Statistics Final Exam: MCQ Quiz!

A bar graph is an appropriate graphical summary for the distribution of a categorical variable because it displays the frequency or proportion of each category as individual bars. This allows for a clear visual comparison of the different categories and their frequencies. It is commonly used to represent qualitative data and is helpful in identifying patterns or trends in the data.

The standard deviation measures the average distance between each data point and the mean. In this case, since the standard deviation is 10 points, it means that on average, the Final scores deviated from their mean by about 10 points. This indicates that there is variability in the scores, with some scores being higher and some lower than the mean by approximately 10 points.

The data set with the largest standard deviation is 301, 304, 306, 308, 311. This is because the standard deviation measures the amount of variation or dispersion in a set of data. In this data set, the numbers are spread out over a wider range compared to the other data sets, resulting in a larger standard deviation.

The population of interest in this study is all BYU single students. The researcher wants to know the average dating expenses for this specific group of students. The researcher obtained a list of single students from the Records Office who live in the BYU dorms and randomly selected 50 students from this list. The average dating expense of these 50 students is used as an estimate for the average dating expenses of all BYU single students.

A lurking variable, such as higher socioeconomic condition, affects the association. This means that there is another variable that is influencing both the number of televisions in the household and the average IQ score of the people in the household. It is likely that households with higher socioeconomic conditions have more resources to afford both more televisions and better education, leading to higher average IQ scores.

As the sample size (n) gets larger, the shape of the sampling distribution of p-hat (the sample proportion) becomes approximately normal. This is due to the Central Limit Theorem, which states that for a large sample size, the sampling distribution of a sample statistic will be approximately normal regardless of the shape of the population distribution. Therefore, as n increases, the distribution of p-hat becomes more symmetric and bell-shaped, resembling a normal distribution.

Procedure is bias when some members of the population cannot be selected at all.

Probability sampling allows us to make inferences about population parameters. This means that by using probability sampling methods, we can draw conclusions about the entire population based on the characteristics of the sample. This is because probability sampling ensures that every member of the population has an equal chance of being selected for the sample, reducing bias and increasing the representativeness of the sample. By making inferences about the sample, we can then generalize those findings to the larger population.

The Central Limit Theorem states that as the sample size increases, the sampling distribution of the sample mean approaches a normal distribution, regardless of the shape of the population distribution. This allows us to use the standard normal table to compute probabilities about sample means and sample proportions from large random samples without knowing the distribution of the population. This is because the standard normal distribution is well-known and its probabilities can be easily calculated.

A comparative experiment where each single student is randomly assigned to one of two treatments can show a cause and effect relationship between the explanatory and response variables. This is because in a comparative experiment, the researcher has control over the assignment of treatments, which allows for the manipulation of the explanatory variable. By randomly assigning each student to one of two treatments, any observed differences in the response variable can be attributed to the effect of the treatment. This helps establish a cause and effect relationship between the variables being studied.

The mean of the sampling distribution of p-hat represents the average value of all possible sample proportions that could be obtained from repeated sampling. It is an important measure in statistics as it provides information about the central tendency of the distribution and helps estimate the true population proportion.

The explanatory variable in this scenario is the time spent practicing free-throws. The researcher wants to determine if the amount of time spent practicing free-throws after practice sessions can be used to predict the percentage of free-throws made in a game. Therefore, the researcher is interested in how the independent variable (time spent practicing free-throws) may have an effect on the dependent variable (percentage of free-throws made in a game).

Statistical significance and practical significance are two different concepts. A result can be statistically significant, meaning that it is unlikely to have occurred by chance, but it may not have practical significance, meaning that it may not have a meaningful or substantial impact in the real world. Therefore, a result that is statistically significant may not necessarily be practically significant.

When testing multiple null hypotheses using the same data set, the probability of making a type I error increases. This is because as more hypotheses are tested, the likelihood of at least one false positive result occurring by chance alone also increases. The more tests conducted, the higher the chance of falsely rejecting a true null hypothesis. Therefore, it is important to consider this increased risk of type I errors when conducting multiple hypothesis tests.

If the assumption of normality of the population is not met and the sample size is less than 40, confidence levels and P-values for t procedures are still approximately correct as long as there are no outliers or strong skewness in the data. This means that even if the data is not normally distributed, the t procedures can still be used to make accurate inferences about the population parameters as long as the data does not deviate significantly from normality due to outliers or strong skewness.

Jeremiah's standardized score (z-score) of 2.5 indicates that his score is 2.5 standard deviations above the average scoring in the league. This means that his score is significantly higher than the average, demonstrating exceptional performance compared to other players.

The middle 95% of all lengths of time to get married from the first date can be found by calculating the range within which 95% of the data falls. In a normal distribution, approximately 95% of the data falls within two standard deviations of the mean. In this case, the mean is 3.5 months and the standard deviation is 0.3 months. Two standard deviations below the mean is 3.5 - (2 * 0.3) = 2.9 months, and two standard deviations above the mean is 3.5 + (2 * 0.3) = 4.1 months. Therefore, the middle 95% of all lengths of time to get married from the first date is between 2.9 and 4.1 months.

