Comparing Two Groups complete solutions correct answers key
For question 3, use the FacultySalaries dataset, obtained from the Bulletin of the American Association of University Professors. The dataset contains the average male and female salaries (in thousands of dollars) of assistant professors from 22 different U.S. colleges and universities.
For question 4, use the Cookies dataset, which contains the number of chocolate chips contained in a sample of cookies for 5 different types of chocolate chip cookies.
1. For each of the following research questions, please indicate if the situation or research question involves investigating (a) two proportions, (b) two means of independent samples, or (c) two means of paired samples.
a. A researcher is interested in comparing the resting pulse rate of people who exercise regularly and people who do not exercise regularly. Simple random samples of sixteen people ages 30-40 who do not exercise regularly, and twelve people ages 30-40 who do exercise regularly, are selected and the resting pulse rate of each person is measured.
b. A company that designs sports shoes has made an improvement to their popular running shoe. The company hopes that athletes wearing the new running shoe will be able to run faster over short distances. To determine this, the company asks a sample of 35 sprinters to run 100 meters using the old shoes, and then to run 100 meters using the new shoes. In each case the time it takes to complete the dash is recorded.
c. A human resource professional wants to know if there is a difference in perceived gender equality in his office between men and women. A random sample containing 47 women and 53 men is taken, and each person is asked, "Do you feel that there is gender equality in your office?" The responses ("yes" or "no") are recorded for each person and the goal is to compare them by gender.
d. A pharmaceutical company wants to determine whether its new anti-anxiety medication has any effect on resting pulse rate. It needs to determine whether the average resting pulse rate for a random sample of 25 adults before the anti-anxiety medication is taken differs from the average resting pulse rate for the same sample of 25 adults after taking the anti-anxiety medication.
e. A convenience store manager is curious to know if caffeinated coffee drinkers are more likely to buy large cups than decaf coffee drinkers. The store sells only "large" and "regular" sizes. She selects a random sample of 30 caffeinated coffee drinkers from her store and records how many of them buy a large coffee. She does the same thing for a random sample of 30 decaf coffee drinkers.
2. (Problem 10.10 from the course text) Two TV commercials are developed for marketing a new product. Group A, consisting of 100 people, watch commercial A in a controlled setting. A total of 25 people from Group A say they would buy the product. Group B, also consisting of 100 people, watch commercial B in a controlled setting. Just 20 from this group say they would buy the product. The marketing manager concludes that commercial A is better.
a. Let p1 = population proportion of people who would buy the product after watching commercial A, and p2 = population proportion of people who would buy the product after watching commercial B. Identify/calculate the following variables:
Sample size of Group A, n1 =
Sample size of Group B, n2 =
Number of people in Group A who would buy the product, x1 =
Number of people in Group B who would buy the product, x2 =
Sample proportion of people who would buy after commercial A, =
Sample proportion of people who would buy after commercial B, =
b. The marketing manager concludes that commercial A is better. Test to see if this conclusion is justified. In other words, test H0: p1 - p2 = 0 versusHa: p1 - p2 > 0. Perform the test using software and the some of the values from part a. Paste the output below. Make sure to select the correct alternative hypothesis!
i. From the output, what is the test statistic, z?
ii. What is the p-value?
iii. Based on the p-value, do you believe the marketing manager's conclusion is justified? Why or why not? Answer assuming α = .10.
c. The 90% confidence interval for the difference in the proportion of people who would buy the product after watching commercial A and B is(-0.0470, 0.1470). Explain how this agrees with your conclusion from the test in part b.
3. Use the FacultySalaries dataset to answer the question, "Is there a difference in the mean male and female salaries of U.S. university assistant professors?"
a. Explain why we can use a paired samples t-test here instead of an independent samples t-test. Be specific.
Hint: Though many times we use paired samples procedures when we measure twice on the same subject, we often use it also when observations in the two samples can be carefully matched together in a logical way.
b. Using proper statistical notation, write down the null and alternative hypothesis for this test. Define µd = µ1 - µ2, where µ1 is the mean salaries of male assistant professors (in thousands of dollars) and µ2 is the mean salaries of female assistant professors (in thousands of dollars).
Null hypothesis: H0:
Alterative hypothesis: Ha:
c. Below are descriptive statistics for "Males" and for "Females" as obtained from Minitab.
Identify the following values from the output:
Sample mean of male salaries, =
Sample mean of female salaries, =
Sample size of each group, n =
d. Perform, by hand, the hypothesis test you defined in part b by following the steps below. Use 95% confidence level (in other words, α = .05). Show all work.
i. Calculate the test statistic (formula given below). Assume that sd = 0.846, and recall that .
ii. What are the degrees of freedom for this test (DF = n - 1)?
iii. Use software to find the p-value of the test by following the instructions below. Don't forget to paste the output from the software.
Hint: Remember that to the find the p-value, we have to look at the alternative hypothesis. In this case, the alternative hypothesis should be two-sided ("not equal to"), so the p-value = 2*P(T > |t|) = 2*[1 - P(T < |t|)] or 2*P(T < -|t|)
e. Use software to confirm your test results from part d, and paste the output below. Use it to double-check your results from above.
f. Decide between the null hypothesis and the alternative hypothesis based on the p-value and significance level, α. Then, write a sentence summarizing the real-world conclusion from your test. Make sure your conclusion is specific and clear.
4. Use the Cookies dataset to answer the question, "Do reduced fat Chips Ahoy chocolate chip cookies contain fewer chocolate chips on average than regular Chips Ahoy chocolate chip cookies?" We'll assume 99% confidence (so α = .01)
a. Complete the correct notation for the null and alternative hypotheses for this test by filling in the two blanks below with either =, ≠, <, or >. Note that µ1 is be the mean number of chocolate chips in reduced fat Chips Ahoy and µ2 is the mean number of chocolate chips in regular Chips Ahoy.
Null hypothesis: H0: μ1 - μ2 ___ 0
Alterative hypothesis: Ha: μ1 - μ2 ___ 0
b. Because the data come from independent samples (instead of paired samples), we should perform an independent two-sample t-test. However, should we perform a pooled or unpooled test? Show how you came to this conclusion. Note that sample standard deviation of chocolate chips in reduced fat cookies is s1 = 2.5515 and the sample standard deviation of chocolate chips in regular cookies is s2 = 3.8351.
Hint: If , use pooled (assume equal variances/standard deviations). Otherwise, use unpooled.
c. Use software to perform the test from part a and paste the output below. Make sure to select the correct alternative hypothesis and the correct test (pooled vs. unpooled).
i. From the output, what is the test statistic, t?
ii. What is the p-value?
iii. Based on the p-value, would you conclude that there are fewer chocolate chips in reduced fat vs. regular Chip Ahoy chocolate chip cookies? Why or why not?