Respond to each item completely, and provide all evidence of your calculations to receive full credit. A researcher has administered an optimism test for graduating college students for the past several years. In the current year, the researcher obtains a sample of $n=36$ people and asks each person to complete the optimism test (higher scores indicate higher levels of optimism about the future). The scores from the sample produce a mean of $M=24$ with $S S=2,835$. Over the past few years the optimism test is known to have a mean of $\mu=20$. Do the sample data support the conclusion that the current graduating students have a different level of optimism than in previous years? Click the link for Tables B1 and B2 $\downarrow$ 1. Perform a null hypothesis test using a two-tailed, single sample $t$ - test and $\mathbf{a}=.05$. Be sure to complete and label all 4 steps in the hypothesis testing process. ( 8 points) 2. Calculate the effect size using Cohen's $d$ if a significant effect exists. (2 points) 3. Calculate $r^{2}$ if a significant effect exists. (2 points) Edit View Insert Format Tools Table

Step 1 ：Set the null hypothesis (H0) as \(\mu = 20\) and the alternative hypothesis (H1) as \(\mu \neq 20\).

Step 2 ：Set the significance level (\(\alpha\)) as 0.05.

Step 3 ：Calculate the sample standard deviation (s) using the formula \(s = \sqrt{\frac{SS}{n-1}}\), where SS is the sum of squares and n is the sample size. Here, \(s = \sqrt{\frac{2835}{35}} = \sqrt{81} = 9\).

Step 4 ：Calculate the standard error (SE) using the formula \(SE = \frac{s}{\sqrt{n}}\). Here, \(SE = \frac{9}{\sqrt{36}} = \frac{9}{6} = 1.5\).

Step 5 ：Calculate the t-score using the formula \(t = \frac{M - \mu}{SE}\), where M is the sample mean and \(\mu\) is the population mean. Here, \(t = \frac{24 - 20}{1.5} = 2.67\).

Step 6 ：Compare the calculated t-score with the critical t-score for a two-tailed test with df = n - 1 = 35 and \(\alpha = 0.05\). The critical t-score from the t-distribution table is approximately ±2.03. Since the calculated t-score (2.67) is greater than the critical t-score, reject the null hypothesis. \(\boxed{\text{Reject } H0}\)

Step 7 ：Calculate Cohen's d using the formula \(d = \frac{M - \mu}{s}\). Here, \(d = \frac{24 - 20}{9} = 0.44\). \(\boxed{d = 0.44}\)

Step 8 ：Calculate r² using the formula \(r² = \frac{t²}{t² + df}\). Here, \(r² = \frac{(2.67)²}{(2.67)² + 35} = 0.20\). \(\boxed{r² = 0.20}\)