Hypothesis Testing – Comparing Two Groups complete solutions correct answers key

Hypothesis Testing – Comparing Two Groups complete solutions correct answers key

1 For each of the following research questions does the situation or research question involve independent samples or paired data?

a. Twenty-five people have their cholesterol measure before eating a Big Mac and again after eating a Big Mac. On average, does eating a Big Mac increase cholesterol? 

b. What is the difference in average ages at which teachers and plumbers retire?

c. What is the difference in average salaries for high school graduates and college graduates?

d. In fifty married couples, the husband and wife each separately take the same test of marital satisfaction. Is there a difference, on average, between the scores of husbands and wives? 

2  In the Datasets folder open the GSS Dataset.   The data are from the 2006 General Social Survey, a federally funded national survey done every other year by the University of Chicago. The variable marital indicates whether the respondent is presently married or not. We’ll compare the mean amount of television watching per typical day (tvhours is the variable) for those who are married versus those who are not. 

a. In words, write a null hypothesis for this situation. We’re comparing two means (television watching for married people versus unmarried people).

b. Using statistical notation for means write null and alternative hypotheses for this problem.

c. Recall from the lecture notes that when doing a two-sample t-test one consideration is whether the two standard deviations (or variances) are equal.  To check, use software to find the standard deviation for tvhours for the two categories of marital status.  In Minitab, you can get these SD by Stat > Basic Statistics > Display Descriptive and select tvhours for Variables and marital for By Variables.  SPSS users go to Analyze > Descriptive Statistics > Explore and use tvhours in the Dependent List and marital for Factor List.

i.                    What are the two standard deviations?

Std.dev for married:

Std.dev for not married:

             ii. Is the larger standard deviation more than twice the smaller standard deviation?

             iii. If your answer to part ii is “Yes” then we will use the unpooled method for calculating the standard error.  If your answer was “No” then we can use the pooled method.  Which method should we use?

d. The two-sample t-test is used to compare means when data is from two independent samples (as it is here). Use software to conduct a 2-sample t test. If your answer to part iii above is “NO” use pooled then click the box for “Assume Equal Variances”.  If you are using SPSS the software will automatically provide results for both the pooled and unpooled methods, i.e. there is no pooled feature to choose.  Read the output to find the values of the t-statistic and the p-value.

      t=                                           p-value =

e. State a conclusion about the hypotheses and about the “real world” situation.

f. The formula for the pooled t-statistic is .  Give values for each of the elements in the formula.  If using SPSS the output does not provide the Sp (i.e. the pooled standard deviation).  Instead, you will find two standard errors.  Enter the correct pooled standard error from SPSS to replace the Sp.

g. The output includes a 95% confidence interval for the difference between means. Write a sentence that interprets this interval in terms of how much difference there is mean television watching for the two groups.

h. Refer again the to the 95% confidence interval of the previous part. Explain why it is evidence that makes it reasonable to conclude that the population means differ.

3 In a national survey of 12^th graders, 254 of 1356 boys said they never or rarely wear a seatbelt when driving. Among 1168 girls, 97 said they never or rarely wear a seatbelt when driving.

a. Let p1 = population proportion that never or rarely wears a seatbelt for boys and p2 = the corresponding proportion for girls. Write null and alternative hypotheses about p1 and p2 for testing if a difference exists.

b. If using Minitab (you can use the Start menu to do this), then use Stat>Basic Stats> 2 proportions.  Click on Summarized data. Use the boys as the first sample and girls as the second sample. “Number of trials” means sample size and “Number of events” means number rarely or never wearing a seatbelt. Next click Options and select your alternative and enter your test difference.  If your hypotheses statements are testing that the difference is 0, then be sure to select the Use Pooled Estimate of p.  If you are using SPSS, then follow the directions in the lecture notes to perform the test on the data set Seatbelts in the data set folder.  Use the output to give values for the following:

For boys, sample proportion =    _______     For girls, sample proportion =  =  ________

The difference between the sample proportions is  =_______

Value of test statistic =  _________                                       p-value = _______

c. Explain whether we can we say there is a difference between the population proportions in this situation.

4  In the Datasets folder click the link for the Class Survey.  Is there a difference between how students performed on their SATM and SATV?  Since we are considering differences between two measurements (i.e. SATM and SATV) on the same individual we can consider the data to be paired.  Use software to conduct a Matched Paired t-test. 

a. Write the null and alternative hypotheses using appropriate statistical notation.

H0:                            Ha:                   

b. Based on the Confidence interval what is your conclusion?

c.  Based on your p-value what is your conclusion?

d. Use the data from the output to calculate the t-statistic by:  t = =