Pennsylvania Refining Company is studying the relationship between the pump price of gasoline and the number of gallons sold. For a sample of 15 stations last Tuesday, the correlation was 0.48.
 

The following hypotheses are given.

H0: ρ ≤ 0

H1: ρ > 0


a. State the decision rule for 0.025 significance level. (Round your answer to 3 decimal places.)

In this problem, you're testing the hypothesis for the correlation between two variables: the pump price of gasoline and the number of gallons sold. You want to test whether the population correlation \( \rho \) is greater than zero.

Here are the hypotheses:

- Null hypothesis \( H_0: \rho \leq 0 \)
- Alternative hypothesis \( H_1: \rho > 0 \)

This is a **one-tailed** test of correlation significance.

To carry out the test, you can follow these steps:

### Step 1: Find the test statistic

You will use the **t-test for correlation**:

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

Where:
- \( r \) is the sample correlation coefficient (0.48 in this case),
- \( n \) is the sample size (15 in this case).

### Step 2: Find the critical value of \( t \) for a one-tailed test at the 0.025 significance level

The degrees of freedom (df) are:

\[
df = n - 2 = 15 - 2 = 13
\]

Using the t-distribution table for a one-tailed test at \( \alpha = 0.025 \) and 13 degrees of freedom, we look up the critical value of \( t \).

The critical value \( t_{0.025, 13} \) from the t-distribution table is approximately **2.160**.

### Step 3: Decision rule

For a one-tailed test at a 0.025 significance level, the decision rule is:

- **Reject** \( H_0 \) if \( t > 2.160 \)
- **Fail to reject** \( H_0 \) if \( t \leq 2.160 \)

This means that if the calculated test statistic is greater than 2.160, you reject the null hypothesis in favor of the alternative hypothesis.

---

Would you like me to calculate the t-statistic for this data, or do you have other questions?

### Related Questions:
1. How do you compute the t-statistic for a correlation test?
2. What does rejecting the null hypothesis in this context mean?
3. What is the interpretation of the correlation coefficient \( r = 0.48 \)?
4. How does a one-tailed test differ from a two-tailed test in hypothesis testing?
5. What are the assumptions behind the t-test for correlation significance?

**Tip:** The correlation coefficient only measures linear relationships, so it's important to check if the data appears to have a linear trend before relying on this test.

Pennsylvania Refining Company is studying the relationship between the pump price of gasoline and the number of gallons sold. For a sample of 15 stations last Tuesday, the correlation was 0.48. The following hypotheses are given. H0: ρ ≤ 0 H1: ρ > 0. a. State the decision rule for 0.025 significance level. (Round your answer to 3 decimal places.)

Explore the hypothesis testing process for determining the significance of the correlation (r = 0.48) between gasoline prices and the number of gallons sold in 15 stations. This solution covers the t-test for correlation significance, hypothesis testing, and decision rules for a 0.025 significance level.

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