Each year the US Environmental Protection Agency (EPA) releases fuel economy dat
ID: 3048532 • Letter: E
Question
Each year the US Environmental Protection Agency (EPA) releases fuel economy data on cars manufactured in that year. Summary statistics on fuel efficiency (in miles/gallon) from random samples of cars with manual and automatic transmissions manufactured in 2012 are shown below. We would like to investigate if 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.
(a) Use the appropriate parametric test to determine, at the 5% level, whether these data provide strong evidence of a difference between the average fuel efficiency of cars with manual and automatic transmissions. Make sure you check the conditions for inference before you proceed and show your manual calculations.
(b) Use MS Excel to perform the hypothesis testing in part (a) and provide the corresponding output.
(c) The table below provides summary statistics on highway fuel economy of cars manufactured in 2012. Use these statistics to manually calculate a 98% confidence interval for the difference between average highway mileage of manual and automatic cars, and interpret this interval in the context of the data. Make sure you check the required conditions first.
(d) Use MS Excel to verify your result from part (c).
Automatic Manual Mean 16.12 19.85 SD 3.58 4.51` n 26 26 35 25 15 automatic manual City MPGExplanation / Answer
Solution:
Part a
Here, we have to use two sample t test for difference in population mean assuming equal population variances. The null and alternative hypotheses are given as below:
Null hypothesis: H0: There is no any statistically significant difference in the average fuel efficiency of cars with manual and automatic transmissions in terms of their average city mileage.
Alternative hypothesis: Ha: There is a statistically significant difference in the average fuel efficiency of cars with manual and automatic transmissions in terms of their average city mileage.
We are given level of significance = = 0.05
Test statistic formula for pooled variance t test is given as below:
t = (X1bar – X2bar) / sqrt[Sp2*((1/n1)+(1/n2))]
Where Sp2 is pooled variance
Sp2 = [(n1 – 1)*S1^2 + (n2 – 1)*S2^2]/(n1 + n2 – 2)
From given data, we have
X1bar = 16.12
X2bar = 19.85
S1 = 3.58
S2 = 4.51
n1 = 26
n2 = 26
df = n1 + n2 – 2 = 50
Sp2 = [(n1 – 1)*S1^2 + (n2 – 1)*S2^2]/(n1 + n2 – 2)
Sp2 = [(26 – 1)*3.58^2 + (26 – 1)*4.51^2]/(26 + 26 – 2)
Sp2 = 16.5783
t = (X1bar – X2bar) / sqrt[Sp2*((1/n1)+(1/n2))]
t = (16.12 – 19.85) / sqrt[16.5783*((1/26)+(1/26))]
t = -3.3030
Critical values = -2.0086 and 2.0086
P-value = 0.0018
= 0.05
P-value < = 0.05
So, we reject the null hypothesis that there is no any statistically significant difference in the average fuel efficiency of cars with manual and automatic transmissions in terms of their average city mileage.
There is a sufficient evidence to conclude that there is a statistically significant difference in the average fuel efficiency of cars with manual and automatic transmissions in terms of their average city mileage.
Part b
Required MS Excel output for the above test is given as below:
Pooled-Variance t Test for the Difference Between Two Means
(assumes equal population variances)
Data
Hypothesized Difference
0
Level of Significance
0.05
Population 1 Sample
Sample Size
26
Sample Mean
16.12
Sample Standard Deviation
3.58
Population 2 Sample
Sample Size
26
Sample Mean
19.85
Sample Standard Deviation
4.51
Intermediate Calculations
Population 1 Sample Degrees of Freedom
25
Population 2 Sample Degrees of Freedom
25
Total Degrees of Freedom
50
Pooled Variance
16.5783
Standard Error
1.1293
Difference in Sample Means
-3.7300
t Test Statistic
-3.3030
Two-Tail Test
Lower Critical Value
-2.0086
Upper Critical Value
2.0086
p-Value
0.0018
Reject the null hypothesis
Part c
Here, we have to find out 98% confidence interval for difference between two population means.
Confidence interval = (X1bar – X2bar) -/+ t*sqrt[Sp2*((1/n1)+(1/n2))]
Where
t is critical value for given confidence level
Sp2 is pooled variance
Sp2 = [(n1 – 1)*S1^2 + (n2 – 1)*S2^2]/(n1 + n2 – 2)
We are given confidence level = 98%
Critical t value = 2.4033
We are given
X1bar = 22.92
X2bar = 27.88
S1 = 5.29
S2 = 5.01
n1 = 26
n2 = 26
df = n1 + n2 – 2 = 50
(X1bar – X2bar) = 22.92 - 27.88 = -4.9600
Sp2 = [(n1 – 1)*S1^2 + (n2 – 1)*S2^2]/(n1 + n2 – 2)
Sp2 = [(26 – 1)*5.29^2 + (26 – 1)*5.01^2]/(26 + 26 – 2)
Sp2 = 26.5421
sqrt[Sp2*((1/n1)+(1/n2))] = sqrt[26.5421*((1/26)+(1/26))]
sqrt[Sp2*((1/n1)+(1/n2))] = 1.4289
Confidence interval = (X1bar – X2bar) -/+ t*sqrt[Sp2*((1/n1)+(1/n2))]
Confidence interval = -4.9600 -/+ 2.4033*1.4289
Confidence interval = -4.9600 -/+ 3.434075
Lower limit = -4.9600 - 3.434075 = -8.3940
Upper limit = -4.9600 + 3.434075 = -1.5260
Difference zero is not lies within above confidence interval. So, we reject the null hypothesis that that there is no any statistically significant difference in the average fuel efficiency of cars with manual and automatic transmissions in terms of their average city mileage.
There is a sufficient evidence to conclude that there is a statistically significant difference in the average fuel efficiency of cars with manual and automatic transmissions in terms of their average city mileage.
Part d
MS excel output for above confidence interval is given as below:
Confidence Interval Estimate
for the Difference Between Two Means
Data
Confidence Level
98%
Intermediate Calculations
Degrees of Freedom
50
t Value
2.4033
Interval Half Width
3.4340
Confidence Interval
Interval Lower Limit
-8.3940
Interval Upper Limit
-1.5260
Pooled-Variance t Test for the Difference Between Two Means
(assumes equal population variances)
Data
Hypothesized Difference
0
Level of Significance
0.05
Population 1 Sample
Sample Size
26
Sample Mean
16.12
Sample Standard Deviation
3.58
Population 2 Sample
Sample Size
26
Sample Mean
19.85
Sample Standard Deviation
4.51
Intermediate Calculations
Population 1 Sample Degrees of Freedom
25
Population 2 Sample Degrees of Freedom
25
Total Degrees of Freedom
50
Pooled Variance
16.5783
Standard Error
1.1293
Difference in Sample Means
-3.7300
t Test Statistic
-3.3030
Two-Tail Test
Lower Critical Value
-2.0086
Upper Critical Value
2.0086
p-Value
0.0018
Reject the null hypothesis
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