Introduction to the Practice of Statistics: w/CrunchIt/EESEE Access Card
Introduction to the Practice of Statistics: w/CrunchIt/EESEE Access Card
8th Edition
ISBN: 9781464158933
Author: David S. Moore, George P. McCabe, Bruce A. Craig
Publisher: W. H. Freeman
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Chapter 11, Problem 8E

(a)

To determine

To test: The hypothesis that the coefficient of variable x1 is zero against the hypothesis that it is not zero.

(a)

Expert Solution
Check Mark

Answer to Problem 8E

Solution: The null hypothesis that the coefficient of variable x1 is zero is true.

Explanation of Solution

Given: The confidence level is provided as 95%. The following information has been provided in the problem:

y^=1.6+6.4x1+5.7x2n=25SEb1=3.3

Explanation:

Calculation: In a regression equation, the coefficient of x1 is denoted as β1. According to the provided problem, the null hypothesis will be formulated as follows:

H0:β1=0

The alternative hypothesis is a two-sided alternative. Hence, it will be formulated as follows:

Ha:β10

The total number of observations has been provided in the problem as

n=25

Since, there are 2 explanatory variables in the regression equation and the degrees of freedom for the provided problem can be calculated as follows:

np1=2521=22

The value of coefficient of x1 is b1=6.4. Since, the test is two-tailed, the value of t* for 95% confidence level for 22 degrees of freedom is obtained from the table as: 2.074. The t-statistic is calculated as:

t=b1SEb1=6.43.3=1.94

Conclusion: Since, the calculated value of t is less than the observed value of t, the null hypothesis does not get rejected. Hence, the null hypothesis that the coefficient of variable x1 is zero is true.

(b)

To determine

To test: The hypothesis that the coefficient of variable x1 is zero against the hypothesis that it is not zero.

(b)

Expert Solution
Check Mark

Answer to Problem 8E

Solution: The null hypothesis that the coefficient of variable x1 is zero is not true.

Explanation of Solution

Given: The confidence level is provided as 95%. The following information has been provided in the problem:

y^=1.6+6.4x1+5.7x2n=43SEb1=2.9

Explanation:

Calculation: In a regression equation, the coefficient of x1 is denoted as β1. According to the provided problem, the null hypothesis will be formulated as follows:

H0:β1=0

And the alternative hypothesis is a two-sided alternative. Hence, it will be formulated as follows:

Ha:β10

The total number of observations has been provided in the problem as:

n=43

Since, there are 2 explanatory variables in the regression equation, the degrees of freedom for the provided problem can be calculated as follows:

np1=4321=40

The value of coefficient of x1 is b1=6.4. Since, the test is two-tailed, the value of t* for 95% confidence level for 40 degrees of freedom is obtained from the table as: 2.021. t-statistic is calculated as:

t=b1SEb1=6.42.9=2.206

Conclusion: Since, the calculated value of t is more than the observed value of t, the null hypothesis gets rejected. Hence, the null hypothesis that the coefficient of variable x1 is zero is not true.

(c)

To determine

To test: The hypothesis that the coefficient of variable x1 is zero against the hypothesis that it is not zero.

(c)

Expert Solution
Check Mark

Answer to Problem 8E

Solution: The null hypothesis that the coefficient of variable x1 is zero is true.

Explanation of Solution

Given: The confidence level is provided as 95%. The following information has been provided in the problem:

y^=1.6+4.8x1+3.2x2+5.2x3n=25SEb1=2.7

Explanation:

Calculation: In a regression equation, the coefficient of x1 is denoted as β1. According to the provided problem, the null hypothesis will be formulated as follows:

H0:β1=0

And the alternative hypothesis is a two-sided alternative. Hence, it will be formulated as follows:

Ha:β10

The total number of observations has been provided in the problem as:

n=25

Since, there are 3 explanatory variables in the regression equation, the degrees of freedom for the provided problem can be calculated as follows:

np1=2531=21

The value of coefficient of x1 is b1=4.8. Since, the test is two-tailed, the value of t* for 95% confidence level for 21 degrees of freedom is obtained from the table as: 2.080. The t-statistic is calculated as:

t=b1SEb1=4.82.7=1.777

Conclusion: Since, the calculated value of t is less than the observed value of t, the null hypothesis does not get rejected. Hence, the null hypothesis that the coefficient of variable x1 is zero is true.

(d)

To determine

To test: The hypothesis that the coefficient of variable x1 is zero against the hypothesis that it is not zero.

(d)

Expert Solution
Check Mark

Answer to Problem 8E

Solution: The null hypothesis that the coefficient of variable x1 is zero is not true.

Explanation of Solution

Given: The confidence level is provided as 95%. The following information has been provided in the problem:

y^=1.6+4.8x1+3.2x2+5.2x3n=104SEb1=1.8

Explanation:

Calculation: In a regression equation, the coefficient of x1 is denoted as β1. According to the provided problem, the null hypothesis will be formulated as follows:

H0:β1=0

And the alternative hypothesis is a two-sided alternative. Hence, it will be formulated as follows:

Ha:β10

The total number of observations has been provided in the problem as:

n=104

Since, there are 3 explanatory variables in the regression equation, the degrees of freedom for the provided problem can be calculated as follows:

np1=10431=100

The value of coefficient of x1 is b1=4.8. Since, the test is two-tailed, the value of t* for 95% confidence level for 100 degrees of freedom is obtained from the table as: 1.984. The t-statistic is calculated as:

t=b1SEb1=4.81.8=2.67

Conclusion: Since, the calculated value of t is more than the observed value of t, the null hypothesis gets rejected. Hence, the null hypothesis that the coefficient of variable x1 is zero is not true.

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