1. **Problem Statement:** We are comparing anxiety levels between students receiving Cognitive Behavioural Therapy (CBT) and those receiving No Therapy. We will test if therapy reduces anxiety using a dependent samples t-test. 2. **Step 1: State the hypotheses** - Null hypothesis ($H_0$): There is no difference in anxiety levels between CBT and No Therapy groups. Formally, $H_0: \mu_D = 0$, where $\mu_D$ is the mean difference. - Alternative hypothesis ($H_A$): CBT reduces anxiety levels, so anxiety levels differ between groups. For two-tailed test: $H_A: \mu_D \neq 0$ (or one-tailed if expecting reduction). 3. **Step 2: Calculate the differences for each pair** $$D = CBT - No\ Therapy = [12-35, 17-37, 20-39, 22-33, 29-26, 30-29] = [-23, -20, -19, -11, 3, 1]$$ 4. **Step 3: Calculate mean and standard deviation of the differences** - Mean difference ($\bar{D}$): $$\bar{D} = \frac{-23 + (-20) + (-19) + (-11) + 3 + 1}{6} = \frac{-69}{6} = -11.5$$ - Calculate squared differences from mean: $$[(-23+11.5)^2, (-20+11.5)^2, (-19+11.5)^2, (-11+11.5)^2, (3+11.5)^2, (1+11.5)^2] = [-11.5^2, -8.5^2, -7.5^2, 0.5^2, 14.5^2, 12.5^2] = [132.25, 72.25, 56.25, 0.25, 210.25, 156.25]$$ - Sum of squared deviations: $$\sum = 132.25 + 72.25 + 56.25 + 0.25 + 210.25 + 156.25 = 627.5$$ - Sample standard deviation of differences ($s_D$): $$s_D = \sqrt{\frac{627.5}{6-1}} = \sqrt{125.5} \approx 11.2$$ 5. **Step 4: Calculate the t-value** Using formula for dependent samples t-test: $$t = \frac{\bar{D}}{s_D / \sqrt{n}} = \frac{-11.5}{11.2 / \sqrt{6}} = \frac{-11.5}{11.2 / 2.449} = \frac{-11.5}{4.57} \approx -2.52$$ 6. **Step 5: Determine degrees of freedom (df)** $$df = n - 1 = 6 - 1 = 5$$ 7. **Step 6: Comment on significance** - For $\alpha = .05$ and two-tailed test with $df=5$, the critical t-value is about $\pm2.571$. - Our calculated $t = -2.52$ is slightly less extreme than critical value, so we *fail to reject* the null hypothesis at $\alpha = .05$. - There is not enough evidence to claim a significant difference in anxiety levels one month after CBT versus no therapy. 8. **Step 7: Assumptions for dependent samples t-test** - The differences are approximately normally distributed. - The pairs are dependent/matched correctly. - The scale of measurement is interval or ratio. - The observations within pairs are independent of other pairs. **Final Answer:** - Calculated t-value: approximately $-2.52$. - At $\alpha = .05$, result is not statistically significant. - Assumptions include normality of differences, paired samples, interval scale, and independence between pairs.