The accompanying multiple regression outputs estimate the following model:
annual cost = 0 + 1(Type of School) + 2(Average Total SAT)
where:
annual cost = annual cost at a college in thousands of dollars
Average Total SAT = average scholastic aptitude scores for students at the school
Type of School = 1 for private schools, = 0 for public schools
a) Is there a significant relationship between annual cost and the independent variables? (use alpha=.05)
b) Find a 90% confidence interval on the coefficient for the variable “school type”.
c) Find the predicted annual cost at a public college, where the SAT scores average 1100. Find an approximate 95% confidence interval for the annual cost.
a) To determine if there is a significant relationship between annual cost and the independent variables, we can look at the p-values associated with the coefficients in the multiple regression outputs.
First, we set up the null and alternative hypotheses:
Null hypothesis (H0): There is no significant relationship between annual cost and the independent variables.
Alternative hypothesis (Ha): There is a significant relationship between annual cost and the independent variables.
To test the hypotheses, we can look at the p-value associated with the regression coefficients. In this case, we are interested in the p-value for the overall model.
If the p-value is less than the chosen significance level (alpha=.05 in this case), we reject the null hypothesis and conclude that there is a significant relationship between annual cost and the independent variables.
b) To find a 90% confidence interval on the coefficient for the variable "school type," we can use the t-distribution.
We start by finding the standard error of the coefficient. The standard error can be obtained from the multiple regression outputs. Once we have the standard error, we can calculate the margin of error by multiplying the critical value from the t-distribution (based on the desired confidence level) with the standard error.
Finally, we can construct the confidence interval by subtracting the margin of error from the coefficient and adding the margin of error to the coefficient.
c) To find the predicted annual cost at a public college, where the SAT scores average 1100, we can substitute the given values into the regression model:
annual cost = β0 + β1(Type of School) + β2(Average Total SAT)
Substitute the given values:
Type of School = 0 (since it is a public college)
Average Total SAT = 1100
By plugging these values into the regression equation, we can find the predicted annual cost at a public college with an SAT score of 1100.
To find an approximate 95% confidence interval for the annual cost, we can use the standard error of the estimate. The standard error can be obtained from the multiple regression outputs. Once we have the standard error, we can calculate the margin of error by multiplying the critical value from the t-distribution (based on the desired confidence level) with the standard error.
Finally, we can construct the confidence interval by subtracting the margin of error from the predicted annual cost and adding the margin of error to the predicted annual cost.