correct interpretation of "95% confidence" is that if this study were to be repeated a large number of times, approximately 95% of the confidence intervals calculated would include the true mean GPA of the population. This means that there is a high level of confidence that the true mean falls within the given interval.