A US consumer lobby wishes to develop a model to predict gasoline usage, as meas

Question

A US consumer lobby wishes to develop a model to predict gasoline usage, as measured by miles per gallon, based on the horsepower of the car's engine. The Excel data file AUTO.xls (contained in a folder under the CML Quizzes tab) contains data on this for fifty recent models. Use Excel Data Analysis to estimate a linear model for the relationship, a 92% confidence interval for the slope coefficient and the residual plot. State all numerical answers below correct to four decimal places using the Excel output results.

1. the intercept ,

2. the slope coefficient , and

3. the standard error of the estimate,

4. Using this model, predict the gasoline mileage for the car with 120 horsepower. Use all decimal places in your calculation by selecting and using the values of b0 and b1 in the output generated by Excel.

5. Does the prediction involve extrapolating the relationship? Type yes or no. no State the 6. lower bound

7. upper bound for the 92% confidence interval for the slope coefficient.

MPG Horsepower Weight 43.1 48 1985 19.9 110 3365 19.2 105 3535 17.7 165 3445 18.1 139 3205 20.3 103 2830 21.5 115 3245 16.9 155 4360 15.5 142 4054 18.5 150 3940 27.2 71 3190 41.5 76 2144 46.6 65 2110 23.7 100 2420 27.2 84 2490 39.1 58 1755 28.0 88 2605 24.0 92 2865 20.2 139 3570 20.5 95 3155 28.0 90 2678 34.7 63 2215 36.1 66 1800 35.7 80 1915 20.2 85 2965 23.9 90 3420 29.9 65 2380 30.4 67 3250 36.0 74 1980 22.6 110 2800 36.4 67 2950 27.5 95 2560 33.7 75 2210 44.6 67 1850 32.9 100 2615 38.0 67 1965 24.2 120 2930 38.1 60 1968 39.4 70 2070 25.4 116 2900 31.3 75 2542 34.1 68 1985 34.0 88 2395 31.0 82 2720 27.4 80 2670 22.3 88 2890 28.0 79 2625 17.6 85 3465 34.4 65 3465 20.6 105 3380

Explanation / Answer

Solution

NOTE: Answers to the point are given below. For better understanding,

Back-up Theory and Details of Excel Calculations are given at the end.

Part (1)

the intercept = 50.0052 ANSWER 1

Part (2)

the slope coefficient – 0.2363 ANSWER 2

Part (3)

the standard error of the estimate

For intercept: 2.5235 ANSWER 3

For slope coefficient: 0.0226 ANSWER 4

Part (4)

Predicted value of the gasoline mileage for the car with 120 horsepower:

21.6522 mpg ANSWER 5

Part (5)

Does the prediction involve extrapolating the relationship? No. ANSWER 6

[because given x-values range from 48 to 165 and 120 is well within that range]

Part (6)

92% confidence interval for the slope coefficient

lower bound: - 0.2839 ANSWER 7

upper bound .- 0.1886 ANSWER 8

Back-up Theory

Let

y represent the gasoline usage, as measured by miles per gallon (i.e., MPG), and

x represent the horsepower of the car's engine.

The linear regression model Y = ?0 + ?1X + ?, ………………………………………..(1)

where ? is the error term, which is assumed to be Normally distributed with mean 0 and variance ?2.

Estimated Regression of Y on X is given by: Y = ?0cap + ?1capX, ………………………….(2)

where

?1cap = Sxy/Sxx and ?0cap = Ybar – ?1cap.Xbar..……………………………………………..(3)

Mean X = Xbar = (1/n)sum of xi ………………………………………….……………….(4)

Mean Y = Ybar = (1/n)sum of yi ………………………………………….……………….(5)

Sxx = sum of (xi – Xbar)2 …………………………………………………..………………………………..(6)

Syy = sum of (yi – Ybar)2 …………………………………………………..………………………………..(7)

Sxy = sum of {(xi – Xbar)(yi – Ybar)} …………………………………………………………………….………(8)

All above sums are over i = 1, 2, …., n

n = sample size ………………………………………………………………………………(9)

Estimate of ?2 is given by s2 = (Syy – ?1cap2Sxx)/(n - 2)……………………………………..(10)

Standard Error of ?1cap is sb, where sb2 = s2/Sxx

Standard Error of ?0cap is sa, where sa2 = sb2{(sum of xi2 over i = 1, 2, …., n)/n}

Standard Error of yicap = s?[(1/n) + {(xi – Xbar)2/Sxx}]

Now to work out the solution,

Details Excel Calculations:

n

50

xbar

90.84

ybar

28.542

Sxx

36408.72

Syy

3271.8418

Sxy

-8602.464

Slope coeff

-0.2363

intercept

50.0052

s^2

25.818669

s

5.0812

SE(slope coeff)

0.0266

SE(intercpt)

2.5235

tn-2,?/2

1.7885

CIbLB

-0.2839

CIbUB

-0.1886

DONE

n

50

xbar

90.84

ybar

28.542

Sxx

36408.72

Syy

3271.8418

Sxy

-8602.464

Slope coeff

-0.2363

intercept

50.0052

s^2

25.818669

s

5.0812

SE(slope coeff)

0.0266

SE(intercpt)

2.5235

tn-2,?/2

1.7885

CIbLB

-0.2839

CIbUB

-0.1886

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