Question

LO LART! 2. (No R required) The director of admissions of a small college selected 120 students at random from the new freshman class in a study to determine whether a student's grade point average (GPA) at the end of the freshman year (y) can be predicted from the ACT test score (x). The director fitted a simple linear regression E(GPA|ACT) = Bo + B1ACT. A lack of fit test was carried out to see if a simple linear regression equation should be used. Unfortunately, the printer was running out of ink and most of the values in the ANOVA table were not printed. (a) Fill in the 9 missing values in the following ANOVA table. Note: You do not need to calculate the mean square for SST. df 1 MS Source of Variation Regression Error Lack of fit Pure error Total 45.818 39.332 49.406 ********* (b) State the null and alternative hypotheses for the lack of fit test. Also state the corresponding F statistic, and conclusion in context. To use R to find critical value for F distribution, type qf (0.95, df1, df 2) with appropriate values for df 1 and df2. ) Carry out a hypothesis test on whether the variables, GPA at end of freshman year and ACT test score, are linearly related. State the null and alternative hypothesis, corresponding F statistic, and conclusion in context.

Solving (c) would be enough! Thanks a lot!

Answer 1

(c)

The null and alternative hypothesis are,

$H_0: \beta_1 = 0$

$H_a: \beta_1 \ne 0$

From table in (a), SS Error = 45.818

SS Pure Error = 39.332

SS Lack of fit = SS Error - SS Pure Error = 45.818 - 39.332 = 6.486

df for Error = Number of observations - Number of coefficient in the model - 1 = 120 - 1 - 1 = 118

df Pure error = 99

df Lack of fit = df Error - df Pure error = 118 - 99 = 19

Test statistic, F = [SS Lack of fit / df Lack of fit ] / [SS Pure Error / df Pure Error]

= (6.486 / 19) / (39.332 / 99)

= 0.8592

Critical value of F at 0.95 and df1 = 19 and df2 = 99 is 1.69

Since the test statistic is less than the critical value, we fail to reject H0 and conclude that there is no significant evidence that the variables, GPA at end of freshman year and ACT test score are linearly related.