Question

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. 26 Movics 4 9 4 0 1 3 Books 12 1 2 13 a. Find the correlation coefficient: r = Round to 2 decimal places. b. The null and alternative hypotheses for correlation are: Ho ? 0 H1? 0 Round to 4 decimal places. The p-value is: c. Use a level of significance of a = 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 significant evidence to conclude that a person who watches fewer movics will read fewer books than a person who watchcs fewer movics. 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 insignificant evidence to conclude that there is a correlation between the number of movies watched per year and the number of books read per ycar. Thus, the use of the regression line not appropriate. d. = (Round to two decimal places) Interpret r2 e. There is a 73% 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, 73% of the population reads the predicted number of books per year. 73% of all people watch about the same number of movies as they rcad books cach year There is a large variation in the number books people read cach ycar, but if you only look at pcople who watch a fixed number of movies cach ycar, this variation on average is reduced by 73%

Answer 1

Solution: The data given in the question statement is as follows. Computing the additional columns: Movies Books y2 XY X Y 4 1 4 16 81 9 4 2 8 16 4 13 169 0 0 25 2 5 10 A 0 36 0 6 12 12 1 1 144 3 15 25 ΣΧ = 29 ΣΥ. ΣΧ-163 ΣΥ-38 ΣΧΥ -49 = 368 Part-A: Correlation Coefficient: ηΣxy - Σx Σy r = η Σx-Σ x)2 x n Σy2 - Σy) 2 8 x 49- 29 x 38 (8 x 163 (29) 2} x /8 x 368 (38) 2 r =-0.852 Part-B: H9: ρ -0 Ha:ρ + 0 8 Degrees of Freedom df =n - 2 2 = 6 Test Statistic: Vn 2 V8-2 = (-0.852) x t = r x -3.986 V1-r2 1 (-0.852)2

P-Value: From t-distribution table; the area under the curve to the left of (t -3.986) and for (df 6), is given by: 0.0036 The test is a two-tailed test. Hence, p-value 0.0036 x 2 = 0.0072 Part-C: a = 0.05 The p-value is< a Hence, we reject the null hypothesis. There is statistically significant evidence to conclude that there is a correlation between the number of movies watched per year and number of books read per year. Thus, the regression line is useful. Part-D: r2 = (-0.852)2 = 0.726 Part-E: There is a 73% 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. Part-F ΣΥ 38 = 4.75 - ΣΧ29 X = 3.625 - = The equation of least squares regression line is: Y = a +bX νΣ ΧΥ-ΣΧΣΥ b = ΝΣΧ-ΣΧ

8 x 49 29 x 38 b 8 x 163 (29) 2 b-1.53 a = Y - bX a 4.75 (-1.53) x 3.625 10.31 The equation of linear regression line is: Y = a +bX >Y=10.31 - 1.53X Part-G: Given, X 7 Hence, books per year: Y = 10.31-1.53X ? = 10.31- 1.53 x 7 Y = -0.425 Part-H: For every additional movie that people watch each year, there tends to be an average decrease of 1.53 books read. Part-l: The best prediction for a person who doesn't watch any movies is that they will read 10 books each year.