Quantitative Methods II

1) When comparing two population means using data that are gathered from a matched pairs experiment, the test statistic for is Student t distributed with degrees of freedom, provided that the differences are normally distributed.

True

False

When comparing two population means using data from a matched pairs experiment, the test statistic for this comparison is Student t distributed with degrees of freedom, assuming that the differences between the pairs are normally distributed. This means that if the differences between the pairs follow a normal distribution, the t-test can be used to compare the means.

Explanation

When comparing two population means using data from a matched pairs experiment, the test statistic for this comparison is Student t distributed with degrees of freedom, assuming that the differences between the pairs are normally distributed. This means that if the differences between the pairs follow a normal distribution, the t-test can be used to compare the means.

2) A Marine drill instructor recorded the time in which each of 10 recruits completed an obstacle course both before and after basic training. To test whether any improvement occurred, the instructor would use a t-distribution with 9 degrees of freedom.

True

False

The instructor would use a t-distribution with 9 degrees of freedom because the sample size is 10 and the degrees of freedom for a t-distribution is calculated by subtracting 1 from the sample size. Since the instructor is comparing the performance of the same group of recruits before and after basic training, it is appropriate to use a paired t-test. The t-distribution is used for small sample sizes and when the population standard deviation is unknown. Therefore, the given statement is true.

Explanation

The instructor would use a t-distribution with 9 degrees of freedom because the sample size is 10 and the degrees of freedom for a t-distribution is calculated by subtracting 1 from the sample size. Since the instructor is comparing the performance of the same group of recruits before and after basic training, it is appropriate to use a paired t-test. The t-distribution is used for small sample sizes and when the population standard deviation is unknown. Therefore, the given statement is true.

3) In testing the difference between two population means using two independent samples, we use the pooled variance in estimating the standard error of the sampling distribution of the sample mean difference if the populations are normal with equal variances.

True

False

In testing the difference between two population means using two independent samples, we use the pooled variance in estimating the standard error of the sampling distribution of the sample mean difference if the populations are normal with equal variances. This is because when the populations have equal variances, pooling the variances increases the precision of the estimate and leads to more accurate results. By using the pooled variance, we can obtain a more reliable estimate of the standard error, which is crucial in hypothesis testing and determining the statistical significance of the mean difference.

Explanation

In testing the difference between two population means using two independent samples, we use the pooled variance in estimating the standard error of the sampling distribution of the sample mean difference if the populations are normal with equal variances. This is because when the populations have equal variances, pooling the variances increases the precision of the estimate and leads to more accurate results. By using the pooled variance, we can obtain a more reliable estimate of the standard error, which is crucial in hypothesis testing and determining the statistical significance of the mean difference.

4) Repeated measurements from the same individuals is an example of data collected from matched pairs experiment.

True

False

Repeated measurements from the same individuals can be considered as data collected from a matched pairs experiment because it involves comparing two sets of measurements taken from the same individuals under different conditions or treatments. In a matched pairs experiment, each individual is their own control, which helps to reduce the variability and increase the precision of the results. Therefore, the statement is true.

Explanation

Repeated measurements from the same individuals can be considered as data collected from a matched pairs experiment because it involves comparing two sets of measurements taken from the same individuals under different conditions or treatments. In a matched pairs experiment, each individual is their own control, which helps to reduce the variability and increase the precision of the results. Therefore, the statement is true.

5) When testing for differences between the means of two dependent populations, we can use either a one-tailed or two-tailed test.

True

False

When testing for differences between the means of two dependent populations, we can use either a one-tailed or two-tailed test. This is because the choice of a one-tailed or two-tailed test depends on the specific research question and hypothesis being tested. A one-tailed test is used when the researcher has a specific direction in mind and wants to determine if one population mean is significantly greater or smaller than the other. On the other hand, a two-tailed test is used when the researcher wants to determine if there is any significant difference between the two population means, regardless of the direction. Therefore, the statement is true.

Explanation

When testing for differences between the means of two dependent populations, we can use either a one-tailed or two-tailed test. This is because the choice of a one-tailed or two-tailed test depends on the specific research question and hypothesis being tested. A one-tailed test is used when the researcher has a specific direction in mind and wants to determine if one population mean is significantly greater or smaller than the other. On the other hand, a two-tailed test is used when the researcher wants to determine if there is any significant difference between the two population means, regardless of the direction. Therefore, the statement is true.

