The accompanying data table shows the engine size, x, in cubic liters and rated
ID: 3313557 • Letter: T
Question
The accompanying data table shows the engine size, x, in cubic liters and rated miles per gallon (MPG), y, for a selection of sedans. a. Using alpha equals=0.05, test for the significance of the regression slope. b. Construct a 95% confidence interval for the population slope. c. What conclusions about engine size and MPG can be drawn from these results?
The accompanying data table shows the engine size x in cutie ste s and eated miles per gato a, using -o 05 test for the signficance of the regression slope b. Construct a 96% confdence intervai for the population siope G. What conclusions about engine size and MPG can be drawn from thes esults? G. y for a selection of sedans Th e ogressor.ne t the data y . 30 5587-4 8395x and SSE 284 199 Cick the icon to wew the data table O Table of Engine Sue and MPO a. Determine the null and atemative hypotheses Choose the correct answer below H 0 Engine Size MPG Car Model A Model 8 Model C Model D Model E Model F Madel G Model H Model I 26 01 24 Calculate the test statistic for the hypothesis test (sDiRound to two decenal places as needed) Find the pvalue p-value = (Roundto three decanal places as needed) is the regression slope signficant 2.7 2.1 2.5 3.4 35 2.1 2.6 2.2 3.7 24 29 23 28 20 Print Done A. No, because the pvalue a less than the level of significance B. Yes because the pvalue 's less than the levul af significance O C. Yes, because the pvalue is greater than or equal to the level of signacance O D. No, because the p-value is greater than or equal to the level of signficance b. Construct a 96% condence interval for the population slope UCl+ Round to two decimal places as needed) LCL LJ Round to two decimal places as needed) ngas mileage af between and mies per gal an Tus there esendence that the population regress on slope equal to zero There c. One can be 95% c mfident that increasing the sizg of an gngne by 1 cubic liter wis result in significant relationship between the size of an engine and its rated miles per galion Use ascending order Round to two decimal places as neededExplanation / Answer
Solution
Back-up Theory
Given X = engine size (in cubic litres) and Y = MPG, the linear regression of y on x is given by Y = + X + , where is the error term, which is assumed to be Normally distributed with mean 0 and variance 2.
Least square estimate of the regression slope, cap = Sxy/Sxx ………………. (1) where
Sxx = sum of (xi – Xbar)2 over i = 1, 2, …., n, ………………. (2)
Syy = sum of (yi – Ybar)2 over i = 1, 2, …., n ………………. (3) and
Sxy = sum of {(xi – Xbar)(yi – Ybar)} over i = 1, 2, …., n ………………. (4)
Estimate of 2 is given by s2 = (Syy – b2Sxx)/(n - 2). ………………. (5)
Standard Error of b is sb, where sb2 = s2/Sxx ………………. (6)
100(1 - )% Confidence Interval (CI) for = b ± SE(b)xtn – 2,/2 ………………. (7)
Preparatory Work
Summary of Excel Calculations
n
9
xbar
2.756
ybar
25.222
Sxx
3.1222
Syy
101.5556
Sxy
-15.1111
cap
-4.84
s^2
4.0600
sb^2
1.3004
s
2.01
sb
1.14
0
0
tb
-4.244
0.05
n-2
7
tn-2,/2
2.364
CIbLB
-7.536
CIbUB
-2.143
Part (a)
Null hypothesis: H0 = 0 Vs Alternative HA 0. Option (C) ANSWER 1
Test statistic: t = cap/SE(cap) = - 4.244 ANSWER 2
p-value = P(|t7| > 4.244) = 0.0038 ANSWER 3
since p-value < (given as 5%), the regression is significant. option (B) ANSWER 4
Part (b)
95% Confidence Interval for population slope = (- 7. 536, - 2.143) [vide (7) of Back-up Theory]
Lower Bound: - 7. 536 and Upper Bound: - 2.143 ANSWER
Part (c)
……… will result in a reduction in gas mileage between 2.143 and 7.536 miles per gallon …….. population slope is not equal to zero. There has to be a significant relationship ……. ANSWER
n
9
xbar
2.756
ybar
25.222
Sxx
3.1222
Syy
101.5556
Sxy
-15.1111
cap
-4.84
s^2
4.0600
sb^2
1.3004
s
2.01
sb
1.14
0
0
tb
-4.244
0.05
n-2
7
tn-2,/2
2.364
CIbLB
-7.536
CIbUB
-2.143
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