# 7.1 A single population mean using the normal distribution -- rrc  (Page 5/19)

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## Try it

Refer back to the pizza-delivery Try It exercise. The mean delivery time is 36 minutes and the population standard deviation is six minutes. Assume the sample size is changed to 50 restaurants with the same sample mean. Find a 90% confidence interval estimate for the population mean delivery time.

(34.6041, 37.3958)

## Working backwards to find the error bound or sample mean

When we calculate a confidence interval, we find the sample mean, calculate the error bound, and use them to calculate the confidence interval. However, sometimes when we read statistical studies, the study may state the confidence interval only. If we know the confidence interval, we can work backwards to find both the error bound and the sample mean.

## Finding the Error Bound

• From the upper value for the interval, subtract the sample mean,
• OR, from the upper value for the interval, subtract the lower value. Then divide the difference by two.

## Finding the Sample Mean

• Subtract the error bound from the upper value of the confidence interval,
• OR, average the upper and lower endpoints of the confidence interval.

Notice that there are two methods to perform each calculation. You can choose the method that is easier to use with the information you know.

Suppose we know that a confidence interval is (67.18, 68.82) and we want to find the error bound. We may know that the sample mean is 68, or perhaps our source only gave the confidence interval and did not tell us the value of the sample mean.

## Calculate the error bound:

• If we know that the sample mean is 68: EBM = 68.82 – 68 = 0.82.
• If we don't know the sample mean: EBM = $\frac{\left(68.82-67.18\right)}{2}$ = 0.82.

## Calculate the sample mean:

• If we know the error bound: $\overline{x}$ = 68.82 – 0.82 = 68
• If we don't know the error bound: $\overline{x}$ = $\frac{\left(67.18+68.82\right)}{2}$ = 68.

## Try it

Suppose we know that a confidence interval is (42.12, 47.88). Find the error bound and the sample mean.

Sample mean is 45, error bound is 2.88

## Calculating the sample size n

If researchers desire a specific margin of error, then they can use the error bound formula to calculate the required sample size.

The error bound formula for a population mean when the population standard deviation is known is
EBM = $\left({z}_{\frac{\alpha }{2}}\right)\left(\frac{\sigma }{\sqrt{n}}\right)$ .

The formula for sample size is n = $\frac{{z}^{2}{\sigma }^{2}}{EB{M}^{2}}$ , found by solving the error bound formula for n .

In this formula, z is ${z}_{\frac{\alpha }{2}}$ , corresponding to the desired confidence level. A researcher planning a study who wants a specified confidence level and error bound can use this formula to calculate the size of the sample needed for the study.

The population standard deviation for the age of Foothill College students is 15 years. If we want to be 95% confident that the sample mean age is within two years of the true population mean age of Foothill College students, how many randomly selected Foothill College students must be surveyed?

• From the problem, we know that σ = 15 and EBM = 2.
• z = z 0.025 = 1.96, because the confidence level is 95%.
• n = $\frac{{z}^{2}{\sigma }^{2}}{EB{M}^{2}}$ = $\frac{{\left(1.96\right)}^{2}{\left(15\right)}^{2}}{{2}^{2}}$ = 216.09 using the sample size equation.
• Use n = 217: Always round the answer UP to the next higher integer to ensure that the sample size is large enough.

Therefore, 217 Foothill College students should be surveyed in order to be 95% confident that we are within two years of the true population mean age of Foothill College students.