To address the hypothesis test, we need to follow a structured approach to calculate the test statistic, determine the critical values, and make a conclusion about the null hypothesis. Here's the step-by-step process:

Step 1: Hypotheses

• Null Hypothesis (H₀): $\rho = 0$
• Alternative Hypothesis (H₁): $\rho \neq 0$

Step 2: Test Statistic

The test statistic for testing the population correlation coefficient $\rho$ is given by: $t = \frac{r\sqrt{n-2}}{\sqrt{1-r^2}}$

where:

• $r$ is the sample correlation coefficient
• $n$ is the sample size

Given:

• $r = 0.188$
• $n = 285$

We can now plug these values into the formula to calculate the test statistic $t$.

$t = \frac{0.188\sqrt{285-2}}{\sqrt{1-0.188^2}}$

First, calculate the components inside the formula: $\sqrt{n-2} = \sqrt{285-2} = \sqrt{283} \approx 16.822$

$\sqrt{1-r^2} = \sqrt{1-(0.188)^2} = \sqrt{1-0.035344} = \sqrt{0.964656} \approx 0.982$

Now, substitute these values into the test statistic formula: $t = \frac{0.188 \times 16.822}{0.982} \approx \frac{3.162}{0.982} \approx 3.220$

Step 3: Determine the Critical Values

Since this is a two-tailed test at $\alpha = 0.05$ with $n-2 = 283$ degrees of freedom, we look up the critical values from the t-distribution table.

For a two-tailed test with $\alpha = 0.05$, the critical values are approximately $\pm 1.969$ for a large degree of freedom (using t-distribution table or calculator for exact values).

Step 4: Conclusion

Compare the calculated test statistic $t = 3.220$ to the critical values $\pm 1.969$:

• If $|t|$ is greater than the critical value, reject the null hypothesis $H_0$.
• If $|t|$ is less than or equal to the critical value, fail to reject the null hypothesis $H_0$.

Since $3.220 > 1.969$, we reject the null hypothesis $H_0$.

Summary:

a. Calculate the test statistic: $t = 3.220$ (rounded to three decimal places)

b. Determine the critical value(s) for the hypothesis test: $\pm 1.969$

c. Conclude whether to reject the null hypothesis or not based on the test statistic: $\text{Reject}$

Would you like further details or have any questions?

Here are some related questions for further understanding:

1. What is the significance of the correlation coefficient $r$ in hypothesis testing?
2. How is the t-distribution used in hypothesis testing?
3. What does rejecting the null hypothesis imply in the context of correlation testing?
4. How does the sample size affect the test statistic?
5. What are Type I and Type II errors in hypothesis testing?
6. How do you interpret the p-value in a hypothesis test?
7. What are the assumptions underlying the use of the t-distribution in correlation testing?
8. How would the hypothesis test change if the sample size were smaller?

Tip: Always ensure your sample meets the assumptions required for the statistical test you are performing to ensure the validity of your results.