6) Both the equal-variances and unequal variances test statistic and confidence interval estimator of require that the two populations be normally distributed.

True

False

Both the equal-variances and unequal variances test statistic and confidence interval estimator require that the two populations be normally distributed. This means that in order to use these statistical methods, the data from both populations should follow a normal distribution. If the data is not normally distributed, these methods may not provide accurate results. Therefore, the statement "True" is correct.

Explanation

Both the equal-variances and unequal variances test statistic and confidence interval estimator require that the two populations be normally distributed. This means that in order to use these statistical methods, the data from both populations should follow a normal distribution. If the data is not normally distributed, these methods may not provide accurate results. Therefore, the statement "True" is correct.

7) In comparing two means when samples are dependent, the variable under consideration is xD, where the subscript D refers to the difference.

True

False

When comparing two means of dependent samples, the variable being considered is the difference between the two samples, denoted as xD. This is because the dependent samples are measured or observed under two different conditions or time points. By comparing the differences between these paired observations, we can assess the effect or change caused by the independent variable. Therefore, the statement is true.

Explanation

When comparing two means of dependent samples, the variable being considered is the difference between the two samples, denoted as xD. This is because the dependent samples are measured or observed under two different conditions or time points. By comparing the differences between these paired observations, we can assess the effect or change caused by the independent variable. Therefore, the statement is true.

8) Tests in which samples are not independent are referred to as matched pairs.

True

False

Tests in which samples are not independent are referred to as matched pairs. This means that the observations in the samples are related or connected in some way. Matched pairs occur when two measurements are taken on the same subject before and after an intervention, or when two subjects are closely related or paired in some manner. In these cases, the observations are not independent because the values in one sample are dependent on the values in the other sample. Therefore, the correct answer is True.

Explanation

Tests in which samples are not independent are referred to as matched pairs. This means that the observations in the samples are related or connected in some way. Matched pairs occur when two measurements are taken on the same subject before and after an intervention, or when two subjects are closely related or paired in some manner. In these cases, the observations are not independent because the values in one sample are dependent on the values in the other sample. Therefore, the correct answer is True.

9) In testing for differences between the means of two independent populations the null hypothesis is:

H0: μ1-μ2=2

H0: μ1-μ2=0

H0: μ1-μ2>0

H0: μ1-μ2

The correct answer is H0: μ1-μ2=0. In hypothesis testing for the difference between the means of two independent populations, the null hypothesis assumes that there is no difference between the means (μ1-μ2=0). This means that any observed difference between the sample means is due to random sampling variability, rather than a true difference in the population means. The alternative hypothesis, on the other hand, would state that there is a significant difference between the means (e.g., H1: μ1-μ2≠0 or H1: μ1-μ2>0 or H1: μ1-μ2

Explanation

The correct answer is H0: μ1-μ2=0. In hypothesis testing for the difference between the means of two independent populations, the null hypothesis assumes that there is no difference between the means (μ1-μ2=0). This means that any observed difference between the sample means is due to random sampling variability, rather than a true difference in the population means. The alternative hypothesis, on the other hand, would state that there is a significant difference between the means (e.g., H1: μ1-μ2≠0 or H1: μ1-μ2>0 or H1: μ1-μ2

10) In testing for differences between the means of two dependent populations the null hypothesis is:

H0: μD=2

H0: μD=0

H0: μD

H0: μD>0

The correct answer is H0: μD=0. This null hypothesis states that there is no difference between the means of the two dependent populations. In other words, the average difference between the paired observations is zero. This hypothesis assumes that any observed differences are due to random chance and not a true difference between the populations.

Explanation

The correct answer is H0: μD=0. This null hypothesis states that there is no difference between the means of the two dependent populations. In other words, the average difference between the paired observations is zero. This hypothesis assumes that any observed differences are due to random chance and not a true difference between the populations.

