Question

determine whether there is a significant linear relationship.
each year u.s. news and world report conducts its \survey of americas best graduate and professional schools\ and ranks the top 25 business schools, as determined by reputation, student selectivity, placement success, and graduation rate.
for each school, three variables were measured: (1) gmat score for the typical incoming student; (2) student acceptance rate (percentage accepted of all students who applied); and (3) starting salary of the typical graduating student.
an academic advisor wants to predict the typical starting salary of a graduate at a top business school using gmat score of the school as a predictor variable. the results of a simple linear regression of salary versus gmat using the 25 data points are shown below. is there sufficient evidence to suggest there is a linear relationship between salary and gmat at the 5% level of significance?
a = -92040 b = 228 s = 3213 r² =.66 r =.81 df = 23 t = 6.67 (the test statistic)
state your conclusion.
there is enough evidence (at α =.05) to suggest that there is a linear relationship between salary and gmat.
only 6.67% of the sample variation in salary can be explained by using gmat in a straight - line model.
there is not enough evidence (at α =.05) of at least a positive linear relationship between salary and gmat.
we estimate salary to increase $6.67 for every 1 - point increase in gmat.

Explanation:

Step1: Hypothesis testing for linear relationship

In simple linear regression, to test for a significant linear relationship between \(x\) (GMAT) and \(y\) (SALARY), the null hypothesis is \(H_0:\beta_1 = 0\) (no linear relationship) and the alternative hypothesis is \(H_a:\beta_1
eq0\) (linear relationship exists). The test statistic is \(t=\frac{b - 0}{s_b}\) (where \(b\) is the slope of the regression line). Given \(t = 6.67\) and degrees of freedom \(df=23\).

Step2: Critical - value comparison

For a two - tailed test with \(\alpha = 0.05\) and \(df = 23\), the critical values are \(t_{\alpha/2,df}=t_{0.025,23}\). From the \(t\) - distribution table, \(t_{0.025,23}=2.069\). Since \(|t|=6.67>2.069\), we reject the null hypothesis.

Step3: Analysis of other options

• Option 2: The coefficient of determination \(r^{2}=0.66\) means \(66\%\) (not \(6.67\%\)) of the sample variation in SALARY can be explained by the straight - line model with GMAT, so this option is wrong.
• Option 3: Since we rejected \(H_0\), there is enough evidence (not not enough) for a linear relationship (the sign of \(b = 228>0\) also suggests a positive relationship), so this option is wrong.
• Option 4: The slope \(b = 228\) means we estimate SALARY to increase \(\$228\) (not \(\$6.67\)) for every 1 - point increase in GMAT, so this option is wrong.

Answer:

There is enough evidence (at \(\alpha = 0.05\)) to suggest that there is a linear relationship between SALARY and GMAT.