Question

In: Statistics and Probability

The accompanying table provides data for​ tar, nicotine, and carbon monoxide​ (CO) contents in a certain...

The accompanying table provides data for​ tar, nicotine, and carbon monoxide​ (CO) contents in a certain brand of cigarette. Find the best regression equation for predicting the amount of nicotine in a cigarette. Why is it​ best? Is the best regression equation a good regression equation for predicting the nicotine​ content? Why or why​ not?

Tar   Nicotine   CO
5   0.5   5
17   1.0   19
16   1.1   18
13   0.7   19
13   0.8   18
15   1.1   13
15   1.0   17
16   1.2   16
15   1.1   15
9   0.7   11
13   0.8   17
12   0.8   17
14   0.7   17
16   1.1   16
3   0.3   3
14   1.0   18
15   1.1   16
13   0.7   19
15   1.0   16
15   1.0   17
16   1.2   15
15   1.1   16
7   0.7   7
17   1.3   15
15   1.0   14

Find the best regression equation for predicting the amount of nicotine in a cigarette. Use predictor variables of tar and/or carbon monoxide​ (CO). Select the correct choice and fill in the answer boxes to complete your choice.

A. Nicotine = __ + (__) CO

B. Nicotine = __ + (__) Tar

C. Nicotine = __ + (__) Tar + (__) CO

Solutions

Expert Solution

A.

        X - Mx         Y - My        (X - Mx)2 (X - Mx)(Y - My)
-9.96 -0.42 99.2016 4.1832
4.04 0.08 16.3216 0.3232
3.04 0.18 9.2416 0.5472
4.04 -0.22 16.3216 -0.8888
3.04 -0.12 9.2416 -0.3648
-1.96 0.18 3.8416 -0.3528
2.04 0.08 4.1616 0.1632
1.04 0.28 1.0816 0.2912
0.04 0.18 0.0016 0.0072
-3.96 -0.22 15.6816 0.8712
2.04 -0.12 4.1616 -0.2448
2.04 -0.12 4.1616 -0.2448
2.04 -0.22 4.1616 -0.4488
1.04 0.18 1.0816 0.1872
-11.96 -0.62 143.0416 7.4152
3.04 0.08 9.2416 0.2432
1.04 0.18 1.0816 0.1872
4.04 -0.22 16.3216 -0.8888
1.04 0.08 1.0816 0.0832
2.04 0.08 4.1616 0.1632
0.04 0.28 0.0016 0.0112
1.04 0.18 1.0816 0.1872
-7.96 -0.22 63.3616 1.7512
0.04 0.38 0.0016 0.0152
-0.96 0.08 0.9216 -0.0768
SS: 428.96 SP: 13.12

Sum of X = 374
Sum of Y = 23
Mean X = 14.96
Mean Y = 0.92
Sum of squares (SSX) = 428.96
Sum of products (SP) = 13.12

Regression Equation = ŷ = bX + a

b = SP/SSX = 13.12/428.96 = 0.0306

a = MY - bMX = 0.92 - (0.03*14.96) = 0.4624

ŷ = 0.0306X + 0.4624

B.

        X - Mx         Y - My        (X - Mx)2 (X - Mx)(Y - My)
-8.36 -0.42 69.8896 3.5112
3.64 0.08 13.2496 0.2912
2.64 0.18 6.9696 0.4752
-0.36 -0.22 0.1296 0.0792
-0.36 -0.12 0.1296 0.0432
1.64 0.18 2.6896 0.2952
1.64 0.08 2.6896 0.1312
2.64 0.28 6.9696 0.7392
1.64 0.18 2.6896 0.2952
-4.36 -0.22 19.0096 0.9592
-0.36 -0.12 0.1296 0.0432
-1.36 -0.12 1.8496 0.1632
0.64 -0.22 0.4096 -0.1408
2.64 0.18 6.9696 0.4752
-10.36 -0.62 107.3296 6.4232
0.64 0.08 0.4096 0.0512
1.64 0.18 2.6896 0.2952
-0.36 -0.22 0.1296 0.0792
1.64 0.08 2.6896 0.1312
1.64 0.08 2.6896 0.1312
2.64 0.28 6.9696 0.7392
1.64 0.18 2.6896 0.2952
-6.36 -0.22 40.4496 1.3992
3.64 0.38 13.2496 1.3832
1.64 0.08 2.6896 0.1312
SS: 315.76 SP: 18.42