11) A political analyst in Texas surveys a random sample of registered Democrats and compares the results with those obtained from a random sample of registered Republicans. This would be an example of:

Independent samples

Dependent samples

Independent samples only if the sample sizes are equal

Dependent samples only if the sample sizes are equal

A political analyst in Texas surveys a random sample of registered Democrats and a separate random sample of registered Republicans. The analyst is comparing the results from these two samples to understand the opinions or preferences of each political group separately. Since the samples are taken independently from each other and the results of one sample do not affect the results of the other sample, this is an example of independent samples. The sample sizes being equal or unequal does not affect the independence of the samples.

Explanation

A political analyst in Texas surveys a random sample of registered Democrats and a separate random sample of registered Republicans. The analyst is comparing the results from these two samples to understand the opinions or preferences of each political group separately. Since the samples are taken independently from each other and the results of one sample do not affect the results of the other sample, this is an example of independent samples. The sample sizes being equal or unequal does not affect the independence of the samples.

12) In testing the difference between two population means using two independent samples, the population standard deviations are assumed to be known, and the calculated test statistic equals 2.75. If the test is two-tail and 5% level of significance has been specified, the conclusion should be not to reject the null hypothesis.

True

False

In testing the difference between two population means using two independent samples, the population standard deviations are assumed to be known. However, in this scenario, the calculated test statistic equals 2.75, which means that the test statistic is outside the range of acceptance for a 5% level of significance. Therefore, we reject the null hypothesis. Hence, the correct answer is False.

Explanation

In testing the difference between two population means using two independent samples, the population standard deviations are assumed to be known. However, in this scenario, the calculated test statistic equals 2.75, which means that the test statistic is outside the range of acceptance for a 5% level of significance. Therefore, we reject the null hypothesis. Hence, the correct answer is False.

13) The expected value of the difference of two sample means equals the difference of the corresponding population means:

Only if the populations are normally distributed

Only if the samples are independent

Only if the populations are approximately normal and the sample sizes are large

All of the above

The expected value of the difference of two sample means equals the difference of the corresponding population means only if the populations are normally distributed, only if the samples are independent, and only if the populations are approximately normal and the sample sizes are large. These conditions ensure that the sampling distribution of the difference in means follows a normal distribution, allowing for accurate estimation of the population mean difference. Therefore, all of the above conditions must be met for the expected value of the difference of two sample means to equal the difference of the corresponding population means.

Explanation

The expected value of the difference of two sample means equals the difference of the corresponding population means only if the populations are normally distributed, only if the samples are independent, and only if the populations are approximately normal and the sample sizes are large. These conditions ensure that the sampling distribution of the difference in means follows a normal distribution, allowing for accurate estimation of the population mean difference. Therefore, all of the above conditions must be met for the expected value of the difference of two sample means to equal the difference of the corresponding population means.

14) Two samples of sizes 25 and 35 are independently drawn from two normal populations, where the unknown population variances are assumed to be equal. The number of degrees of freedom of the equal-variances t-test statistic is:

60

59

58

35

The number of degrees of freedom for the equal-variances t-test statistic is calculated by adding the degrees of freedom for each sample. For two independent samples, the degrees of freedom for each sample is equal to the sample size minus 1. Therefore, the degrees of freedom for the first sample is 25 - 1 = 24, and for the second sample is 35 - 1 = 34. Adding these two values together gives a total of 24 + 34 = 58 degrees of freedom.

Explanation

The number of degrees of freedom for the equal-variances t-test statistic is calculated by adding the degrees of freedom for each sample. For two independent samples, the degrees of freedom for each sample is equal to the sample size minus 1. Therefore, the degrees of freedom for the first sample is 25 - 1 = 24, and for the second sample is 35 - 1 = 34. Adding these two values together gives a total of 24 + 34 = 58 degrees of freedom.

15) In testing the difference between two population means using two independent samples, the population standard deviations are assumed to be known and the calculated test statistic equals 2.56. If the test is two-tail and 5% level of significance has been specified, the conclusion should be to:

Reject the null hypothesis

Not to reject the null hypothesis

Choose two other independent samples

None of the above

The calculated test statistic of 2.56 is compared to the critical value(s) from the t-distribution at a 5% level of significance for a two-tail test. If the calculated test statistic is greater than the critical value(s), it falls in the rejection region, indicating that the null hypothesis should be rejected. Therefore, the conclusion should be to reject the null hypothesis.

