Question

5.32 Fuel efficiency of manual and automatic cars, Part I: Each year the US Environmental Protection Agency (EPA) releases fuel economy data on cars manufactured in that year. Below are summary statistics on fuel efficiency (in miles/gallon) from random samples of cars with manual and automatic transmissions manufactured in 2012. Do these data provide strong evidence of a difference between the average fuel efficiency of cars with manual and automatic transmissions in terms of their average city mileage? Assume that conditions for inference are satisfied.

	City MPG, Automatic	City MPG, Manual
Mean	16.12	19.85
SD	3.58	4.51
n	26	26

The hypotheses for this test are:

• Ho: μ_automatic = μ_manual
  Ha: μ_automatic < μ_manual
• Ho: μ_automatic = μ_manual
  Ha: μ_automatic > μ_manual
• Ho: μ_automatic = μ_manual
  Ha: μ_automatic ≠ μ_manual

The test statistic is:  (please round to two decimal places)
The p-value is:  (please round to four decimal places)
Interpret the result of the hypothesis test in the context of the problem:

• The data do not provide sufficient evidence that there is a difference between the average fuel efficiency of manual and automatic cars in terms of their average city mileage
• The data provide sufficient evidence that there is a difference between the average fuel efficiency of manual and automatic cars in terms of their average city mileage
• The data provide sufficient evidence that there is no difference between the average fuel efficiency of manual and automatic cars in terms of their average city mileage

Answer 1

n1 = 26 , x1 = 16.12 , s1 = 3.58

n2 = 26 , x2 = 19.85 , s2 = 4.51

.

H0: u1 = u2

H1: u1 is not equal to u2

.

pooled variance Sp^2 = = ((26-1)*(3.58^2) + (26-1)*(4.51^2)) / (26+26-2)

Sp^2 = 16.5783

Sp = 4.0716

.

standard error,SE = Sp * sqrt((1/n1)+(1/n2))

SE = 1.1293

.

test statistic, t = (x1-x2) / SE

t = -3.303

degree of freedom = n1+n2-2 = 50

Since it is a 2-tailed test so

P-value = 0.0018

since P-value < 0.05, hence reject H0

Yes, these data provide strong evidence of a difference between the means.