T test for independent groups and dependent groups (Two-group designs)
1. An LU professor is interested in whether there is a difference between undergraduate students and graduate students in the amount of time spent praying each day. The professor gathers information from random samples of undergraduate and graduate students on the LU campus. The amount of time praying is normally distributed and is measured on an interval/ratio scale.
Graduate Undergraduate
15 9
17 11
10 9
13 6
11 5
17 6
a. What statistical test should be used to analyze the data?
b. Is this a one- or two tailed test?
c. Identify H0 and Ha for this study.
d. Conduct the appropriate analysis. Should H0 be rejected?
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Sra
a. The appropriate statistical test to analyze the data in this study is the t-test for independent groups because the professor is comparing the means of two independent groups (graduate students and undergraduate students).
b. This is a two-tailed test because the professor is interested in determining if there is any difference between the amount of time spent praying by undergraduate and graduate students, without specifying a specific direction for the difference.
c. The null hypothesis (H0) for this study would be: There is no difference in the amount of time spent praying each day between undergraduate and graduate students. The alternative hypothesis (Ha) would be: There is a difference in the amount of time spent praying each day between undergraduate and graduate students.
d. To conduct the appropriate analysis, you will need to calculate the t-statistic and compare it to the critical value from the t-distribution table. Here are the steps to perform the t-test for independent groups:
1. Calculate the sample means and sample variances for both groups.
- For the graduate student group: Mean = 14.6, Variance = 9.7
- For the undergraduate student group: Mean = 7.5, Variance = 4.5
2. Calculate the pooled variance:
- Pooled variance = ((n1-1) * var1 + (n2-1) * var2) / (n1 + n2 - 2)
- Pooled variance = ((5-1) * 9.7 + (6-1) * 4.5) / (5 + 6 - 2) = 5.47
3. Calculate the standard error of the difference between means:
- Standard error = sqrt((var1/n1) + (var2/n2))
- Standard error = sqrt((9.7/5) + (4.5/6)) = 1.64
4. Calculate the t-statistic:
- t = (mean1 - mean2) / standard error
- t = (14.6 - 7.5) / 1.64 = 4.35
5. Determine the degrees of freedom:
- Degrees of freedom = n1 + n2 - 2 = 5 + 6 - 2 = 9
6. Look up the critical value of t for the desired significance level and degrees of freedom. For a two-tailed test with 9 degrees of freedom and a significance level of 0.05, the critical value is approximately 2.262.
7. Compare the calculated t-value to the critical value. If the calculated t-value is greater than the critical value, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.
In this case, the calculated t-value (4.35) is greater than the critical value (2.262). Therefore, we reject the null hypothesis and conclude that there is a significant difference in the amount of time spent praying each day between undergraduate and graduate students.