Question

The heights of dogs, in inches, in a city are normally distributed with a population standard deviation of 7 inches and an unknown population mean. If a random sample of 20 dogs is taken and results in a sample mean of 21 inches, find a 95% confidence interval for the population mean.

Answer 1

Answer:

The 95% confidence interval for the population mean is between 17.93 inches and 24.07 inches.

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.

$M = 1.96*\frac{7}{\sqrt{20}} = 3.07$

The lower end of the interval is the sample mean subtracted by M. So it is 21 - 3.07 = 17.93 inches.

The upper end of the interval is the sample mean added to M. So it is 21 + 3.07 = 24.07 inches

The 95% confidence interval for the population mean is between 17.93 inches and 24.07 inches.