Mathematical Statistics with Applications
Mathematical Statistics with Applications
7th Edition
ISBN: 9780495110811
Author: Dennis Wackerly, William Mendenhall, Richard L. Scheaffer
Publisher: Cengage Learning
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Chapter 11.3, Problem 9E

a.

To determine

Draw the data points on a graph paper.

a.

Expert Solution
Check Mark

Answer to Problem 9E

The plot of data points on graph paper is represented as follows:

Mathematical Statistics with Applications, Chapter 11.3, Problem 9E , additional homework tip  1

Explanation of Solution

Step-by-step procedure to obtain the graph:

  • Mark the values of x along the horizontal axis from 1.5 to 2.5 at intervals of 0.25 units.
  • Mark the values of y along the vertical axis from 50 to 150 at intervals of 25 units.
  • Plot the point for each value of x with the corresponding value of y.

Thus, the graph is obtained.

b.

To determine

Obtain the least-square line for the data.

b.

Expert Solution
Check Mark

Answer to Problem 9E

The least-square line for the data is y^=15.452+65.172x.

Explanation of Solution

Calculation:

The formula for the mean of y is y¯=1ni=1nyi. The formula for the mean of x is x¯=1ni=1nxi.

The formula for Sxy is Sxy=i=1n(xix¯)(yiy¯). The formula for Sxx is Sxx=i=1n(xix¯)2. The formula for β^1 is β^1=SxySxx. The formula for β^0 is β^0=y¯β^1x¯.

 The calculation is shown below:

Horsepower (y)Cylinder Volume (x)
511.8
511.5
1152
1502.5
1261.8
1502.5
1181.6
1061.5
i=18yi=867i=18x=15.2

Thus, the mean of y and mean of x are obtained as follows:

y¯=1ni=1nyi=8678=108.375

x¯=1ni=1nxi=15.28=1.9

The value of y¯ is 108.375; the value of x¯ is 1.9.

The values of Sxy and Sxx are calculated below:

Sxy=i=1n(xix¯)(yiy¯)=[(1.81.9)(51108.375)+(1.51.9)(51108.375)+...+(1.51.9)(106108.375)]=75.6

Sxx=i=1n(xix¯)2=[(1.81.9)2+(1.51.9)2+..+(1.51.9)2]=1.16

The value of Sxy is 75.6; the value of Sxx is 1.16.

Thus, the values of β^1 and β^0 are obtained as follows:

β^1=SxySxx=75.61.16=65.172

β^0=y¯β^1x¯=108.375(65.172)1.9=108.375123.827=15.452

The value of β^0 is –15.452; the value of β^1 is 65.172.

The straight line is obtained as follows:

y^=β^0+β^1x=15.452+65.172x

Thus, the least-square line for the data is y^=15.452+65.172x.

c.

To determine

Draw the least-square line on a graph paper.

c.

Expert Solution
Check Mark

Answer to Problem 9E

The graph of a least-square line is represented as follows:

Mathematical Statistics with Applications, Chapter 11.3, Problem 9E , additional homework tip  2

Explanation of Solution

Step-by-step procedure to obtain the straight line:

  • Mark the values of x along the horizontal axis from 1.5 to 2.5 at intervals of 0.25 units.
  • Mark the values of y along the vertical axis from 50 to 150 at intervals of 25 units.
  • Plot the point for each value of x with the corresponding value of y.
  • Draw the line based on the estimated values of β0 and β1.

Thus, the graph is obtained.

d.

To determine

Obtain the estimated value of the mean horsepower rating for a fuel-efficient automobile with cylinder volume 1.9 L.

d.

Expert Solution
Check Mark

Answer to Problem 9E

The estimated value of the mean horsepower rating for a fuel-efficient automobile with cylinder volume 1.9 L is 108.375 Horsepower.

Explanation of Solution

Calculation:

From the part (b), the straight line is y^=15.452+65.172x.

The estimated value of the mean horsepower rating for a fuel-efficient automobile with cylinder volume 1.9 L is calculated as follows:

y^=15.452+65.172x=15.452+65.172(1.9)=15.452+123.827=108.375

Therefore, the estimated value of the mean horsepower rating for a fuel-efficient automobile with cylinder volume 1.9 L is 108.375 Horsepower.

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Chapter 11 Solutions

Mathematical Statistics with Applications

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