STAT 110 - Practice Questions for Exam 2
True/False Questions: Circle the correct answer. Explain your reasoning.
1. The t-test is never valid if the sample size is small, say less than 25. (True or False)
2. Suppose you are comparing two data sets. Data Set A contains 5 observations, and Data Set B contains
10 observations. The standard deviation will always be larger for Data Set B since it has a larger
sample size. (True or False)
3. Holding all else constant, an increase in the sample size will lead to a narrower confidence interval
(True or False).
4. The smaller the p-value, the more evidence we have that the null hypothesis is true (True or False).
5. Dependent samples can only come from situations where we take two measurements on the same
experimental unit. (True or False).
6. Acetaminophen is an active ingredient found in many over-the-counter and prescription medicines, such as pain relievers and cold medications. It is safe and effective when used correctly, but taking too much can lead to liver damage. A researcher hypothesizes the mean amount of acetaminophen in a particular brand of cold tablets is greater than the 600 mg claimed by the manufacturer. A random sample of 25 tablets was taken, and the data were analyzed in JMP as follows.
Research question: Is the mean amount of acetaminophen greater than 600 mg?
a. Explain why the t-test can be used for these data to test a claim about the population mean (i.e., check the assumptions behind the t-test). (3 pts)
b. Set up the null and alternative hypotheses to test the research question of interest. Use both statements in words and using appropriate statistical notation. (3 pts)
c. Find the p-value for testing the research question of interest. (2 pts.) ______
d. Write a conclusion in the context of the problem. (2 pts.)
7. Researchers used magnetic resonance imaging to measure the volumes of various regions of the brain for a sample of 15 identical twins, where one twin was affected by schizophrenia and the other not (“unaffected”). The twins were found in a search through the U.S. and Canada, and ages ranges from 25 to 44 years, with 8 male and 7 female pairs. The data (in cubic centimeters) for the left hippocampus region of the brain appear here and in the file Hippocampus.JMP.
a. Explain why these two samples are dependent. (2 pts.)
b. What is the response variable in this study? (1 pt.)
c. What is the explanatory or predictor variable in this study? (1 pt.)
d. Complete the following sentence to explain why this is an observational study and not a designed experiment. “The subjects were…”
e. What is the name of a test we could use to investigate this research question?
Use the JMP output below to answer (f) – (h).
f. Which group had the larger hippocampus based on these data, the schizophrenics or normal individuals? How much larger was it? (2 pts.)
g. Is there a significant difference mean hippocampus size between the twin with schizophrenia and one without? Justify your answer. (2 pts.)
h. Interpret the confidence interval for the mean difference in hippocampus size. (2 pts.)
Questions 8 - 10 relate to the following scenario: Five-hundred randomly selected inmates in U.S. prisons were surveyed. An analysis was carried out to see if the average sentence length (measured in years) differs between inmates who served in the armed forces and those who did not. The 95% confidence interval for the true difference in means is .48≤ μyes armed forces – μno armed forces ≤ 6.24. Identify whether the following statements about this interval are correct by circling whether the statement is true or false.
8. We are 95% certain that the average sentence length for all inmates in U.S. prisons is between .48 and 6.24 years. (TRUE OR FALSE)
9. This interval provides evidence that inmates in U.S. prisons that have served in the armed forces have a longer average sentence length than those who have not. (TRUE OR FALSE)
10. We have evidence that 95% of all inmates in U.S. prisons are sentenced to serve between .48 and 6.24 years in prison. (TRUE OR FALSE)
______
General TRUE or FALSE Questions
11. The smaller the p-value, the more evidence we have that the alternative hypothesis is true.
(TRUE OR FALSE)
12. The two-sample t-test for comparing two population means is never valid if the sample sizes of both groups are small. (TRUE OR FALSE)
13. The results of an observational study indicate that people who use vitamin supplements get fewer colds than people who don’t. However, we can’t conclude that vitamin supplements prevent colds because this type of study design does not allow us to infer causation. (TRUE OR FALSE)
14. Suppose that both males and females were asked the question “What is the fastest you have ever driven a car (in mph)?” The sample mean for females was 90.5 mph, and the sample mean for males was 96.1 mph. The p-value for testing for a difference in means was .0644. Identify (by circling the label) which of the following 95% confidence intervals is most likely correct for µfemales - µmales.
15. Using the WSU STAT 110 Student Survey results one potential question of interest is whether or not the average WSU GPA differs between students who drink alcohol and those who don’t. A 95% CI for the difference in these population means is found to be .10 ≤ μNondrinker- μDrinker≤.35. Which of the following statements would be true about the p-value for comparing these two population means using a t-test? (2 pts.)
a. Less than .05, since the 95% confidence interval does not include zero.
b. Greater than .05, since the 95% confidence interval does not include zero.
c. Greater than .05, because the lower confidence limit is .10.
d. It is impossible to tell.
Short Answer Questions:
16. A study was conducted in which the mean number of hangover symptoms was compared for students whose parents have alcohol problems (the YES group) and students whose parents do not (the NO group). There were 282 students in the study whose parents did have alcohol problems, and there were 945 students whose parents did not.
a. Is there evidence that the mean number of hangover symptoms is higher for the population of students whose parents have alcohol problems than for the population whose parents do not? Carry out a formal hypothesis test to answer this research question.
Step 0: / You can assume that all assumptions for this test are met.Step 1: / Convert the research question into H0 and Ha. (3 pts)
Ho :
Ha :
Step 2: / Determine α, the significance level. (1 pt.)
