To evaluate the statistical significance of the treatment effect on depression scores, we can perform a z-test for a population mean. Here's the process step by step:

Step 1: Define the hypotheses

• Null hypothesis (H₀): The treatment has no effect on depression scores, i.e., the mean depression score after treatment is equal to the population mean.

  • $H₀: \mu = 24$

• Alternative hypothesis (H₁): The treatment has an effect on depression scores, i.e., the mean depression score after treatment is not equal to the population mean.

  • $H₁: \mu \neq 24$

This is a two-tailed test since we're checking for any difference (increase or decrease) from the population mean.

Step 2: Identify the known values

• Population mean ($\mu$) = 24
• Population standard deviation ($\sigma$) = 10
• Sample size ($n$) = 36
• Sample mean ($M$) = 22.5

Step 3: Calculate the standard error of the mean

The standard error (SE) is calculated using the formula: $SE = \frac{\sigma}{\sqrt{n}}$ Substitute the values: $SE = \frac{10}{\sqrt{36}} = \frac{10}{6} = 1.67$

Step 4: Calculate the test statistic (z)

The z-score is calculated using the formula: $z = \frac{M - \mu}{SE}$ Substitute the values: $z = \frac{22.5 - 24}{1.67} = \frac{-1.5}{1.67} \approx -0.90$

Step 5: Make a decision

Now, we compare the calculated z-value with the critical z-values for a two-tailed test at a typical significance level ($\alpha = 0.05$):

• For $\alpha = 0.05$ (two-tailed), the critical z-values are approximately $\pm 1.96$.

Since the calculated z-value of $-0.90$ falls within the range of $-1.96$ to $+1.96$, we fail to reject the null hypothesis.

Step 6: Interpretation of the results

• The z-test results indicate that the observed sample mean (22.5) is not significantly different from the population mean of 24.
• Therefore, the treatment does not appear to have a significant effect on depression scores based on this sample.

Conclusion:

The researcher cannot conclude that the treatment had an impact on depression scores. There is not enough evidence to suggest that the treatment changed the depression scores from the population mean.

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Would you like a more detailed explanation of any of these steps, or do you have any follow-up questions? Here are 5 related questions to explore further:

1. What is the difference between a one-tailed and a two-tailed test in hypothesis testing?
2. How does increasing the sample size impact the test statistic and decision?
3. What are some other statistical tests you could use to analyze treatment effects, such as t-tests?
4. How do you interpret p-values in hypothesis testing?
5. What is the impact of using a significance level ($\alpha$) of 0.01 instead of 0.05?

Tip: Always check your sample size and standard deviation. Larger sample sizes lead to more precise estimates and smaller standard errors, making it easier to detect significant differences.