The fact that the mean is less than the median suggests that there are some smaller values in the dataset that are pulling the mean down. This indicates that the distribution is skewed to the left, as the tail of the distribution is on the left side.

The probability that a randomly chosen car traveling on this highway is less than 48 miles per hour can be calculated using the standard normal distribution. We can convert the given value of 48 miles per hour into a z-score by subtracting the mean (60) and dividing by the standard deviation (5). This gives us a z-score of -2.4. Looking up this z-score in the standard normal distribution table, we find that the probability corresponding to this z-score is 0.0082. Therefore, the probability that a randomly chosen car traveling on this highway is less than 48 miles per hour is 0.0082.

By decreasing the sample size, the administration will increase the margin of error. This means that the results of the student opinion poll will be less precise, but it will also allow for a larger margin of error, which is what the administration wants in this case. Increasing the sample size would actually decrease the margin of error, which is not desired. Decreasing the confidence level or the standard deviation would not address the issue of the margin of error being too small.

The given answer correctly interprets the margin of error. It states that if the survey is conducted multiple times, 95% of the sample proportions will differ from the true proportion by no more than 4%. This means that the reported proportion of people opposing the gay marriage initiative may vary within a range of 4% due to sampling variability. The answer accurately explains the concept of margin of error and its application in estimating population proportions.

The standard deviation of the sampling distribution of p-hat represents the variability of the sample proportion. It measures how much the sample proportions from different samples are likely to vary from each other. A larger standard deviation indicates a greater spread of sample proportions, while a smaller standard deviation indicates a more consistent and reliable estimate of the population proportion. Therefore, option C is the correct answer.

A chi-square analysis procedure is appropriate for this set of data because all expected counts are greater than 5. This ensures that the assumptions for conducting a chi-square test are met, and the results can be considered reliable.

The chi-square program was used to analyze the data and test the hypothesis that the proportions of people who are well satisfied financially are the same for all educational levels. The analysis concluded that at an alpha level of 0.05, the proportions of people who are well satisfied financially are not all equal for all educational levels. This means that there is evidence to suggest that there is a difference in the level of financial satisfaction among different educational levels.

The correct answer is NO, the sample size is too small to apply the Central Limit theorem. The Central Limit theorem states that as the sample size increases, the sampling distribution of the sample mean approaches a normal distribution, regardless of the shape of the population distribution. However, for this question, the sample size is only 16, which is considered small. Therefore, the Central Limit theorem cannot be applied, and we cannot assume that the sampling distribution of the sample mean is approximately normal.

If a result is statistically significant, it means that the observed data is unlikely to have occurred by chance alone. This suggests that there is evidence to reject the null hypothesis, which assumes that there is no relationship or difference between variables. Therefore, if a result is statistically significant, it is likely that the null hypothesis is false. Alternatively, there is a possibility that a type I error was committed, which means that the null hypothesis was incorrectly rejected.

The statement "Where r is close to 1, there is good evidence that x and y have a strong positive linear relationship" is true. The correlation coefficient, r, measures the strength and direction of the linear relationship between two variables, x and y. When r is close to 1, it indicates a strong positive linear relationship, meaning that as x increases, y also tends to increase in a consistent manner. The closer r is to 1, the stronger the relationship between x and y.

The statement is true because the mean of the theoretical sampling distribution of x-bar is always equal to the population mean. This is known as the Central Limit Theorem, which states that as the sample size increases, the distribution of sample means approaches a normal distribution with a mean equal to the population mean. Therefore, the mean of the theoretical sampling distribution of x-bar will always be equal to the population mean.

The primary purpose of a confidence interval for a population mean is to give a range of plausible values for the population mean. Confidence intervals provide a range of values within which the true population mean is likely to fall, based on the sample data. This range allows for uncertainty and variability in the estimate, giving researchers a sense of how precise their estimate is and the level of confidence they can have in it.

The statement is false because we assume Ho (null hypothesis) is true whenever we perform a test of significance. The null hypothesis represents the default assumption or the status quo, and the purpose of the test is to gather evidence to either support or reject the null hypothesis. Therefore, we assume Ho is true until proven otherwise through the test.

The p-value for a hypothesis test represents the probability of obtaining a sample mean as extreme as the one observed, assuming that the null hypothesis is true. In this case, the null hypothesis states that the mean increase in scores is equal to 15. The alternative hypothesis suggests that the mean increase is less than 15.

To find the p-value, we can use a one-sample t-test. With a sample mean of 13, a sample size of 25, and a sample standard deviation of 7, we can calculate the t-statistic. Using the t-distribution table or a statistical software, we find that the t-statistic corresponds to a p-value between .10 and .05. Therefore, the correct answer is ".10>p-value>.05".

The appropriate Ho and Ha for this experiment are Ho: µ = 12 versus Ha: µ

We declare a result to be statistically significant when it has a small probability of occurring by chance. This means that the observed result is unlikely to have happened by random chance alone, suggesting that there is a real relationship or effect present. Statistical significance helps us determine if the observed data is reliable and can be generalized to a larger population. It provides evidence for the validity of our hypothesis and supports the idea that the observed result is not due to random variation.

