# Interval Estimate of Population Mean with Known Variance

After we found a point estimate of the population mean, we would need a way to quantify its accuracy. Here, we discuss the case where the population variance σ2 is assumed known.

Let us denote the 100(1 α∕2) percentile of the standard normal distribution as zα∕2. For random sample of sufficiently large size, the end points of the interval estimate at (1 α) confidence level is given as follows:

#### Problem

Assume the population standard deviation σ of the student height in survey is 9.48. Find the margin of error and interval estimate at 95% confidence level.

#### Solution

We first filter out missing values in survey\$Height with the na.omit function, and save it in height.response.

> library(MASS)                  # load the MASS package
> height.response = na.omit(survey\$Height)

Then we compute the standard error of the mean.

> n = length(height.response)
> sigma = 9.48                   # population standard deviation
> sem = sigma/sqrt(n); sem       # standard error of the mean
[1] 0.65575

Since there are two tails of the normal distribution, the 95% confidence level would imply the 97.5th percentile of the normal distribution at the upper tail. Therefore, zα∕2 is given by qnorm(.975). We multiply it with the standard error of the mean sem and get the margin of error.

> E = qnorm(.975)sem; E         # margin of error
[1] 1.2852

We then add it up with the sample mean, and find the confidence interval as told.

> xbar = mean(height.response)   # sample mean
> xbar + c(E, E)
[1] 171.10 173.67

#### Answer

Assuming the population standard deviation σ being 9.48, the margin of error for the student height survey at 95% confidence level is 1.2852 centimeters. The confidence interval is between 171.10 and 173.67 centimeters.

#### Alternative Solution

Instead of using the textbook formula, we can apply the z.test function in the TeachingDemos package. It is not a core R package, and must be installed and loaded into the workspace beforehand.

> library(TeachingDemos)         # load TeachingDemos package
> z.test(height.response, sd=sigma)

One Sample ztest

data:  height.response
z = 262.88, n = 209.000, Std. Dev. = 9.480,
Std. Dev. of the sample mean = 0.656, pvalue < 2.2e16
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
171.10 173.67
sample estimates:
mean of height.response
172.38