B) A college bookstore must order books two months before euch semester starts. They
believe that the number of books that will ultimately be sold for any particular course is
related to the number of students registered for the course when the books are ordered.
They would like to develop a linear regression equation to help plan how many books to
order. From past records, the bookstore obtains the number of students registered, X, and
the number of books actually sold for a course, Y, for 8 different semesters. These data
are below.
Semester
Students
Books
1
36
31
2
28
29
3
36
34
4
38
38
5
30
29
6
29
29
7
38
33
8
39
35
Construct a linear Regression for the above data,
Answers
Answer:
SOLUTION
A. The following scatterplot with the fitted line was obtained using StatCrunch.
As the number of students registered for the course increases, the number of books sold by the bookstore appears to increase in a straight-line manner.
B. H0: The number of students registered and the number of books sold are not correlated
Ha: The number of students registered and the number of books sold are correlated
Decision Rule: Accept Ha if the calculated p-value < .01.
Test Statistic: r = the Pearson coefficient of correlation
Calculations from StatCrunch: r = 0.8997, p-value < 0.0001
Interpretation: At the .01 level of significance I conclude that as the number of students registered increases, the number of books sold increases in a straight-line manner.
C. Since the p-value is less than 0.0001, this indicates that if the number of students registered and the number of books sold are not correlated (if the null hypothesis is true), then there is virtually no chance that the observed points in the scatterplot would exhibit such an obvious straight-line pattern.
D. r 2 = .809 (80.9%). 80.9% of the variability in the number of books sold is explained by the straight-line relationship with the number of registered students. 19.1% of this variability is unexplained, and due to error. This relationship is quite strong.
When no students have registered for a course, the number of books sold is 9.30 (or about 9). This is the starting point of the straight-line when x = 0. It is not particularly meaningful in this problem since all the classes sampled had more than 25 students registered. For each additional student registered for a course, the number of books sold increases by 0.673.
F. Since 30 students is within the range of the sampled number of students, it is appropriate to make this prediction. From Minitab the calculated prediction interval is (25.865078, 33.09856). I am 95% confident that for a course that has 30 students registered the bookstore will sell between 25.9 and 33.1 books.
G. Since 30 students is within the range of the sampled number of students, it is appropriate to make this estimation. From Minitab the calculated confidence interval is (28.279491, 30.684145). I am 95% confident that for all courses that have 30 students registered the bookstore will sell an average of between 28.3 and 30.7 books per semester.
H. Since 5 students is not within the range of the sampled number of students, it is not appropriate to use the regression equation to make this prediction. We do not know if the straight-line model would fit data at this point, and we should not extrapolate.