Hypothesis Testing - Comparing Two Groups
Datafiles used: Class Survey, IQ
1 For each of the following research questions does the situation or research question involve independent samples or paired data?
a. A researcher was interested in comparing the resting pulse rate of people who exercise regularly and people who do not exercise regularly. Simple random samples of 16 people ages 30-40 who do not exercise regularly and 12 people ages 30-40 who do exercise regularly were selected and the resting pulse rate of each person was measured.
b. A company which designs sports shoes has made an improvement to their popular running shoe. They hope that athletes wearing the new running shoe will be able to run faster over short distances. They need to determine whether the mean time to run 100 meters for a random sample of 35 sprinters wearing the new running shoe is less than the mean time to run the 100 meters for the same sample of 35 sprinters wearing the old running shoe.
c. A pharmaceutical company wants to determine whether its new antianxiety medication has any effect on resting pulse rate. They need to determine whether the mean resting pulse rate for a random sample of 25 adults before the antianxiety medication is taken differs from the mean resting pulse rate for the same sample of 25 adults after taking the antianxiety medication.
d. A researcher was interested in comparing the heights of women in two different countries. Independent simple random samples of 9 women from country A and 9 women from country B yielded the following heights (in inches).
3 In the Datasets folder open the Class Survey dataset. The variable Gender indicates whether respondents were female or male students. We'll compare the mean number of parties attended per month (Monthly Parties is the variable) for female students versus male students to see if males have a higher propensity for partying.
a. In words, write a null hypothesis for this situation. We're comparing two means (number of parties attended monthly for female students versus male students.)
b. Consider female students to be population 1 and male students to be population 2. 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, consideration should be made to determine whether the two standard deviations (or variances) are equal. To check, use software to find the standard deviation for Monthly Parties for the two categories of Gender status. In Minitab, you can get these SD by Stat > Basic Statistics > Display Descriptive and select Monthly Parties for Variables and Gender for By Variables. SPSS users go to Analyze > Descriptive Statistics > Explore and use Monthly Parties in the Dependent List and Gender for Factor List.
i. What are the two standard deviations?
Std.dev for female students:
Std.dev for male students:
ii. Is the larger standard deviation more than twice the smaller standard deviation?
ii. 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.
e. State a conclusion about the hypotheses and about the "real world" situation.
f. The formula for the pooled t-statistic is . The formula for the non-pooled t-statistic is
Give values for each of the elements in the formula.
If you selected the pooled method, provide a value for the pooled standard deviation. 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 .
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 attendance at parties between the genders.
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. Using the same Class Survey datafile, we want to determine if there is increased experimental marijuana use (variable is Try Weed) for male students versus that of female students (variable is Gender.)
a. Let p1 = population proportion of those who tried weed for female students and p2 = the corresponding proportion for male students. Write null and alternative hypotheses about p1 and p2 for testing to see if the proportion for male students is higher.
b. If using Minitab (you can use the Start menu to do this), then use Stat>Basic Stats> 2 proportions. Since we are not testing if the difference is 0, we would not select the Use Pooled Estimate of p (this is done only for confidence intervals and two tailed tests.) 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 female students, sample proportion = = _________
For male students, sample proportion = = ________
The difference between the sample proportions is = _______
Value of test statistic = _________ p-value = _______
c. What is your conclusion from this test? Explain whether we can we say the proportion of males who tried weed is greater than that of female students.
4. In the Datasets folder click the link for the IQ datafile. This data set contains the IQ scores for twenty individuals, along with their scores on both a Mathematics Competency test as well as Abstract Reasoning test, both of which are measured on a 0 - 50 point scale. Is there a difference how those tested scored on the Mathematics Competency test and Abstract Reasoning test? Since we are considering differences between two measurements (i.e. Math Exam score and Abstract Reasoning score) 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 = =