An online marketplace retailer performed a study that compared the average domestic shipping time of 100 ﻿sellers with their overall seller satisfaction rating (as a percent). ﻿The study yielded a sample correlation coefficient of r = -0.172. ﻿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.

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.

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.

Learn how to test the correlation coefficient using a study from an online marketplace. Calculate the test statistic, determine critical values, and interpret results at the 2% significance level.

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