From a random sample of 6 college students, a school counselor obtained the students' high school GPAs(x) and their college GPAs(y) (show in the table below). She proposed to use the simple linear regression to develop a model that will allow her to predict a student's college GPA from his or her own high school GPA.

Obsx y

1 2.78 2.95

2 3.82 3.64

3 3.16 3.55

4 3.53 3.89

5 2.96 3.06

6 3.37 3.31

You can copy the following R code to fit the simple linear regression model to the data in R.

y=c(2.95,3.64,3.55,3.89,3.06,3.31) x=c(2.78,3.82,3.16,3.53,2.96,3.37) result=lm(y~x)

summary(result)

anova(result)

Based on the output in R, answer the following questions. Please provide R output in your solutions.

(a) Write out the estimated regression line1.

(b) Find a 95% confidence interval for .(Note: t critical value can be found by R function

qt(.975,dfe) )

(c) Find SSR, SSE, and TSS.

(d) Based on the ANOVA F test, is the simple linear regression model useful to fit the data at the 0.05 significance level? Justify your answer clearly.

(e) Find a 95% confidence interval for the mean college GPA for all students who have 3.5 high school GPA. (Note: x and Sxx can be calculated by R functions mean(x)=3.27; var(x)*(6- 1)=0.7284, respectively.

(e) Find a 95% prediction interval for a student's college GPA who has 3.5 high school GPA. (You will notice the prediction interval is wider than the confidence interval)

The R-output