To test the claim that there is no linear correlation between the seller's average shipping time and their overall satisfaction rating at the 2% level of significance, we will follow these steps:

Step 1: Calculate the Test Statistic

The test statistic for testing the correlation coefficient $\rho$ can be calculated using the following formula:

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

Given:

• Sample correlation coefficient $r = -0.172$
• Sample size $n = 100$

Substitute these values into the formula:

$t = \frac{-0.172 \sqrt{100 - 2}}{\sqrt{1 - (-0.172)^2}}$ $t = \frac{-0.172 \sqrt{98}}{\sqrt{1 - 0.029584}}$ $t = \frac{-0.172 \sqrt{98}}{\sqrt{0.970416}}$ $t = \frac{-0.172 \times 9.89949493742}{0.985086}$ $t = \frac{-1.702835928}{0.985086}$ $t \approx -1.729$

So, the test statistic is:

$t \approx -1.729$

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

The critical values for a two-tailed test at the 2% significance level ($\alpha = 0.02$) and degrees of freedom $df = n - 2 = 98$ can be found using the t-distribution table. For a two-tailed test, 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 $df = 98$ and $\alpha/2 = 0.01$, we get:

$t_{\text{critical}} \approx \pm 2.364$

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

We compare the calculated test statistic $t \approx -1.729$ with the critical value $t_{\text{critical}} = \pm 2.364$:

• Since $-1.729$ is within the range of $-2.364$ to $2.364$, we fail to reject the null hypothesis $H_0$.

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

Summary

a. The test statistic is approximately $t = -1.729$.

b. The critical values are $\pm 2.364$.

c. Since the test statistic falls within the range of the critical values, we fail to 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 null hypothesis in a correlation test?
2. How do you interpret the p-value in a hypothesis test?
3. What does a negative correlation coefficient indicate about the relationship between two variables?
4. How does sample size affect the test statistic in a correlation test?
5. What are the assumptions of the Pearson correlation coefficient?
6. How would you interpret a correlation coefficient of -0.172 in the context of the study?
7. What is the importance of degrees of freedom in a t-test?
8. How can you use software tools to perform a hypothesis test for correlation?

Tip: When performing hypothesis tests, always verify that the data meets the assumptions of the test being used. For correlation tests, this includes checking for linearity and homoscedasticity.