Pennsylvania Refining Company is studying the relationship between the pump price of gasoline and the number of gallons sold. For a sample of 20 stations last Tuesday, the correlation was .78.
At the .01 significance level, is the correlation in the population greater than zero? (Round your answer to 3 decimal places.)
What is the test statistic?
To determine if the correlation in the population is greater than zero at the 0.01 significance level, we can perform a hypothesis test.
H0 (null hypothesis): The correlation in the population is equal to zero (ρ = 0).
H1 (alternative hypothesis): The correlation in the population is greater than zero (ρ > 0).
To conduct this hypothesis test, we will use the sample correlation coefficient (r) and its associated t-test statistic. The formula for the t-test statistic is:
t = (r * sqrt(N - 2)) / sqrt(1 - r^2)
Where:
r = sample correlation coefficient
N = sample size
In this case, the sample correlation coefficient is 0.78, and the sample size is 20.
Calculating the test statistic:
t = (0.78 * sqrt(20 - 2)) / sqrt(1 - 0.78^2)
= (0.78 * sqrt(18)) / sqrt(1 - 0.6084)
= (0.78 * sqrt(18)) / sqrt(0.3916)
= (0.78 * 4.24264) / 0.6255
= 3.314
The calculated test statistic is 3.314.
Now, to determine if the test statistic is statistically significant at the 0.01 significance level, we need to compare it with the critical value from the t-distribution.
Degrees of freedom (df) = sample size - 2 = 20 - 2 = 18
Using a t-table or a statistical software, we find that the critical value for a one-tailed t-test with a significance level of 0.01 and 18 degrees of freedom is approximately 2.878.
Since the calculated test statistic (3.314) is greater than the critical value (2.878), we reject the null hypothesis.
Therefore, at the 0.01 significance level, we can conclude that the correlation in the population is greater than zero.
In summary:
- Null hypothesis (H0): The correlation in the population is equal to zero.
- Alternative hypothesis (H1): The correlation in the population is greater than zero.
- Test statistic: 3.314
- Decision: Reject the null hypothesis. The correlation in the population is greater than zero.