Final answer:
The regression equation explains approximately 44.89% of the total variation in the average fare (FARE). The bivariate correlation coefficient between FARE and DISTANCE is 0.6700, indicating a moderately strong positive correlation. By rejecting the null hypothesis, it can be concluded that longer air routes tend to have higher fares than shorter air routes, on average.
Step-by-step explanation:
a. What percent of the total variation in FARE does the regression equation, or DISTANCE, explain or determine?
The coefficient of determination, R², represents the percentage of the total variation in the response variable (FARE) that is explained by the regression equation (DISTANCE). In this case, R² is 0.4489, which means that approximately 44.89% of the total variation in FARE is explained by DISTANCE.
b. What is the value of the bivariate correlation coefficient between FARE and DISTANCE in the sample?
The bivariate correlation coefficient, R, measures the strength and direction of the linear relationship between FARE and DISTANCE. In this case, R is 0.6700, indicating a moderately strong positive correlation between the two variables.
c. State the null and alternative hypotheses to test whether the slope coefficient for DISTANCE is significantly greater than zero
The null hypothesis (H0) states that the slope coefficient for DISTANCE is equal to zero, indicating no linear relationship between FARE and DISTANCE. The alternative hypothesis (Ha) states that the slope coefficient is significantly greater than zero, indicating a positive linear relationship between FARE and DISTANCE.
d. If α = 0.01, determine the critical value for the test statistic associated with the hypotheses in Part c
The critical value depends on the significance level (α) and the degrees of freedom. Since α = 0.01, and the degrees of freedom for this regression analysis is 636 (638 - 2), you can consult a t-table or a statistical software to find the critical value. It is usually a t-distribution value associated with a two-tailed test at α/2 (0.005) for a given degrees of freedom.
e. Given the MS Excel output and Part d, state your decision regarding the null hypothesis in Part c
To make a decision regarding the null hypothesis, you need to calculate the test statistic (t-value) for the slope coefficient and compare it with the critical value determined in Part d. If the t-value is greater than the critical value, then you reject the null hypothesis, indicating that there is a significant linear relationship between FARE and DISTANCE. If the t-value is less than the critical value, you cannot reject the null hypothesis.
f. You are an airline industry executive. Can you conclude longer air routes have higher fares than shorter air routes, on average?
Based on the regression analysis, which showed a positive slope coefficient for DISTANCE, and the decision to reject the null hypothesis that the slope coefficient is zero, you can conclude that there is a significant positive relationship between the length of an air route (DISTANCE) and the average fare (FARE). Therefore, longer air routes tend to have higher fares than shorter air routes, on average.