Question

For the data set shown below, complete parts (a) through (d) below.

X 20 30 40 50 60

Y 98 95 91 81 68

(a) Find the estimates of Bo and B1.

Bo=bo= _____ (Round to three decimal places as needed.)

B1=b1= ______(Round to four decimal places as needed.)

(b) Compute the standard error the point estimate for

se= ____

(c) Assuming the residuals are normally distributed, determine

Sb1=____

(Round to four decimal places as needed.)

(d) Assuming the residuals are normally distributed, test HoB1=0 versus H1:B1/=0 at the a=0.05 level of significance. Use the P-value approach.

The P-value for this test is _____.

(Round to three decimal places as needed.)

Make a statement regarding the null hypothesis and draw a conclusion for this test. Choose the correct answer below.

A. Reject Ho. There is sufficient evidence at the a= 0.05 level of significance to conclude that a linear relation exists between x and y.

B. Reject Ho. There is not sufficient evidence at the a= 0.05 level of significance to conclude that a linear relation exists between x and y.

C. Do not reject Ho. There is not sufficient evidence at the a= 0.05 level of significance to conclude that a linear relation exists between x and y.

D. Do not reject Ho. There is sufficient evidence at the a= 0.05 level of significance to conclude that a linear relation exists between x and y.

Answer 1

(a) The formula for calculating the estimate of the population intercept and slope is and which is given by-

20	98	-20	400	11.4	129.96	-228
30	95	-10	100	8.4	70.56	-84
40	91	0	0	4.4	19.36	0
50	81	10	100	-5.6	31.36	-56
60	68	20	400	-18.6	345.96	-372

So the estimate of and is calculated as and

______________________________

(b) The formula for calculating the standard error of the estimate, is given by-

where,

; ; ;

So the standard error of the estimate is calculated as

______________________________

(c) The formula for calculating the standard deviation of is given by-

where, and

hence,

____________________________

(d) For the given null and alternative hypothesis we need to test this at significance level.

Null hypothesis,

Alternative hypothesis,

Test-statistic: The formula for calculating the test statistic is

and it follows a t-distribution with , where n is the number of observations, is the estimate of the slope of the regression line and is the sample standard deviation of .

We have already calculated, and

So the test statistic is calculated as

P-value: Since we are testing a two-tailed hypothesis and the test statistic is calculated as , then the p-value is calculated as-

So the p-value is calculated as

Decision: The significance level is given and the p-value is

Conclusion: The correct option is (A), i.e., Reject . There is sufficient evidence at level of significance to conclude that a linear relation exists between x and y.