Determine the critical value(s) for the hypothesis test.H0 :  ρ =0H0 :  ρ =0

H1 :  ρ  ≠ 0H1 :  ρ  ≠ 0

rn α =0.188=285=0.05r=0.188n=285 α =0.05

To determine the critical values for the hypothesis test, we need to find the critical t-values for a two-tailed test with a significance level (\(\alpha\)) of 0.05 and \(n-2\) degrees of freedom.

### Hypotheses
- **Null Hypothesis (H₀):** \(\rho = 0\)
- **Alternative Hypothesis (H₁):** \(\rho \neq 0\)

### Given
- Sample correlation coefficient \(r = 0.188\)
- Sample size \(n = 285\)
- Significance level \(\alpha = 0.05\)

### Degrees of Freedom
For a correlation hypothesis test, the degrees of freedom (df) are calculated as:
\[ \text{df} = n - 2 \]

Substituting the given values:
\[ \text{df} = 285 - 2 = 283 \]

### Critical t-Values
For a two-tailed test with \(\alpha = 0.05\) and 283 degrees of freedom, we look up the critical t-values in the t-distribution table.

Since the degrees of freedom are relatively large, we can approximate the critical t-values using the t-distribution table for large degrees of freedom, or use a t-distribution calculator.

The critical t-values for a two-tailed test with \(\alpha = 0.05\) and 283 degrees of freedom are approximately:

\[ t_{\text{critical}} = \pm 1.969 \]

### Conclusion
The critical values for this hypothesis test are:

\[ \text{Critical values} = \pm 1.969 \]

These values indicate the cutoff points beyond which we would reject the null hypothesis.

Would you like further details or have any questions?

Here are some related questions for further understanding:
1. How do critical values change with different significance levels (\(\alpha\))?
2. Why do we use \(\pm\) values for a two-tailed test?
3. What is the difference between a one-tailed and a two-tailed test?
4. How does the degrees of freedom affect the critical t-values?
5. What are some common significance levels used in hypothesis testing?
6. How can software tools be used to find critical values for t-tests?
7. What is the relationship between the p-value and critical values in hypothesis testing?
8. How would the critical values change if the sample size was different?

**Tip:** Always verify the degrees of freedom and significance level before looking up or calculating the critical values for accurate hypothesis testing.

Learn how to determine critical values for a correlation hypothesis test with a sample correlation coefficient of 0.188 and a sample size of 285 at a significance level of 0.05. Understand the significance of degrees of freedom and the use of critical t-values in hypothesis testing. Suitable for undergraduate students and researchers.

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