H0 : ρ =0 H1 : ρ ≠ 0 rn α =0.18=217=0.02 T-Distribution Table a. Calculate the test statistic. t= 0.000 Round to three decimal places if necessary b. Determine the critical value(s) for the hypothesis test. + c. Conclude whether to reject the null hypothesis or not based on the test statistic. Reject Fail to Reject

To solve the given hypothesis test, let's go through the steps in detail:

### Step 1: Calculate the Test Statistic

The test statistic for testing the correlation coefficient (ρ) can be calculated using the following formula:

\[ t = \frac{r\sqrt{n-2}}{\sqrt{1-r^2}} \]

Given:
- Sample correlation coefficient \( r = 0.18 \)
- Sample size \( n = 217 \)

Let's substitute these values into the formula:

\[ t = \frac{0.18 \sqrt{217 - 2}}{\sqrt{1 - 0.18^2}} \]
\[ t = \frac{0.18 \sqrt{215}}{\sqrt{1 - 0.0324}} \]
\[ t = \frac{0.18 \sqrt{215}}{\sqrt{0.9676}} \]
\[ t = \frac{0.18 \times 14.6628782984}{0.9836612304} \]
\[ t = \frac{2.6393180937}{0.9836612304} \]
\[ t \approx 2.684 \]

So, the test statistic is:

\[ t \approx 2.684 \]

### Step 2: Determine the Critical Value(s) for the Hypothesis Test

The critical values for a two-tailed test can be found using the t-distribution table. For a two-tailed test with a significance level \( \alpha = 0.02 \) and degrees of freedom \( df = n - 2 = 215 \), we need to find the critical t-values that correspond to \( \alpha/2 = 0.01 \) in each tail.

Looking up the t-distribution table or using a calculator for \( df = 215 \) and \( \alpha/2 = 0.01 \), we get:

\[ t_{\text{critical}} \approx \pm 2.326 \]

### Step 3: Conclude Whether to Reject the Null Hypothesis or Not

We compare the calculated test statistic \( t \approx 2.684 \) with the critical value \( t_{\text{critical}} = \pm 2.326 \):

- Since \( |2.684| > 2.326 \), we reject the null hypothesis \( H_0 \).

Therefore, based on the test statistic, we **reject the null hypothesis**.

### Summary

a. The test statistic is approximately \( t = 2.684 \).

b. The critical values are \( \pm 2.326 \).

c. Since the test statistic falls outside the range of the critical values, we **reject** the null hypothesis \( H_0 \).

Would you like more details or have any questions?

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**Related Questions:**

1. What is the formula for the test statistic used in testing correlation coefficients?
2. How do you interpret the critical values in hypothesis testing?
3. What does it mean to reject the null hypothesis in the context of correlation?
4. How can the degrees of freedom affect the critical value in a t-test?
5. What is the significance level in hypothesis testing, and how is it chosen?
6. How would the test statistic change if the sample size were smaller?
7. What is the impact of a higher correlation coefficient on the test statistic?
8. How do you find the critical values using a t-distribution table?

**Tip:** Always check the degrees of freedom and significance level when looking up critical values in a t-distribution table.

Explore how to perform hypothesis testing for correlation coefficients using a detailed example. Calculate the test statistic and determine critical values with a significance level of 0.02. Understand when to reject or fail to reject the null hypothesis based on the test statistic.

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