Question

Researchers have collected data on the hours of television watched in a day and the age of a person. They have taken 6 samples. A linear regression analysis has already been run on this data. Here are the partial results:

Analysis of variance (Hours of Television):
Source	DF	Sum of squares	Mean squares	F	Pr > F
Model		10.939
Error
Total		13.333
Model parameters (Hours of Television):
Source	Value	Standard error	t	Pr > |t|	Lower bound (95%)	Upper bound (95%)
Intercept	6.910	0.894	7.727	0.002	4.427	9.392
Age	-0.115	0.027

Fill out the remaining blanks in the above tables. The df may be answered as whole numbers, but all other responses should be rounded to three decimal places.

What would be the result of the hypothesis test: H0: β 1 = 0, HA: β 1 ≠ 0? (Choose one: reject / fail to reject)  the null. Let α = 5 %.

Based on the result above, can age be considered a useful predictor of hours of television watched? (Choose one: Y/N)

What would be the value of R^2?  % Round to the nearest whole number.

What would be the value of the correlation coefficient?  Round to three decimal places.

What would be the equation of the line of best fit? Round each coefficient to three decimal places.

Hours of Television =  +  *Age

For someone that is 15 years old, how many hours of television would we expect that they watch per day?  Round to the nearest whole number.

For every additional year of age, what would be the expected decrease in the hours of television watched? Round to three decimal places.

Answer 1

Analysis of variance (Hours of Television):
Source	DF	Sum of squares	Mean squares	F	Pr > F
Model	1	10.939	10.939	18.277	0.013
Error	4	2.394	0.5985
Total	5	13.333
Model parameters (Hours of Television):
Source	Value	Standard error	t	Pr > |t|	Lower bound (95%)	Upper bound (95%)
Intercept	6.910	0.894	7.727	0.002	4.427	9.392
Age	-0.115	0.027	-4.259	0.013	-0.190	-0.040

DF Model = Number of predictors (k) = 1

DF Error = n - k - 1 = 6 -1 -1 = 4

DF Total = n-1 = 6 - 1 = 5

SS Error = SS Total - SS Model = 13.333 - 10.939 = 2.394

Mean Square = SS / DF

F = MS Model / MS Error = 10.939 / 0.5985 = 18.277

Degree of freedom for F statistic, = DF Model, DF Error = 1, 4

P(F > 18.277) = 0.013

t = Coeff / Std Error = -0.115 / 0.027 = -4.259

Degree of freedom of t statistic = DF Error = 4

P-value = 2 * P(t < -4.259) = 0.013

Critical value of t at df = 4 and 95% confidence interval is  2.776

Lower limit = -0.115 - 2.776 * 0.027 = -0.190

Upper limit = -0.115 + 2.776 * 0.027 = -0.040

Since P-value (0.013) is less than 0.05 significance level, we reject the null hypothesis H0. Yes, age be considered a useful predictor of hours of television watched.

R^2 = SS Model / SS Total = 10.939 / 13.333 = 0.820 = 82%

Correlation coefficient r = = 0.906

The the equation of the line of best fit is,

Hours of Television = 6.910 - 0.115 *Age

For Age = 15, expected Hours of Television is,

Hours of Television = 6.910 - 0.115 * 15   5 hours per day

For every additional year of age, the expected decrease in the hours of television watched is 0.115 (the absolute value of slope coefficient).