Question

1.     A particular brand of petrol was used in 80 randomly chosen cars of the same model and age. The petrol consumption, x miles per gallon, was obtained for each car. The results are summarized by

∑x=1896 and ∑x²=45959.

Calculate a 99% confidence interval for the mean petrol consumption of all cars of this model and age.

 

Answer 1

Since the population value of σ is unknown, we'll use the confidence interval for σ unknown.

Thus the 1 − α confidence interval is

$\bar{x}-t_{\alpha/2}\cfrac{s}{\sqrt{n}}<\mu<\bar{x}+t_{\alpha/2}\cfrac{s}{\sqrt{n}}$

where $t_{\alpha/2}$ is the value from the t-distribution table for

$P(T>t_{\alpha/2})=\alpha/2.$

For a 99% confidence interval we have α = 0.01 and α/2 = 0.005.

Using the table for df = n - 1 = 80 - 1 = 79 (degrees of freedom, n = 80 - the sample size):

$t_{0.005;79}=2.639.$

The sample mean:

$\bar{x}=\cfrac{\sum x}{n}=\cfrac{1896}{80}=23.7.$

The sample variance:

$s^2=\cfrac{\sum x^2-\cfrac{(\sum x)^2}{n}}{n-1}=\\ =\cfrac{45959-\cfrac{1896^2}{80}}{80-1}=12.96.$

The sample standard deviation:

$s=\sqrt{12.96}=3.60.$

Thus the 99% CI is

$23.7-2.639\cdot\cfrac{3.60}{\sqrt{80}}<\mu<23.7+2.639\cdot\cfrac{3.60}{\sqrt{80}}\\ 22.64<\mu<24.76.$