Question

We are interested in determining if the amount of sleep someone gets per night affects how much they exercise. The following data are provided:

(Sleep- Exercise) 4-20, 5-0, 5.5- ,7 6 -14, 6.5- 7, 7 -12, 7.5- 4, 8- 6, 8.5- 1, 9- 8

i. Find s^2.

j. Find the test statistic for testing if b_1 is significant or not.

k. What conclusion would you make based on the test statistic found above (use .05 confidence level)?

l. Find the 95% confidence interval for β_1.

m. Assuming someone sleeps 8 hours a night, how much would you expect them to exercise?

n. Assuming someone sleeps 8 hours a night, what is the 90% confidence interval for the expected amount of exercise?

o. Assuming someone sleeps 8 hours a night, what is the 90% prediction interval for the expected amount of exercise?

Answer 1

As the interest is whether the amount of sleep afftects how much one exercises, hence we take exercise time as response (y) variable , which depends thon the amount of sleep, the independent (X) variable .

The model of regression wiil be : y= b0+b1x

where b0 is the intercept coefficient( interpreted as mean response value under x=0) and b1 is slope coefficient( interpreted as the change in the mean response for unit change in x value )

1. the s^2 is estimated as the sum of square of the residuals,which is calculated as 0.07381373

2. The estimated value of the slope coefficient b1 =-0.02799

To test that b1 is significant or not, a t-test is carried out. The p value of the test =   0.1989 >0.05 = the alpha level of significance. Thus we cant reject the null hypothesis of b1=0 and accept that the slope is significant at 5% level of significance.

3. From the above result we can say that the variable "amount of sleep" significantly affects the amount of exercise one gets.

4. The 95% CI for b_1 is given by : (0.1617186 , 0.2360814 )

5. Assuming someone sleeps 8 hours a night, how much would you expect them to exercise

ANs : y= y= 0.31923-(0.02799 *8)= 0.09531 hour