Pennsylvania Refining Company is studying the relationship between the pump price of gasoline and the number of gallons sold. For a sample of 14 stations last Tuesday, the correlation was 0.65.
 

The following hypotheses are given.

H0: ρ ≤ 0

H1: ρ > 0


a. State the decision rule for 0.010 significance level. (Round your answer to 3 decimal places.)

To state the decision rule, we need to perform a hypothesis test for the population correlation coefficient \( \rho \), using the sample correlation coefficient \( r \). We're testing the following hypotheses:

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

Since we're given a significance level \( \alpha = 0.010 \), and we have a sample size of 14, we can proceed with the following steps:

### Step 1: Test Statistic
For a correlation test, we use the test statistic for \( t \), given by:

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

where:
- \( r = 0.65 \) is the sample correlation coefficient,
- \( n = 14 \) is the sample size.

### Step 2: Degrees of Freedom
The degrees of freedom for this test is \( n - 2 = 14 - 2 = 12 \).

### Step 3: Critical Value
Since this is a one-tailed test, we need the critical value from the t-distribution at \( \alpha = 0.010 \) with 12 degrees of freedom. We denote this critical value as \( t_{\alpha, 12} \).

We can look up the critical value for a one-tailed test with \( \alpha = 0.010 \) and 12 degrees of freedom in a t-distribution table, or calculate it using statistical software.

From a t-distribution table, the critical value \( t_{\alpha, 12} \) for \( \alpha = 0.010 \) and 12 degrees of freedom is approximately:

\[
t_{\alpha, 12} = 2.681
\]

### Step 4: Decision Rule
The decision rule is:
- **Reject \( H_0 \)** if the calculated \( t \)-value is greater than 2.681.
- Otherwise, **fail to reject \( H_0 \)**.

Would you like help calculating the test statistic or verifying the decision rule?

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Here are 5 related questions to expand your understanding:
1. How is the test statistic calculated in hypothesis testing for correlation?
2. What does the correlation coefficient of 0.65 imply about the relationship between the pump price and gallons sold?
3. How would the decision rule change if we used a different significance level?
4. What does it mean to reject the null hypothesis in this context?
5. How can we interpret the strength of the relationship between two variables based on the correlation coefficient?

**Tip**: Always ensure the hypothesis test is appropriate for the sample size to avoid inaccurate conclusions.

Pennsylvania Refining Company is studying the relationship between the pump price of gasoline and the number of gallons sold. For a sample of 14 stations last Tuesday, the correlation was 0.65. The following hypotheses are given. H0: ρ ≤ 0 H1: ρ > 0 a. State the decision rule for 0.010 significance level. (Round your answer to 3 decimal places.)

Discover how to perform a hypothesis test on the correlation between the pump price of gasoline and gallons sold, using a significance level of 0.010. This example includes the correlation coefficient, test statistic formula, and critical value calculation for 14 stations. Ideal for undergraduate statistics students.

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