Independent random samples of 17 sophomores and 13 juniors attending a large uni
ID: 2958390 • Letter: I
Question
Independent random samples of 17 sophomores and 13 juniors attending a large university yield the following data on grade point averages (GPA). Assume that GPAs are normally distributed.Mean SD Sample size
sophomor 2.84 0.52 17
juniors 2.98 0.31 13
a)At the 5% significance level, do the data provide sufficient evidence to conclude that the mean GPAs of sophomores and juniors at the university differ?
b)Construct a 95% confidence interval for the difference between the mean times?
c)Write down and check the assumptions of the procedures in (a) and (b).
Explanation / Answer
Independent random samples of 17 sophomores and 13 juniors attending a large university yield the following data on grade point averages: Sophomores Juniors 3.04 2.92 2.86 2.56 3.47 2.65 1.71 3.60 3.49 2.77 3.26 3.00 3.30 2.28 3.11 2.70 3.20 3.39 2.88 2.82 2.13 3.00 3.19 2.58 2.11 3.03 3.27 2.98 2.60 3.13 At the 5% significance level, do the data provide sufficient evidence to conclude that the mean GPAs of sophomores and juniors at the university differ? Check assumption 1: Are these independent samples? Yes, the students selected from the sophomores are not related to the students selected from juniors. Check assumption 2: Is this a normal population or large samples? Since we don't have large samples from both populations, we need to check the normal probability plots of the two samples: Now, we need to determine whether to use the pooled t-test or the non-pooled (separate variances) t-test. • Using SPSS • Variable N Mean Median TrMean StDev sophomor 17 2.840 2.920 2.865 0.520 juniors 13 2.9808 3.0000 2.9745 0.3093 Variable Minimum Maximum Q1 Q3 sophomor 1.710 3.600 2.440 3.200 juniors 2.5600 3.4700 2.6750 3.2300 Note: The standard deviations are 0.520 and 0.3093 respectively; both the sample sizes and the standard deviations are quite different from each other. We, therefore, decide to use a non-pooled t-test. Step 1. Set up the hypotheses: Ho: - = 0 Ha: - 0 Step 2. Write down the significance level. = 0.05 Step 3. Perform the 2-sample t-test in SPSS with the appropriate alternative hypothesis. Note: The default for the 2-sample t-test in Minitab is the non-pooled one: Two sample T for sophomores vs juniors N Mean StDev SE Mean sophomor 17 2.840 0.520 0.13 juniors 13 2.981 0.309 0.086 95% CI for mu sophomor - mu juniors: ( -0.45, 0.173) T-Test mu sophomor = mu juniors (Vs no =): T = -0.92 P = 0.36 DF = 26 Conclusion in words: At 5% level of significance, the data does not provide sufficient evidence that the mean GPAs of sophomores and juniors at the university are different. Question #2 Continuing with the previous example, give a 95% confidence interval for the difference between the mean GPA of Sophomores and the mean GPA of Juniors. Using SPSS: 95% CI for mu sophomor - mu juniors is: ( -0.45, 0.173) Interpreting the above result: We are 95% confident that the difference between the mean GPA of sophomores and juniors is between -0.45 and 0.173.