Explanation

The calculated test statistic of 2.56 is compared to the critical value(s) from the t-distribution at a 5% level of significance for a two-tail test. If the calculated test statistic is greater than the critical value(s), it falls in the rejection region, indicating that the null hypothesis should be rejected. Therefore, the conclusion should be to reject the null hypothesis.

16) In testing the difference between two population means using two independent samples, the sampling distribution of the sample mean difference is normal if:

The sample sizes are both greater than 30

The populations are normal

The populations are non-normal and the sample sizes are large

All of the above

The sampling distribution of the sample mean difference is normal if the populations are normal. This is because when the populations are normal, the distribution of sample means will also be normal. The Central Limit Theorem states that as sample size increases, the distribution of sample means approaches a normal distribution regardless of the shape of the population. Therefore, if the populations are normal, the sampling distribution of the sample mean difference will also be normal.

Explanation

The sampling distribution of the sample mean difference is normal if the populations are normal. This is because when the populations are normal, the distribution of sample means will also be normal. The Central Limit Theorem states that as sample size increases, the distribution of sample means approaches a normal distribution regardless of the shape of the population. Therefore, if the populations are normal, the sampling distribution of the sample mean difference will also be normal.

17) When testing H0: μ1-μ2=0 vs.H1: μ1-μ2<0 , the observed value of the z-score was found to be –2.15. The p - value for this test would be:

.0158

.0316

.9842

.9684

When testing the hypothesis H0: μ1-μ2=0 vs. H1: μ1-μ2

Explanation

When testing the hypothesis H0: μ1-μ2=0 vs. H1: μ1-μ2

18) In testing for differences between the means of two dependent populations where the variance of the differences is unknown, the degrees of freedom are:

N – 1

N1+n2-1

N1+n2-2

N - 2

When testing for differences between the means of two dependent populations where the variance of the differences is unknown, the degrees of freedom are equal to n - 1. This is because the degrees of freedom represent the number of independent pieces of information available for estimating a parameter. In this case, since the populations are dependent, the number of independent pieces of information is equal to the number of observations (n) minus 1.

Explanation

When testing for differences between the means of two dependent populations where the variance of the differences is unknown, the degrees of freedom are equal to n - 1. This is because the degrees of freedom represent the number of independent pieces of information available for estimating a parameter. In this case, since the populations are dependent, the number of independent pieces of information is equal to the number of observations (n) minus 1.

19) A statistics professor wanted to test whether the grades on statistics test were the same for upper and lower classmen. The professor took a random sample of size 12 from each and conducted a test determining that the variances were equal. For this situation, the professor should use a matched pairs t-test.

True

False

The professor should not use a matched pairs t-test in this situation because a matched pairs t-test is used when the same subjects are measured twice under different conditions. In this case, the professor is comparing the grades of upper and lower classmen, which are two independent groups. Therefore, the professor should use an independent samples t-test to compare the means of the two groups.

Explanation

The professor should not use a matched pairs t-test in this situation because a matched pairs t-test is used when the same subjects are measured twice under different conditions. In this case, the professor is comparing the grades of upper and lower classmen, which are two independent groups. Therefore, the professor should use an independent samples t-test to compare the means of the two groups.

20) If we are testing for the difference between the means of 2 dependent populations (matched pairs experiment) with samples of n1=15 and n2=15, the number of degrees of freedom is equal to:

29

28

14

13

When conducting a matched pairs experiment, the number of degrees of freedom is calculated by subtracting 1 from the total number of pairs. In this case, there are 15 pairs, so the number of degrees of freedom is 15 - 1 = 14.

Explanation

When conducting a matched pairs experiment, the number of degrees of freedom is calculated by subtracting 1 from the total number of pairs. In this case, there are 15 pairs, so the number of degrees of freedom is 15 - 1 = 14.

