Question

A study was done to look at the relationship between 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	10	6	9	0	9	8	6	5	6
Books	0	3	0	8	0	1	3	1	0

1. Find the correlation coefficient: r= Round to 2 decimal places.
2. The null and alternative hypotheses for correlation are:
   H0: ? r ρ μ   = 0
   H1: ? μ r ρ   ≠ 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.
   • There is statistically significant evidence to conclude that a person who watches more movies will read fewer books than a person who watches fewer movies.
   • There is statistically 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.
   • There is statistically 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.
   • There is statistically significant evidence to conclude that a person who watches fewer movies will read fewer books than a person who watches fewer movies.
4. r2 =  (Round to two decimal places)
5. Interpret r2r2 :
   • There is a 78% chance that the regression line will be a good predictor for the number of books people read based on the number of movies they watch each year.
   • Given any fixed number of movies watched per year, 78% of the population reads the predicted number of books per year.
   • There is a large variation in the number books people read each year, but if you only look at people who watch a fixed number of movies each year, this variation on average is reduced by 78%.
   • 78% of all people watch about the same number of movies as they read books each year.
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:
   • The slope has no practical meaning since people cannot read a negative number of books.
   • For every additional movie that people watch each year, there tends to be an average decrease of 0.77 books read.
   • As x goes up, y goes down.
   
   
9. Interpret the y-intercept in the context of the question:
   • The best prediction for a person who doesn't watch any movies is that they will read 7 books each year.
   • The average number of books read per year is predicted to be 7 books.
   • The y-intercept has no practical meaning for this study.
   • If someone watches 0 movies per year, then that person will read 7 books this year.

  

**** Can you please type or print clearly. Thank you so much

Answer 1

Let denote the Pearson's Linear Correlation Coefficient between the No. of Movies watched and Books read. We have to test:

Vs   

The appropriate test statistic to test the above hypothesis would be:

where, r is the linear correlation coefficient based on the sample.

It is computed using the formula:

No. of Movies watched (X)	No. of Books read (Y)	(X-X bar)2	(Y-Y bar)2	(X-X bar)(Y-Y bar)
10	0	11.86	3.16	-6.12
6	3	0.31	1.49	-0.68
9	0	5.98	3.16	-4.35
0	8	42.98	38.72	-40.79
9	0	5.98	3.16	-4.35
8	1	2.09	0.60	-1.12
6	3	0.31	1.49	-0.68
5	1	2.42	0.60	1.21
6	0	0.31	3.16	0.99
X bar = 6.56	Y bar = 1.78	SUM
		72.22	55.56	-55.89

Substituting the values,

Correlation coefficient: r= -0.88 ...................................................................(1)

We find that the coefficient is negative and close to unity. Hence, we may infer that there exists a strong negative correlation between the No. of Movies watched and Books read.

The test statistic is obtained as:

= -4.90

To obtain the exact p-value, we may make use of the excel function:

We get P-value = 0.0018

Since, P-value = 0.0018 < 0.05, we may reject the null hypothesis. We may conclude that the data provide sufficient evidence to support the claim that there is a significant positive correlation between Ticket Price and Attendence.The correct option would be:

There is statistically 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.

This is nothing but the goodness of fit measure called the coefficient of determination.It measures the amount of variation in the dependent variable - No. of books read per year by the predictor - No. of movies watched per year. Here, the fitted regression model with predictor No. of movies watched per year explains about 78% of the variation in the dependent variable - No. of books read per year.

The correct option would be:

There is a 78% chance that the regression line will be a good predictor for the number of books people read based on the number of movies they watch each year.

The fitted regression model can be expressed as:

=Predicted number of books people read per year; x = Number of movies people watch per year

where the intercept coefficient is estimated by the formula:

and slope coefficient estimate can be obtained using the formula:

Computing the values,

Hence, the fitted regression equation is expressed as:

For x = 4,

Books per year = 4

The slope can be interpreted as:

• For every additional movie that people watch each year, there tends to be an average decrease of 0.77 books read.
• As x goes up, y goes down.

The y-intercept can be interpreted as:

The best prediction for a person who doesn't watch any movies is that they will read 7 books each year.