Sum of X = 334
Sum of Y = 23
Mean X = 13.36
Mean Y = 0.92
Sum of squares (SSX) = 315.76
Sum of products (SP) = 18.42

Regression Equation = ŷ = bX + a

b = SP/SSX = 18.42/315.76 = 0.0583

a = MY - bMX = 0.92 - (0.06*13.36) = 0.1406

ŷ = 0.0583X + 0.1406

C.

X1-Mx1 X2-Mx2 Y-My (X1-Mx1)2 (X2-Mx2)2 SPx1y SPx2y SPx1x2
-9.96 -8.36 -0.42 99.202 69.89 4.183 3.511 83.266
4.04 3.64 0.08 16.322 13.25 0.323 0.291 14.706
3.04 2.64 0.18 9.242 6.97 0.547 0.475 8.026
4.04 -0.36 -0.22 16.322 0.13 -0.889 0.079 -1.454
3.04 -0.36 -0.12 9.242 0.13 -0.365 0.043 -1.094
-1.96 1.64 0.18 3.842 2.69 -0.353 0.295 -3.214
2.04 1.64 0.08 4.162 2.69 0.163 0.131 3.346
1.04 2.64 0.28 1.082 6.97 0.291 0.739 2.746
0.04 1.64 0.18 0.002 2.69 0.007 0.295 0.066
-3.96 -4.36 -0.22 15.682 19.01 0.871 0.959 17.266
2.04 -0.36 -0.12 4.162 0.13 -0.245 0.043 -0.734
2.04 -1.36 -0.12 4.162 1.85 -0.245 0.163 -2.774
2.04 0.64 -0.22 4.162 0.41 -0.449 -0.141 1.306
1.04 2.64 0.18 1.082 6.97 0.187 0.475 2.746
-11.96 -10.36 -0.62 143.042 107.33 7.415 6.423 123.906
3.04 0.64 0.08 9.242 0.41 0.243 0.051 1.946
1.04 1.64 0.18 1.082 2.69 0.187 0.295 1.706
4.04 -0.36 -0.22 16.322 0.13 -0.889 0.079 -1.454
1.04 1.64 0.08 1.082 2.69 0.083 0.131 1.706
2.04 1.64 0.08 4.162 2.69 0.163 0.131 3.346
0.04 2.64 0.28 0.002 6.97 0.011 0.739 0.106
1.04 1.64 0.18 1.082 2.69 0.187 0.295 1.706
-7.96 -6.36 -0.22 63.362 40.45 1.751 1.399 50.626
0.04 3.64 0.38 0.002 13.25 0.015 1.383 0.146
-0.96 1.64 0.08 0.922 2.69 -0.077 0.131 -1.574
SSX1: 428.96 SSX2: 315.76 SPX1Y: 13.12 SPX2Y: 18.42 SPX1X2: 306.36

Sum of X1 = 374
Sum of X2 = 334
Sum of Y = 23
Mean X1 = 14.96
Mean X2 = 13.36
Mean Y = 0.92
Sum of squares (SSX1) = 428.96
Sum of squares (SSX2) = 315.76
Sum of products (SPX1Y) = 13.12
Sum of products (SPX2Y) = 18.42
Sum of products (SPX1X2) = 306.36

Regression Equation = ŷ = b1X1 + b2X2 + a

b1 = ((SPX1Y)*(SSX2)-(SPX1X2)*(SPX2Y)) / ((SSX1)*(SSX2)-(SPX1X2)*(SPX1X2)) = -1500.38/41591.96 = -0.0361

b2 = ((SPX2Y)*(SSX1)-(SPX1X2)*(SPX1Y)) / ((SSX1)*(SSX2)-(SPX1X2)*(SPX1X2)) = 3882/41591.96 = 0.0933

a = MY - b1MX1 - b2MX2 = 0.92 - (-0.04*14.96) - (0.09*13.36) = 0.2127

ŷ = -0.0361X1 + 0.0933X2 + 0.2127


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