H0 :  ρ =0
H1 :  ρ  ≠ 0
rn α =0.188=285=0.05
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 this hypothesis test problem, we need to calculate the test statistic and determine whether to reject or fail to reject the null hypothesis based on the test statistic and critical value. Here's the step-by-step process:

### a. Calculate the Test Statistic

Given:
- Null Hypothesis \( H_0: \rho = 0 \)
- Alternative Hypothesis \( H_1: \rho \neq 0 \)
- Correlation coefficient \( r = 0.188 \)
- Sample size \( n = 285 \)
- Significance level \( \alpha = 0.05 \)

The test statistic for testing the significance of the correlation coefficient is calculated using the formula:

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

Substituting the given values:

\[
t = \frac{0.188 \times \sqrt{285-2}}{\sqrt{1-0.188^2}}
\]

Now let's calculate this.

\[
t = \frac{0.188 \times \sqrt{283}}{\sqrt{1-0.035344}}
\]

\[
t = \frac{0.188 \times 16.8226}{\sqrt{0.964656}}
\]

\[
t = \frac{3.1639}{0.9822}
\]

\[
t = 3.221
\]

So, the test statistic \( t \) is approximately **3.221**.

### b. Determine the Critical Value(s) for the Hypothesis Test

Given the significance level \( \alpha = 0.05 \), and considering that this is a two-tailed test (because \( H_1: \rho \neq 0 \)), we need to find the critical value from the t-distribution table for \( n-2 = 283 \) degrees of freedom.

For \( \alpha/2 = 0.025 \), and 283 degrees of freedom, the critical t-value is approximately **±1.969** (using a t-distribution table or calculator).

### c. Conclude Whether to Reject the Null Hypothesis

To determine whether to reject or fail to reject the null hypothesis, we compare the test statistic to the critical value:

- If \( |t| > t_{critical} \), reject \( H_0 \).
- If \( |t| \leq t_{critical} \), fail to reject \( H_0 \).

In this case, \( |t| = 3.221 \) is greater than \( t_{critical} = 1.969 \).

**Conclusion:** Since \( |t| > t_{critical} \), we **reject** the null hypothesis.

This indicates that there is significant evidence to suggest that the population correlation coefficient \( \rho \) is not equal to zero.

---

Do you want more details or have any questions?

### 8 Related Questions:
1. What is the interpretation of the correlation coefficient \( r = 0.188 \)?
2. How does the sample size affect the test statistic in hypothesis testing?
3. What are the assumptions behind using the t-distribution for testing correlation?
4. How would the result change if the significance level \( \alpha \) was 0.01?
5. What is a two-tailed test, and why is it used here?
6. How would you conduct this test if the sample size was much smaller?
7. Can this method be applied to non-linear relationships?
8. How does the p-value approach compare with the critical value approach?

### Tip:
Always check the assumptions of normality and linearity before conducting a correlation hypothesis test.

Learn how to perform hypothesis testing for correlation coefficient with a detailed step-by-step solution. Calculate the test statistic, find critical values, and interpret results based on provided data. Suitable for undergraduate students and researchers.

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