Question

"2. A NAPA Auto Parts supplier wants information about how long car owners plan to keep their cars. A simple random sample of 24 car owners had a mean of 7.01 years and a standard deviation of 3.74 years (based on data from a Roper poll). Assume that the sample is drawn from a normally distributed population. Find and interpret a 95% confidence interval estimate of the population mean."

Answer 1

Answer:

The 95% confidence interval estimate of the population mean is between 5.51 and 8.51 years. This means that we are 95% sure that the true mean time that car owners plan to keep their cars is between 5.51 and 8.51 years.

Step-by-step explanation:

We have that to find our $\alpha$ level, that is the subtraction of 1 by the confidence interval divided by 2. So:

$\alpha = \frac{1-0.95}{2} = 0.025$

Now, we have to find z in the Ztable as such z has a pvalue of $1-\alpha$.

So it is z with a pvalue of $1-0.025 = 0.975$, so $z = 1.96$

Now, find M as such

$M = z*\frac{\sigma}{\sqrt{n}}$

In which $\sigma$ is the standard deviation of the population and n is the size of the sample.

$M = 1.96*\frac{3.74}{\sqrt{24}} = 1.50$

The lower end of the interval is the sample mean subtracted by M. So it is 7.01 - 1.50 = 5.51 years.

The upper end of the interval is the sample mean added to M. So it is 7.01 + 1.50 = 8.51 years.

The 95% confidence interval estimate of the population mean is between 5.51 and 8.51 years. This means that we are 95% that the true mean time that car owners plan to keep their cars is between 5.51 and 8.51 years.