The following data represent the age in years of various peo

The following data represent the age (in years) of various people and the number of days per week they exercise. We are interested in doing a regression analysis on this data to see if age affects how many days per week someone works out. Please show your work for your answers.

               Age        Days they Exercise

24                          4

18                          6

29                          6

17                          4

61                          5

51                          2

30                          6

24                          4

18                          2

21                          1

               a. Which one is the independent variable and which one is the dependent variable?

               b. Find b₁.

               c. Find b₀.

               d. Find SST.

               e. Find SSR.

               f. Find SSE.

               g. Find the coefficient of determination.

               h. Find the correlation coefficient.

               i. Find s².

               j. Find the test statistic for testing if b₁ is significant or not.

               k. What conclusion would you make based on the test statistic found above?

               l. Find the 95% confidence interval for B₁.

m. Assuming a person is 40 years old, how many days per week are they expected/predicted to exercise?

n. Assuming a person is 40 years old, what is the 90% confidence interval for the expected number of days of exercise for them?

o. Assuming a person is 40 years old, what is the 90% prediction interval for the expected number of days of exercise for them?

Solution

The following data represent the age (in years) of various people and the number of days per week they exercise. We are interested in doing a regression analysis on this data to see if age affects how many days per week someone works out.

Let dependent variable X is age.

and independent variable Y is days they exercise.

We can obtain regression output by using MINITAB.

steps :

STAT -->regression --> regression -->response :Y -->predictors :X --> Result : second option --> ok

This will gives us following output.

Regression Analysis: Y versus X

The regression equation is
Y = 3.80 + 0.0070 X

Predictor Coef SE Coef T P
Constant 3.796 1.404 2.70 0.027
X 0.00697 0.04314 0.16 0.876

S = 1.93334 R-Sq = 0.3% R-Sq(adj) = 0.0%

Analysis of Variance

Source DF SS MS F P
Regression 1 0.098 0.098 0.03 0.876
Residual Error 8 29.902 3.738
Total 9 30.000

b1 = slope = oefficient of X = 0.0070

b0 = constant = 3.80

SST = total summ of squares = 30.000

SSR = regression sum of squares = 0.098

SSE = error sum of squares = 29.902

coefficient of determination (R²) = 0.3% = 0.3/100 = 0.003

correlation coefficient (r) = sqrt(0.003) = 0.05477

The test of hypothesis is,

H0: B1=0 Vs H1 : B1 not=0

the test statistic for testing if b1 is = 0.16

P-value = 0.876

Alpha = 0.05

p-value > alpha

Accept H0 at 5% level of significance.

Conclusion :slope is 0.

SEb = 0.16

critical value (tc) = 2.3060

The confidence interval for B1 is b - E < B1 < b + E

where E is the margin of error.

E = tc * SEb = 2.3060 * 0.16 = 0.3690

lower limit = b - E = 0.0070 - 0.3690 = - 0.362

upper limit = b + E = 0.0070 + 0.3690 = 0.376

assume that x = 40,

y = 3.80 + 0.0070* x

y = 3.80 + (0.0070*40) = 4.08

Assuming a person is 40 years old, what is the 90% confidence interval for the expected number of days of exercise for them?

when x=40 then y = 4.08

90% confidence interval for y is,

Yp - E < y < Yp + E

Where E = tc*se sqrt [ 1+1/n + (x-xbar)² / SSx ]

se = sqrt [ (SSy - bSSxy) / n-2

se = 1.9333

E = 4.7955

lower limit = Yp - E = 4.08 - 4.7955 = -0.7155

upper limit = Yp + E = 4.08 + 4.7955 = 8.8755

n	10
xbar	29.3
ybar	4
SSx	2008.1
Ssy	30
Ssxy	14
b	0.006972
a	3.795727
	3.737799
se	1.933339