Please do problem 2.13 Please do problem 2.13 1.19. Grade point average. The dir
ID: 3304966 • Letter: P
Question
Please do problem 2.13
Please do problem 2.13
1.19. Grade point average. 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 results of the study follow. Assume that first-order regression model (1.1) is appropriate. 2 3 118 119 120 21 Xi: Yi: 14 28 28 16 28 3.897 3.885 3.778 3.914 1.860 2.948 a. Obtain the least squares estimates of Po and Bi, and state the estimated regression function. b. Plot the estimated regression function and the data. Does the estimated regression function appear to fit the data well? c. Obtain a point estimate of the mean freshman GPA for students with ACT test score X = 30 d. What is the point estimate of the change in the mean response when the entrance test score increases by one point?Explanation / Answer
I am using R software to solve this problem.
First we can copy the data into a .txt file and then import it into R using read.table() function.
Data <- read.table("Data.txt",sep=" ",header = T)
A linear model can be created in R using the lm() function as below.
model <- lm(GPA ~ ACT, data = Data)
a) Prediction can be done using the predict() function in R as below:
predict(model, data.frame(ACT=28), interval ="confidence", level = 0.95)
fit lwr upr
3.201209 3.061384 3.341033
We can say that we are 95% confident that the GPA score will lie between 3.061384 to 3.341033.
b) We can obtain the prediction interval as below:
predict(model, data.frame(ACT=28), interval ="prediction", level = 0.95)
fit lwr upr
3.201209 1.959355 4.443063
We can say that there is 95% probability that the GPA score of Mary Jones will be contained within the prediction interval from 1.959355 to 4.443063.
c) Yes the prediction interval is wider. That is always the case because prediction intervals are conceerned with the individual observations in a population as well as parameter estimates.