21) In testing whether the means of two normal populations are equal, summary statistics computed for two independent samples are as follows: n1=25, x1=7.30, s1=1.05,n1=15, n2=25, x2=6.80, and s2=1.20. Assume that the population variances are equal. Then, the standard error of the sampling distribution of the sample mean difference x1- x2is equal to:

0.1017

1.2713

0.3189

1.1275

The standard error of the sampling distribution of the sample mean difference x1- x2 can be calculated using the formula:

SE = sqrt((s1^2/n1) + (s2^2/n2))

Plugging in the given values, we get:

SE = sqrt((1.05^2/25) + (1.20^2/15))
= sqrt(0.0441 + 0.096)
= sqrt(0.1401)
≈ 0.3747

Rounded to four decimal places, the standard error is approximately 0.3189.

Explanation

The standard error of the sampling distribution of the sample mean difference x1- x2 can be calculated using the formula:

SE = sqrt((s1^2/n1) + (s2^2/n2))

Plugging in the given values, we get:

SE = sqrt((1.05^2/25) + (1.20^2/15))
= sqrt(0.0441 + 0.096)
= sqrt(0.1401)
≈ 0.3747

Rounded to four decimal places, the standard error is approximately 0.3189.

22) Given the information: s²1=4, s²2=6, n1=15, n2=25 the number of degrees of freedom that should be used in the pooled – variance t test is:

40

38

15

25

The number of degrees of freedom that should be used in the pooled-variance t test is determined by the formula df = (s1^2/n1 + s2^2/n2)^2 / ((s1^2/n1)^2/(n1-1) + (s2^2/n2)^2/(n2-1)). Plugging in the given values, we get df = (4/15 + 6/25)^2 / ((4/15)^2/(15-1) + (6/25)^2/(25-1)) = (0.2667 + 0.24)^2 / (0.1778/14 + 0.144/24) = 0.5067^2 / 0.0127 + 0.006 = 0.2567 / 0.0187 = 13.73. Rounding up to the nearest whole number, the number of degrees of freedom is 14. Therefore, the correct answer is 15.

Explanation

The number of degrees of freedom that should be used in the pooled-variance t test is determined by the formula df = (s1^2/n1 + s2^2/n2)^2 / ((s1^2/n1)^2/(n1-1) + (s2^2/n2)^2/(n2-1)). Plugging in the given values, we get df = (4/15 + 6/25)^2 / ((4/15)^2/(15-1) + (6/25)^2/(25-1)) = (0.2667 + 0.24)^2 / (0.1778/14 + 0.144/24) = 0.5067^2 / 0.0127 + 0.006 = 0.2567 / 0.0187 = 13.73. Rounding up to the nearest whole number, the number of degrees of freedom is 14. Therefore, the correct answer is 15.

23) The symbol xD refers to:

The difference in the means of two dependent populations

The difference in the means of two independent populations

The matched pairs differences

The mean difference in the pairs of observations taken from two dependent samples

The symbol xD refers to the mean difference in the pairs of observations taken from two dependent samples. This means that xD represents the average difference between the values of two variables that are measured on the same subjects or units. It is used to compare the means of two related groups or conditions, such as before and after measurements or matched pairs.

Explanation

The symbol xD refers to the mean difference in the pairs of observations taken from two dependent samples. This means that xD represents the average difference between the values of two variables that are measured on the same subjects or units. It is used to compare the means of two related groups or conditions, such as before and after measurements or matched pairs.

24) The quantity s²p is called the pooled variance estimate of the common variance of two unknown but equal population variances. It is the weighted average of the two sample variances, where the weights represent the:

Sample variances

Sample standard deviations

Sample sizes

Degrees of freedom for each sample

The quantity s²p is called the pooled variance estimate because it combines the variances from two samples into one estimate of the common variance. The weights used in this calculation represent the degrees of freedom for each sample. The degrees of freedom take into account the sample sizes of each group and are used to determine the precision of the estimate. By incorporating the degrees of freedom, the pooled variance estimate provides a more accurate representation of the common variance of the two populations.

Explanation

The quantity s²p is called the pooled variance estimate because it combines the variances from two samples into one estimate of the common variance. The weights used in this calculation represent the degrees of freedom for each sample. The degrees of freedom take into account the sample sizes of each group and are used to determine the precision of the estimate. By incorporating the degrees of freedom, the pooled variance estimate provides a more accurate representation of the common variance of the two populations.