Answer to Question #121483 in Statistics and Probability for Sebastian

Question

A car manufacturer claims that the average fuel consumption is 25 mpg (miles per gallon). To check the truth of this hypothesis, a study of 55 randomly selected cars. The results regarding the fuel consumption of the car are as follows:

30 27 22 25 24 25 24 15 35 35 33 52 49 10 27 18
20 23 24 25 30 24 24 24 18 20 25 27 24 32 29 27
24 27 26 25 24 28 33 30 13 13 21 28 37 35 32 33
29 31 28 28 25 29 31

(a). Estimate average fuel consumption for the entire population with a 95% confidence level.

(b). Conduct a hypothesis test to support or reject the claim of the producer, with a 95% confidence level.

Answer 1

a)95% $CI= \bar x ±z_\alpha \frac{s}{\sqrt n}$

$\bar x=\frac{\sum x}{n}$

=$\frac{30+27+...+31}{55}$

=26.8546

$s=\sqrt {\frac {(x-\bar x)^2}{n-1}}$

=$\sqrt{\frac{(30-26.8546)^2+(27-26.8546)^2+...+(31-26.8546)^2}{54}}$

=7.329

$CI=26.8546±1.96*\frac{7.329}{54}$

[24.9176, 28.7916]

b) we test the following null and alternative hypothesis

$H_0:\mu=25, \, H_1:\mu \ne 25$

The critical value is 1.96 from the z tables

The test statistic

$z=\frac{x-\mu}{\frac{s}{\sqrt n}}$

=$\frac{25-26.8546}{\frac{7.329}{\sqrt{55}}}$ =-1.8766

If the the absolute value of the test statistic is greater than the critical value, we reject the null hypothesis

Since the absolute value of the test statistic is less than the critical value, we fail to reject the null hypothesis.

The average fuel consumption is 25 mpg with a 95% confidence level.