Question 23 10 pts Respond to each item completely, and provide all evidence of your calculations to receive full credit. A cognitive psychologist studies the effects of popular media on language. In one study, a sample of college students served as a control group and was not exposed to popular media, while a second sample of students was exposed to the entire first season of Keeping Up with the Kardashians (KUWTK). The researcher tested the students by recording the number of times "um" and "uh" were used by each of the respondents during follow-up interviews. A summary of the responses are presented below. Does KUWTK have a significant effect on language functioning? Click the link for Tables B1 and B2 $\downarrow$ 1. Perform a null hypothesis test using a two-tailed, independent samples $t$-test and $\alpha=.05 . \underline{B e}$ sure to complete and label all 4 steps in the hypothesis testing.process. (8 points) 2. Calculate the effect size using Cohen's dif a significant effect exists. (2 points) 3. Calculate $r^{2}$ if a significant effect exists. (2 points) KUWTK \[ \mathrm{n}=4 \] \[ M=22 \] \[ \mathrm{SS}=88 \] Control Group \[ \begin{array}{l} n=6 \\ M=14 \\ S S=112 \end{array} \] Edit View Insert Format Tools Table $12 \mathrm{pt}$ Paragraph B I U A $\mathrm{T}^{2}$
Solution
Step 1 ：1. Null Hypothesis Test Step 1: State the hypotheses. The null hypothesis is that there is no difference in the mean number of "um" and "uh" used by the two groups. The alternative hypothesis is that there is a difference. H0: μ1 = μ2 H1: μ1 ≠ μ2 Step 2: Set the criteria for a decision. If the p-value is less than the significance level (0.05), we reject the null hypothesis. Step 3: Compute the test statistic. We need to calculate the pooled variance, the standard error, and finally the t-value. Pooled variance (s^2) = (SS1 + SS2) / (n1 + n2 - 2) = (88 + 112) / (4 + 6 - 2) = 200 / 8 = 25 Standard error (SE) = sqrt [ s^2 * (1/n1 + 1/n2) ] = sqrt [ 25 * (1/4 + 1/6) ] = sqrt [ 25 * (0.25 + 0.1667) ] = sqrt [ 25 * 0.4167 ] = sqrt [ 10.4175 ] = 3.23 t = (M1 - M2) / SE = (22 - 14) / 3.23 = 2.47 Step 4: Make the decision. We need to find the critical t-value for a two-tailed test with df = n1 + n2 - 2 = 4 + 6 - 2 = 8 at α = .05. The critical t-value is approximately ±2.306. Since our calculated t-value (2.47) is greater than the critical t-value, we reject the null hypothesis. 2. Effect Size (Cohen's d) Cohen's d = (M1 - M2) / sqrt(s^2) = (22 - 14) / sqrt(25) = 8 / 5 = 1.6 3. R-squared r^2 = t^2 / (t^2 + df) = 2.47^2 / (2.47^2 + 8) = 6.1 / (6.1 + 8) = 6.1 / 14.1 = 0.43 In conclusion, the results suggest that exposure to KUWTK has a significant effect on language functioning, with a large effect size (d = 1.6) and a moderate proportion of variance explained (r^2 = 0.43).
