Estimation of a Population Mean when \(\sigma\) is known
In estimation, a sample is used to learn about an unknown population parameter. For a population mean, the parameter is
\(\mu\). When the population standard deviation \(\sigma\) is known, the sampling distribution of the sample mean
\(\bar{x}\) is (approximately) normal, and we use z-critical values to build a confidence interval.
Point estimate
The point estimate of the population mean \(\mu\) is the sample mean:
\[
\hat{\mu} = \bar{x}
\]
Standard error of \(\bar{x}\)
The standard deviation of the sampling distribution of \(\bar{x}\) is called the standard error:
\[
\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}
\]
where n is the sample size and \(\sigma\) is the known population standard deviation.
Confidence level and \(\alpha\)
A confidence interval is built for a chosen confidence level:
\[
(1-\alpha)\times 100\%
\]
The total probability in the two tails is \(\alpha\), so each tail has area \(\alpha/2\).
z critical value
The z critical value (often written z*) is defined by the standard normal distribution so that the
area to the left is \(1 - \alpha/2\):
\[
z^{*} = z_{1-\alpha/2}
\]
Common values include approximately:
\[
90\%: 1.645,\quad 95\%: 1.96,\quad 99\%: 2.576
\]
Margin of error
The quantity added to and subtracted from the point estimate is the margin of error:
\[
E = z^{*}\cdot \sigma_{\bar{x}} = z^{*}\cdot \frac{\sigma}{\sqrt{n}}
\]
Confidence interval for \(\mu\) (\(\sigma\) known)
The \((1 - \alpha)100\%\) confidence interval for \(\mu\) is:
\[
\bar{x} \pm z^{*}\cdot \frac{\sigma}{\sqrt{n}}
\]
or equivalently:
\[
\Big[\bar{x} - E,\ \bar{x} + E\Big]
\]
Interpreting a confidence interval
The interval is random (because it depends on the sample), while \(\mu\) is a fixed constant. The correct interpretation is:
if the same procedure were repeated many times, then about \((1 - \alpha)100\%\) of the intervals produced would contain \(\mu\).
How confidence level and sample size affect width
-
Increasing the confidence level increases z*, so the interval becomes wider.
-
Increasing the sample size n decreases \(\sigma/\sqrt{n}\), so the interval becomes narrower.
Planning the sample size (\(\sigma\) known)
If a maximum margin of error E is desired at a given confidence level, solve
\[
E = z^{*}\cdot \frac{\sigma}{\sqrt{n}}
\]
for n:
\[
n = \left(\frac{z^{*}\sigma}{E}\right)^2
\]
Always round up to the next integer to guarantee the margin of error is at most E.
Assumptions
- \(\sigma\) is known.
-
The sampling distribution of \(\bar{x}\) is normal or approximately normal:
typically reasonable when n >= 30, and for n < 30 it is most reliable when the population is
approximately normal (without strong skew or extreme outliers).
