Statistics for Engineers and Scientists
Statistics for Engineers and Scientists
4th Edition
ISBN: 9780073401331
Author: William Navidi Prof.
Publisher: McGraw-Hill Education
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Textbook Question
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Chapter 7, Problem 20SE

Use Equation (7.34) (page 545) to show that μ β ^ 1 = β 1 .

Expert Solution & Answer
Check Mark
To determine

Show that μβ^1=β1 by using Equation 7.34.

Explanation of Solution

Calculation:

The equation 7.34 is,

β^1=i=1n[(xix¯)i=1n(xix¯)2]yi

The denominator of the equation is,

i=1n(xix¯)=i=1nxinx¯=nx¯nx¯=0

Since x¯ is constant,

i=1nx¯(xix¯)=x¯i=1n(xix¯)=x¯(0)=0

Express that i=1nxi(xix¯)=i=1n(xix¯)2.

i=1nxi(xix¯)=i=1nxi(xix¯)i=1nx¯(xix¯)=i=1n[xi(xix¯)x¯(xix¯)]=i=1n[xi2xix¯x¯xi+x¯2]=i=1n[xi22xix¯+x¯2]

=i=1n(xix¯)2

The mean of yi is,

μyi=β0+β1xi

Therefore,

μβ^1=i=1n(xix¯)i=1n(xix¯)2μyi=i=1n(xix¯)i=1n(xix¯)2(β0+β1xi)=1i=1n(xix¯)2i=1n(xix¯)(β0+β1xi)=1i=1n(xix¯)2[β0i=1n(xix¯)+β1i=1n(xi)(xix¯)]

On simplification,

μβ^1=1i=1n(xix¯)2[β0(0)+β1i=1n(xix¯)2][Since,i=1n(xix¯)=0andi=1nxi(xix¯)=i=1n(xix¯)2 ]=1i=1n(xix¯)2[β1i=1n(xix¯)2]=β1

Hence, μβ^1=β1.

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Chapter 7 Solutions

Statistics for Engineers and Scientists

Ch. 7.1 - Prob. 12ECh. 7.1 - Prob. 13ECh. 7.1 - A scatterplot contains four points: (2, 2), (1,...Ch. 7.2 - Each month for several months, the average...Ch. 7.2 - In a study of the relationship between the Brinell...Ch. 7.2 - A least-squares line is fit to a set of points. If...Ch. 7.2 - Prob. 4ECh. 7.2 - In Galtons height data (Figure 7.1, in Section...Ch. 7.2 - In a study relating the degree of warping, in mm....Ch. 7.2 - Moisture content in percent by volume (x) and...Ch. 7.2 - The following table presents shear strengths (in...Ch. 7.2 - Structural engineers use wireless sensor networks...Ch. 7.2 - The article Effect of Environmental Factors on...Ch. 7.2 - An agricultural scientist planted alfalfa on...Ch. 7.2 - Curing times in days (x) and compressive strengths...Ch. 7.2 - Prob. 13ECh. 7.2 - An engineer wants to predict the value for y when...Ch. 7.2 - A simple random sample of 100 men aged 2534...Ch. 7.2 - Prob. 16ECh. 7.3 - A chemical reaction is run 12 times, and the...Ch. 7.3 - Structural engineers use wireless sensor networks...Ch. 7.3 - Prob. 3ECh. 7.3 - Prob. 4ECh. 7.3 - Prob. 5ECh. 7.3 - Prob. 6ECh. 7.3 - The coefficient of absorption (COA) for a clay...Ch. 7.3 - Prob. 8ECh. 7.3 - Prob. 9ECh. 7.3 - Three engineers are independently estimating the...Ch. 7.3 - In the skin permeability example (Example 7.17)...Ch. 7.3 - Prob. 12ECh. 7.3 - In a study of copper bars, the relationship...Ch. 7.3 - Prob. 14ECh. 7.3 - In the following MINITAB output, some of the...Ch. 7.3 - Prob. 16ECh. 7.3 - In order to increase the production of gas wells,...Ch. 7.4 - The following output (from MINITAB) is for the...Ch. 7.4 - The processing of raw coal involves washing, in...Ch. 7.4 - To determine the effect of temperature on the...Ch. 7.4 - The depth of wetting of a soil is the depth to...Ch. 7.4 - Good forecasting and control of preconstruction...Ch. 7.4 - The article Drift in Posturography Systems...Ch. 7.4 - Prob. 7ECh. 7.4 - Prob. 8ECh. 7.4 - A windmill is used to generate direct current....Ch. 7.4 - Two radon detectors were placed in different...Ch. 7.4 - Prob. 11ECh. 7.4 - The article The Selection of Yeast Strains for the...Ch. 7.4 - Prob. 13ECh. 7.4 - The article Characteristics and Trends of River...Ch. 7.4 - Prob. 15ECh. 7.4 - The article Mechanistic-Empirical Design of...Ch. 7.4 - An engineer wants to determine the spring constant...Ch. 7 - The BeerLambert law relates the absorbance A of a...Ch. 7 - Prob. 2SECh. 7 - Prob. 3SECh. 7 - Refer to Exercise 3. a. Plot the residuals versus...Ch. 7 - Prob. 5SECh. 7 - The article Experimental Measurement of Radiative...Ch. 7 - Prob. 7SECh. 7 - Prob. 8SECh. 7 - Prob. 9SECh. 7 - Prob. 10SECh. 7 - The article Estimating Population Abundance in...Ch. 7 - A materials scientist is experimenting with a new...Ch. 7 - Monitoring the yield of a particular chemical...Ch. 7 - Prob. 14SECh. 7 - Refer to Exercise 14. Someone wants to compute a...Ch. 7 - Prob. 16SECh. 7 - Prob. 17SECh. 7 - Prob. 18SECh. 7 - Prob. 19SECh. 7 - Use Equation (7.34) (page 545) to show that 1=1.Ch. 7 - Use Equation (7.35) (page 545) to show that 0=0.Ch. 7 - Prob. 22SECh. 7 - Use Equation (7.35) (page 545) to derive the...

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