Step 3: / Find the test statistic from your data. (2 pts)
Test Statistic =
Step 4: / Determine the p-value and make a decision regarding H0. (2 pts)
p-value =
Step 5: / Write a conclusion in terms of the original research question. (3 pts)
b. Give a 95% confidence interval for μYes – μNo. That is, provide the lower and upper endpoints of this confidence interval, using the appropriate JMP output from above. (2 pts)
c. Interpret the confidence interval from part b in the context of the problem. (3 pts)
17. The following data give the diastolic blood pressures for 15 patients with moderate essential hypertension, immediately before and two hours after taking a drug, Captopril. Our interest is in investigating the response to the drug treatment.
Patient / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10 / 11 / 12 / 13 / 14 / 15Before / 130 / 122 / 124 / 104 / 112 / 101 / 121 / 124 / 105 / 102 / 98 / 119 / 106 / 107 / 100
After / 125 / 121 / 121 / 106 / 101 / 85 / 108 / 115 / 103 / 98 / 90 / 98 / 110 / 103 / 95
Dependent Samples where Difference = Before - After
Independent Samples Assuming Equal Variances
a. Which of these analyses is appropriate for these data—Dependent or Independent Samples? Explain. (3 pts)
b. Use the correct JMP output to find the 95% confidence interval of interest. You do not need to interpret this interval; you need only provide the lower and upper endpoints. (2 pts)
c. Do we have enough evidence to conclude that the drug Captopril lowers blood pressure levels? Explain. (3 pts)
18. The data presented below were collected to compare the shell width (measured in mm) of zebra mussels in the back channel of the Mississippi to those in the main channel. All specimens were collected within a few miles of Winona.
Back Channel / 8.33 / 2.36
Main Channel / 9.56 / 2.6
THE RESEARCH QUESTION WAS AS FOLLOWS: "Does the average shell width of zebra mussels that live in a back channel of the Mississippi differ from the average shell width of those that live in the main channel?”
a. Describe the population(s) of interest in this study. (2 pts)
b. One variable of interest is Shell Width, measured in mm. As measured and reported in this study, is this variable categorical, or numeric? (1 pt)
c. The other variable of interest is Location; i.e., whether the zebra mussel was collected in the main channel or a back channel of the Mississippi. Is this variable categorical, or numeric? (1 pt)
d. Which variable is the predictor – Location or Shell Width? (2 pts)
e. Complete the following sentence to explain why this was an observational study and not a designed experiment. “The zebra mussels were…” (2 pts)
f. Given the predictor variable and response variable in this study, which of the following tests could we use to determine statistical significance? (Circle only ONE answer.) (2 pts)
i. The binomial exact test
ii. The two-sample t-test (independent samples)
iii. The paired-sample t-test (dependent samples)
iv. The chi-square test
g. Evaluate each of the sentences below as "True" or "False" assuming that the p-value from the appropriate analysis is < 0.0001. Note that this is not a multiple choice problem! (1 pt. each)
i. Based on the p-value, there is evidence (using a 5% level of significance) that the average shell width of zebra mussels that live in a back channel of the Mississippi differ from the average shell width of those that live in the main channel. (TRUE or FALSE)
ii. Based on the p-value, there is evidence (using a 5% level of significance) that the location of the zebra mussels (i.e., whether they live in the main channel or a back channel) causes a change in shell width. (TRUE or FALSE)
Questions 19-22 concern the following scenario: You want to investigate a claim that the average time it takes to fall asleep differs between men and women. You take a random sample of men and a random sample of women (in your community) and measure the time it takes to them to fall asleep.
Note: a “statistically significant” difference provides convincing evidence of a difference between men and women.
19. If the difference in means between the two groups turns out not to be statistically significant, which of the following is the best conclusion to draw? (2 pts)
a. You have found strong evidence that there is no difference between males and females.
b. You have not found enough evidence to conclude that there is a difference between males and females.
c. Because the result is not significant, your study is invalid.
20. If the difference in means between the two groups does turn out to be statistically significant, which of the following is a valid conclusion? (2 pts)
a. It would not be surprising to obtain the observed sample results if there is really no difference between men and women.
b. It would be very surprising to obtain the observed sample results if there is really no difference between men and women.
c. It would be very surprising to obtain the observed sample results if there is really a difference between men and women.
21. Suppose that the difference between men and women turns out not to be significant in your study, but several additional studies addressing the same research question suggest that there really is a difference between men and women. Which of the following conclusions is most reasonable? (2 pts)
a. Something went wrong with your analysis.
b. These additional studies must be wrong.
c. Your sample size might have been too small.
22. If the difference in means between men and women does turn out to be statistically significant, which of the following is a possible explanation for this result? (2 pts)
a. Men and women do not differ on this issue, but there is a small chance that random sampling alone led to the difference you observed between men and women.
b. Men and women differ on this issue.
c. Either (a) or (b) are possible explanations for this result.
23. Text Messages Per Day and Gender for WSU Students
The plots and summary statistics below are for the number texts per day sent by WSU male and female students who took the survey this Fall Semester.
a) How would you characterize the shape of the distribution for the number of text messages sent per
day for both males and females? (1 pt.)
b) What is the 75th percentile for the number of text messages sent by females? ______(1 pt.)
c) What is the correct interpretation of the value from part (b)? (2 pts.)
d) Which of the t-tests (A or B) below would you use to answer the research question: Is there evidence to suggest that WSU female students send more texts per day on average than do WSU male students? Justify your answer. (2 pts.)