The standard error of x-bar refers to the standard deviation of the sample mean. It measures the variability or dispersion of the sample means from the population mean. A smaller standard error indicates that the sample means are closer to the population mean, suggesting greater precision and accuracy in estimating the population mean. Therefore, option B is the correct answer as it accurately defines the standard error of x-bar.

The mean of the sampling distribution of x-bar is the same as the population mean. This means that, on average, the sample means will be equal to the population mean. This is because the sampling distribution of x-bar is created by taking multiple random samples from the population and calculating the mean of each sample. As the number of samples increases, the distribution of sample means will approach a normal distribution centered around the population mean. Therefore, the mean of the sampling distribution of x-bar is a good estimate of the population mean.

The correct answer is 31%. This means that out of the 226 people surveyed, 31% of them had a religiosity level in the range of 56-60.

When basketballs X, Y, and Z are added to the group of five balls, the new set of eight balls will have more variability in volume compared to the original set of five balls. This is because the addition of the three basketballs introduces more diversity in sizes, resulting in a larger range of volumes. As a result, the standard deviation of the volume of the new set of eight balls will be greater than the standard deviation of the volume of the original five balls.

The sampling distribution of a statistic tells us the possible values of the statistic and their frequencies from all possible samples. This means that it provides information about the range of values that the statistic can take and how often each value occurs when multiple samples are taken from the population. It helps us understand the variability and distribution of the statistic across different samples, which is important for making inferences about the population based on the sample.

The expected count for people who completed high school and are not financially satisfied is 104.6. This is calculated based on the assumption that the proportions of people who are well satisfied financially are the same for all educational levels. The chi-square test is used to determine if there is a significant difference between the observed and expected counts, and in this case, the expected count for this particular category is 104.6.

The probability that a randomly chosen car traveling on this highway has a speed between 75 and 63 mph can be calculated by finding the area under the normal distribution curve between these two speeds. This can be done by calculating the z-scores for both speeds using the formula z = (x - μ) / σ, where x is the speed, μ is the mean, and σ is the standard deviation. Then, using a z-table or a calculator, we can find the probability associated with these z-scores. The correct answer of .2729 represents the probability that falls within this range.

The symbol α in hypothesis testing represents the probability of Type I error. Type I error occurs when the null hypothesis (Ho) is rejected, even though it is true. In other words, it is the probability of incorrectly rejecting a true null hypothesis.

The slope of the least-squares line represents the change in the response variable (Final scores) for every 1 unit increase in the explanatory variable (Test3 scores). In this case, the slope is 1.4, which means that, on average, the Final scores increase by about 1.4 units when the Test3 scores increase by 1 unit.

The conditional distribution for college Majors for students whose last Math class taken was College Algebra is represented by option D.

The value 80% is a parameter. In statistics, a parameter is a numerical value that describes a population characteristic. In this case, the population is all students who took Stats221 at the BYU Salt Lake Center. The 80% represents the proportion of these students who worked full-time. Since it is based on data from the entire population, it is considered a parameter rather than a statistic, which would be based on a sample of the population.

The correct answer explains that a 95% confidence interval means that for 95% of all possible samples, the procedure used to obtain the confidence interval will provide an interval that contains the population mean. This means that if the same procedure is repeated multiple times, 95% of the intervals obtained will contain the true population mean. It is a measure of the level of confidence we have in the accuracy of the interval estimate.

The degrees of freedom for the chi-square statistic in this case would be 8. This is because the degrees of freedom for a chi-square test is calculated by subtracting 1 from the number of categories in each variable and then multiplying those values together. In this case, there are 5 categories for educational levels (no high school, high school, some college, bachelor degree, some graduate education) and 3 categories for satisfaction levels (well satisfied, somewhat satisfied, not satisfied), so the degrees of freedom would be (5-1) * (3-1) = 8.

The histogram shows the distribution of religiosity among 226 people. The numbers on the y-axis represent the frequency of people falling into each religiosity range. The x-axis represents the different religiosity ranges. The answer is 12 because the histogram shows that there are 12 people who had a religiosity less than the 34 religiosity range.

Randomization in an experiment is done to help avoid selection bias. Selection bias occurs when the participants or subjects in the experiment are not representative of the target population, leading to biased results. By randomly assigning participants to different groups or treatments, randomization ensures that the groups are similar in terms of their characteristics and reduces the likelihood of any systematic differences. This helps to minimize the impact of selection bias and increases the validity and generalizability of the results obtained from the experiment.

About 25% of the students participated in more than 100 dates before getting married. This can be determined by looking at the Q1 (the first quartile) value, which represents the 25th percentile. Since the Q1 value is 40, it means that 25% of the students had less than or equal to 40 dates. Therefore, the remaining 75% of the students had more than 40 dates, indicating that about 25% of the students participated in more than 100 dates.

Figure A indicates that all the assumptions are met.