n order to estimate the difference between the average Miles per Gallon of two different models
of
cars
, samples are taken and the following information is collected.
Model
A
Model
B
Sample Size
13
14
Sample Mean
35
32
Sample Variance
13
7
a)
Find a
95% confidence
interval
for the difference between the average
Miles per Gallon for the two models.
b)
From your answer in Part a)
, i
s there conclusive evidence that one model
gets a
significantly
higher
average
MPG than the other? Why or why not? Explain
a) To find a 95% confidence interval for the difference between the average Miles per Gallon (MPG) for the two models, we can use the formula:
Confidence interval = (sample mean of Model A - sample mean of Model B) ± (critical value * standard error)
1. Compute the difference between the sample means:
Difference in sample means = 35 - 32 = 3
2. Calculate the standard error of the difference:
Standard error of the difference = sqrt((sample variance of Model A / sample size of Model A) + (sample variance of Model B / sample size of Model B))
Standard error of the difference = sqrt((13 / 13) + (7 / 14)) ≈ 1.322
3. Determine the critical value for a 95% confidence interval. The critical value depends on the sample sizes and the desired confidence level. For this case, since the sample sizes are relatively small (n < 30), we can use the t-distribution instead of the standard normal distribution. With 13 + 14 - 2 = 25 degrees of freedom, the critical value for a 95% confidence interval is approximately 2.06 (look this value up in a t-distribution table or use a statistical calculator).
4. Calculate the confidence interval:
Confidence interval = 3 ± (2.06 * 1.322)
Confidence interval ≈ ( 0.16, 6.84)
Therefore, the 95% confidence interval for the difference between the average MPG for the two models is approximately (0.16, 6.84).
b) To determine if there is conclusive evidence that one model gets a significantly higher average MPG than the other, we need to check whether the confidence interval includes the value of 0 (no difference).
In this case, the confidence interval (0.16, 6.84) does not include 0, which means there is evidence that the average MPG of one model is significantly higher than the other. Specifically, we can conclude that Model A, on average, has a higher MPG than Model B.