Question

The population standard deviation for the heights of dogs, in inches, in a city is 3.7 inches. If we want to be 95% confident that the sample mean is within 2 inches of the true population mean, what is the minimum sample size that can be taken?

Answer 1

Answer:

The minimum sample size that can be taken is of 14 dogs.

Step-by-step explanation:

The formula for calculating the minimum sample size to estimate a population mean is given by:

$n=\frac{z^{2}\sigma^{2}}{e^{2}}$

The first step is obtaining the values we're going to use to replace in the formula.

Since we want to be 95% confident, $1-\alpha=0.95 \Rightarrow \alpha=0.05$.

Therefore we look for the critical value $z_{\alpha/2}=1.96$.

Then we calculate the variance:

$\sigma = 3.7 \Rightarrow \sigma^{2}=13.69$

And we have that:

$e=2 \Rightarrow e^{2}=4$

Now we replace in the formula with the values we've just obtained:

$n=\frac{1.96^{2}*13.69}{4}=13.1479\approx 14$

Therefore the minimum sample size that can be taken to guarantee that the sample mean is within 2 inches of the population mean is of 14 dogs.