Answer to Question #157311 in Statistics and Probability for SL 1

Question

It is claimed that watching television reduces the amount of physical exercise, causing weight gains. A sample of 15 children was taken. The number of kilograms each child was overweight (a negative number indicates the child is underweight) and the number of hours the television viewing per week were recorded. The data are listed in the table.

Television Viewing (hours)

42

34

25

35

37

38

31

33

19

29

Overweight (kg)

18

6

-1

13

14

7

7

-9

8

Additional Information: SST = Σ(𝑌−𝑌̅)2 = 572.10, SSE = Σ(𝑌−𝑌̂)2 = 149.75

a) Compute the coefficients of the sample regression line.

b) Interpret the estimated slope coefficient.

c) Determine and interpret the coefficient of determination and test whether the model is good or not?

d) Test whether there is evidence of a negative linear relationship between the number of hours of television viewing per week and the child's overweight using 1 per cent level of significance.

Answer 1

$$SST=\sum (Y_i-\bar{Y})^2=572.10$$

$$SSE=\sum (Y_i-\hat{Y})^2=149.75$$

a)

$$\sum X_i=323,\sum Y_i=63, n=10$$

$$\bar{X}=32.3, \bar{Y}=6.3$$

$$S_{XX}=402.1,S_{XY}=412.1, S_{YY}=572.10$$

$$B=\dfrac{S_{XY}}{S_{XX}}=\dfrac{412.1}{402.1}=1.02487$$

$$A=\bar{Y}-B\bar{X}=6.3-\dfrac{412.1}{402.1}\cdot32.3=-26.80328$$

$$Y=-26.80328+1.02487X$$

b)

$B=1.02487>0,$ positive relationship

c)

$$r^2=1-\dfrac{SSE}{SST}=1-\dfrac{149.75}{572.10}=0.738245$$

$$r=0.8592, strong\ positive\ correlation$$ $$s_B=\dfrac{\sqrt{\dfrac{SSE}{n-2}}}{\sqrt{S_{XX}}}=\dfrac{\sqrt{\dfrac{149.75}{10-2}}}{\sqrt{402.1}}=0.21576$$

The following needs to be tested:

$H_0:B=0$

$H_1:B\not=0$

The sample size is $n=10,$ so then the number of degrees of freedom is $df=n-2=10-2=8.$

The corresponding critical correlation value $r_c$ for a significance level of $\alpha=0.01,$ for a two-tailed test is: $t_c=3.35536$

$$t=\dfrac{B}{s_B}=\dfrac{1.02487}{0.21576}=4.75$$

We have that $|t|=4.75>3.35536=t_c,$  from which is concluded that the null hypothesis is rejected.

d) The following needs to be tested:

$H_0:\rho\geq0$

$H_1:\rho<0$

where $\rho$  corresponds to the population correlation.

The sample size is $n=10,$ so then the number of degrees of freedom is $df=n-2=10-2=8.$

The corresponding critical correlation value $r_c$ for a significance level of $\alpha=0.01,$ for a left-tailed test is: $r_c=-0.715$

Based on the sample correlation provided, we have that $r=0.8592>-0.715=r_c,$   from which is concluded that the null hypothesis is not rejected.