Statistics II Midterm Exam 1: Key Concepts and Practice Questions

STAT212 STATSITICS II Fall 2023 Midterm Exam 1 Name: 1. (3 points) (Choose two answers) Which statements are true about t distribution? ( ) A. The larger the degrees of freedom, the larger the variance B. The larger the degrees of freedom, the smaller the variance C. If ࠵? follows t distribution, then ࠵? ଶ follows F distribution D. If ࠵? follows t distribution, then ࠵? ଶ follows Chi-square distribution 2. (3 points) (choose two answers) What are true for confidence level? ( ) A. It is the frequency of possible confidence intervals that contain the true value of the corresponding parameter B. It is for interval estimate of a population parameter C. It is for interval estimate of a sample statistic D. The desired confidence level is determined by data 3. (3 points, choose all that applies) For the same data set and same hypothesis, which of the following statements could be true? ( ) A. You reject it at α=0.01 and reject it at α=0.05 B. You fail to reject it at α=0.01 and reject it at α=0.05 C. You reject it at α=0.01 and fail to reject it at α=0.05 D. You fail to reject it at α=0.01 and fail to reject it at α=0.05 4. (2 points, choose two) Which statement is true regarding type I error? ( ) (2 points, choose two) Which statement is true regarding type II error? ( ) A. It occurs when the null is rejected while it is in fact true B. It occurs when the null is not rejected while it is in fact true C. It occurs when the null is not rejected while it is in fact false D. It occurs when the null is rejected while it is in fact false E. It is called false positive F. It is called false negative 5. (3 points) Which of the following statement is true regarding p-value of the test? ( ) A. If p-value is less than significance level, we reject the null B. If p-value is bigger than significance level, we reject the null C. p-value is the probability of observing a statistic that is as extreme as or even more extreme than what you observe D. p-value is the probability of observing a statistic that is smaller than what you observed

6. (7 points) If sample1: 53 successes out of 400 observations, sample 2: 78 successes out of 600 observations, calculate the test statistic for hypothesis testing for the population proportion difference 7. (7 points) If ࠵? = 10, ࠵? ଶ = 8, what is the 95% confidence interval for ࠵? , under normality assumption? 8. (7 points) If the 90% CI of a population mean is (2, 8), what would be the 95% CI of the same parameter using the same data, assuming sample size is large enough?

9. (15 points) A large university provides housing for 10% of its graduate students to live on campus. The university’s housing office thinks that the percentage of graduate students looking for housing on campus may be more than 10%. A random sample of graduate students was surveyed, and 62 out of 481 say that they are looking for housing on campus. A. Based on the data and most commonly used ࠵? level, would you recommend that the housing office consider increasing the amount of housing on campus? Formulate hypotheses and test it, report p-value. B. In addition to the 481 students who responded, there were 19 who didn’t respond. If these 19 had responded, is it possible that your recommendation would have changed (assuming same ࠵? level)? Explain

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10. (13 points) The department head decided to conduct a study to determine the proportion of time the secretaries spend assisting students. He collected 100 random observations and the results showed that the secretaries assisted students 9 times. A. What is 95% confidence interval for the proportion of time that the secretaries assist students? B. How many fewer or additional observations would be required for the department head to be at least 95% confident that the sample result is within 3% of the true proportion?

11. (20 points) Two groups of children were shown a 60-second commercial. One group was shown the ad with both audio and video, and the other was shown only the video. Children were then asked to recall 10 specific items from the ad. The number of correctly recalled items is recorded. The data is summarized below. Sample size Mean Standard deviation Video only group 10 3.7 1.98 Audio/Video group 9 3.3 2.13 A. Under normality assumption, construct the 98% confidence interval of the ratio of population variances from the two groups. B. Based on your CI in part A, can you conclude that the two population variances are same, using ࠵? = 0.02 ? Explain C. Get the 95% confidence interval for the population mean difference. D. Based on your CI in part C, can you conclude that ad with video only will help children recognize the items, using ࠵? = 0.05 ? Explain

12. (15 points) Suppose we want to compare the mean daily sales of two restaurants in the same city. A. We randomly selected 7 days during two-month period and on each day, we went to the two restaurants and recorded the daily sales, as listed below If no normal assumption is made, does the data provide evidence that on average restaurant 1 daily sales is higher than that of restaurant 2, with significance level 0.1? Report the p-value B. If Abby randomly selected 7 days during two-month period and on each day, she went to restaurant 1 and recorded the daily sales. Unknown What Abby did, Betsy also randomly selected 7 days during the two-month period and on each day, she went to restaurant 2 and recorded the daily sales. If the data happen to be the same as in the above table, no normal assumption is made, does the data provide evidence that on average restaurant 1 daily sales is higher than that of restaurant 2, with ࠵? = 0.05? Report the p-value Restaurant 1 773 905 974 1238 1345 1832 1973 Restaurant 2 825 918 999 1281 1302 1827 1971

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