Question

A study was done to look at the relationship between the number of movies people watch at the theater each year and the number of books that they read each year. The results of the survey are shown below.

Movies	2	2	2	2	7	10	2	2	9	7	6	1
Books	10	6	6	6	3	0	8	5	1	6	2	6

x value y value

1. Find the correlation coefficient: r=r=    Round to 2 decimal places.
2. The null and alternative hypotheses for correlation are:
   H0:H0: ? ρ r μ  == 0
   H1:H1: ? μ ρ r   ≠≠ 0
   The p-value is:      Round to 4 decimal places.
   
3. Use a level of significance of α=0.05α=0.05 to state the conclusion of the hypothesis test in the context of the study.
   Select an answer At the 5% significance level, the data provides significant evidence to conclude that a person who watches more movies will read fewer books than a person who watches fewer movies. At the 5% significance level, the data provided significant evidence to conclude that a person who watches fewer movies will read fewer books than a person who watches fewer movies. At the 5% significance level, the data provides insignificant evidence to conclude that there is a correlation between the number of movies watched per year and the number of books read per year. Thus, the use of the regression line is not appropriate. At the 5% significance level, the data provides significant evidence to conclude that there is a correlation between the number of movies watched per year and the number of books read per year. Thus, the regression line is useful.
4. r2r2 =  (Round to two decimal places)
5. Interpret r2r2 : Select an answer Approximately 66% of the variation in the number books people read each year can be explained by the variation in the number of movies that the regression line is a good predictor. Approximately 66% of the variation in the number of movies viewed each year can be explained by the variation in number of books people read each years that the regression line can be explained by the variation is a good predictor.
6. The equation of the linear regression line is:   
   ˆyy^ =   + xx    (Please show your answers to two decimal places)   
   
7. Use the model to predict the number of books read per year for someone who watches 4 movies per year.
   Books per year =  (Please round your answer to the nearest whole number.)   
   
8. Interpret the slope of the regression line in the context of the question: Select an answer For every additional movie that people watch each year, there tends to be an average decrease of 0.73 books read. As x goes up, y goes down. The slope has no practical meaning since people cannot read a negative number of books.
   
9. Interpret the y-intercept in the context of the question: Select an answer The average number of books read per year is predicted to be 8 books. The y-intercept has no practical meaning for this study. If someone watches 0 movies per year, then that person will read 8 books this year. The best prediction for a person who doesn't watch any movies is that they will read 8 books each year.

Answer 1

Sol:

∑X=86 , ∑Y=66 , ∑X⋅Y=206 , ∑X2=756 , ∑Y2=644

Use the following formula to work out the correlation coefficient.

$r=\frac{n\sum XY-\sum X\sum Y}{\sqrt{[n\sum X^2-(\sum X)^2][n\sum Y^2-(\sum Y)^2]}}=-0.5654$

B. R^2=$R^2=(-0.5654)^2=0.3197$

C.

Step 1: Find X⋅Y and X2 as it was done in the table below.

X	Y	X⋅Y	X⋅X
3	7	21	9
6	0	0	36
0	18	0	0
9	0	0	81
14	1	14	196
5	4	20	25
7	5	35	49
12	0	0	144
2	6	12	4
1	2	2	1
7	10	70	49
0	9	0	0
5	2	10	25
11	2	22	121
4	0	0	16

Step 2: Find the sum of every column:

∑X=86 , ∑Y=66 , ∑X⋅Y=206 , ∑X2=756

Step 3: Use the following equations to find a and b:

$a=\frac{\sum Y\sum X^2-\sum X\sum XY}{n\sum X^2-(\sum X)^2}=8.159$

$b=\frac{n\sum XY-\sum X\sum Y}{n\sum X^2-(\sum X)^2}=-0.656$

Step 4: Substitute a and b in regression equation formula

$y = a + bx= 8.159 - 0.656x$

D. $x=8, y = a + bx= 8.159 - 0.656*8=2.911$