Answer to Question #283677 in Statistics and Probability for willie

Question

You are working at a firm that conducts independent testing for heavy industry. Recently, an automobile manufacturer has been in the news for complaints about the highway gas mileage of their latest model minivan. You receive a contract from a consumer action group to test and write a report on the company’s claim that its minivans get 28 miles per gallon on the highway. The car company agrees to allow you to select randomly 35 low-mileage fleet minivans to test their highway mileage. Your test results gave you the following data:

29.7

24.5

27.1

29.8

29.2

27.0

27.8

24.1

29.3

25.9

26.2

24.5

32.8

26.8

27.8

24.0

23.6

29.2

26.5

27.7

27.1

23.7

24.1

27.2

25.9

26.7

27.8

27.3

27.6

22.8

25.3

26.6

26.4

27.1

26.1

 

Sample Mean, Standard deviation, confidence interval

 

Calculate a 95% confidence interval around your sample mean. (75 words, or 1 paragraph)

What is the null and alternative hypothesis?

Answer 1

To find the sample mean, we enter the following commands in R

> x=c(29.7,24.5,27.1,29.8,29.2,27.0,27.8,24.1,29.3,25.9,26.2,24.5,32.8,26.8,27.8,24.0,23.6,29.2,26.5,27.7,27.1,23.7,24.1,27.2,25.9,26.7,27.8,27.3,27.6,22.8,25.3,26.6,26.4,27.1,26.1)

> mean(x)

[1] 26.72

Therefore, the sample mean is 26.72.

To determine the standard deviation, we enter the following commands in R.

> x=c(29.7,24.5,27.1,29.8,29.2,27.0,27.8,24.1,29.3,25.9,26.2,24.5,32.8,26.8,27.8,24.0,23.6,29.2,26.5,27.7,27.1,23.7,24.1,27.2,25.9,26.7,27.8,27.3,27.6,22.8,25.3,26.6,26.4,27.1,26.1)

> sd(x)

[1] 2.100392

Therefore, the sample mean$(\bar{x})$ and the standard deviation$(s)$ are 26.72 and 2.100392 respectively.

The 95% confidence interval is given as,

$CI=\bar{x}\pm t_{{\alpha\over2},n-1}({s\over \sqrt{ n}})$ where $\alpha=0.05$ and $t_{{\alpha\over2},n-1}=t_{0.025,34}= 2.032245$

Therefore,

$CI=26.72\pm 2.032245({2.100392\over\sqrt{35}})=26.72\pm0.7215$

Thus, the 95% confidence is,

(26.00 , 27.44)

The hypotheses are,

$H_0:\mu=28\space vs\space H_1:\mu\not=28$

Since the claimed value of 28 miles per gallon does not lie in the confidence interval obtained above, we reject the null hypothesis and conclude that the there is no sufficient evidence to show that the minivans get 28 miles per gallon on the highway at 5% level of significance.