Answer to Question #215194 in Statistics and Probability for mel

Question

James kept careful records of the fuel efficiency of his car. After the first 100 times he filled up the tank, he found the mean was 23.4 miles per gallon (mpg) with a population standard deviation of 0.9mpg. Compute the 95 percent confidence interval for his mpg.

Answer 1

The critical value for $\alpha=0.05$ is $z_c=z_{1-\alpha/2}=1.96.$  

The corresponding confidence interval is computed as shown below:

$$CI=(\bar{x}-z_c\times\dfrac{\sigma}{\sqrt{n}}, \bar{x}+z_c\times\dfrac{\sigma}{\sqrt{n}})$$

$$=(23.4-1.96\times\dfrac{0.9}{\sqrt{100}}, 23.4+1.96\times\dfrac{0.9}{\sqrt{100}})$$

$$=(23.2236, 23.5764)$$

Therefore, based on the data provided, the 95% confidence interval for the population mean is $23.2236<\mu < 23.5764,$ which indicates that we are 95% confident that the true population mean $\mu$  is contained by the interval $(23.2236, 23.5764).$