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Estimating the Population Mean Using Intervals

Problem revisited: Estimate the average GPA $\mu$  of the population of approximately 23000 WMU undergraduates.

Consider estimating the population average $\mu$  by the sample average of, say n=25 randomly selected students. Suppose this sample average, which we denote by $\overline{X}$, equals 3.05. Now chances are the true average $\mu$ is not equal to 3.05. For sure, the true WMU average GPA is between 1.00 and 4.00, and with high confidence between (2.50, 3.50); but what level of confidence do we have that it is between say, (2.75, 3.25) or (2.95, 3.15)? Even better, can we find an interval (a, b) which will contain $\mu$ with , say, 95% certainty?

Recall that if $\sigma$  is the standard deviation for individuals, then $\sigma /\sqrt {n}$ is the standard deviation for averages (also called the SE of the sample average). More precisely, n-member averages form an approximate normal histogram with mean $\mu$ and standard error $\sigma /\sqrt {n}$. This means that there is 68% likelihood that the sample average falls within $\sigma /\sqrt {n}$ of $\mu$, and 95% likelihood that the sample average falls within $2(\sigma/\sqrt{n})$ of $\mu$; see figure below.

Selected areas under histogram for $\overline{X}$
\epsfig{file=emp68.ps, height=3in, angle=-90} \epsfig{file=emp95.ps, height=3in, angle=-90}

In statistical shorthand, we write

\begin{displaymath}P( \overline{X}\;\; \mbox{is inside the interval }
\mu \pm 2(\sigma/\sqrt{n})) = .95
\end{displaymath}

where $P(\;\cdot\;)$ is read as ``the probability that $(\;\cdot\;)$".

Of course, the distance of $\overline{X}$ from $\mu$ is also the distance of $\mu$ from $\overline{X}$, so it is equally true that

 \begin{displaymath}
P( \mu \;\; \mbox{is inside the interval }
\overline{X} \pm 2(\sigma/\sqrt{n})) = .95.
\end{displaymath} (7.1)

Recall the question posed earlier: ``Can we find an interval which we know contains $\mu$  with 95% confidence?'' Answer: $\overline{X} \pm 2(\sigma/\sqrt{n})$.  

Example 1: Suppose that the SD of student GPA is .30. If a random sample of n=25 students has average GPA equal to 3.05, then we are 95% confident that $\mu$ is contained within $3.05 \pm 2(.30/\sqrt{25})$ or (2.93, 3.17).

Replacing $2(\sigma/\sqrt{n})$ by $1(\sigma/\sqrt{n})$ in Equation ( 7.1) results in a 68% confidence interval. In general, we can create an interval with any desired confidence level by replacing the multiplier with the appropriate standard normal percentile z (also called the z-critical value).   

 \begin{displaymath}
P( \mu \;\;\mbox{is inside the interval }
\overline{X} \pm z(\sigma/\sqrt{n})) = 1-\alpha
\end{displaymath} (7.2)

where z is determined from $1 - \alpha$ as follows:

\epsfig{file=zalpha2.ps, height=4in, width=3in, angle=-90}

The following table gives the appropriate critical value z for typical values of $1 - \alpha$.  


 \begin{displaymath}
\begin{array}{c\vert cccccc}
1-\alpha & .68 & .80 & .90 & ....
...\hline
z & 1.00 & 1.28 & 1.64 & 1.96 & 2.00 & 2.58
\end{array}\end{displaymath} (7.3)

  % latex2html id marker 4820
\fbox{ \parbox{5.5in}{
\vspace*{1ex}
{\bf $z$ -Confi...
... given by
\begin{equation}
\overline{X} \pm z(\sigma/\sqrt{n})
\end{equation}} }
   

Example 1 (con't.):
a. Construct a 99% confidence interval for WMU GPA.
b. Construct a 75% confidence interval for WMU GPA.
c. Construct a 95.4% confidence interval for WMU GPA.
d. Construct a 95.0% confidence interval for WMU GPA.

Theoretically speaking, $\overline{X} \pm 2(\sigma/\sqrt{n})$ has a confidence level of 95.4%. Applied statisticians usually just round off and say 95%. We can, of course, choose the more precise critical value of 1.96 to get an exact 95% confidence level. However, normality of $\overline{X}$ is at best approximate anyway, so the precision seems spurious. However, the distinction is important enough to merit a reminder.

% latex2html id marker 4830
\fbox{ \parbox{5.5in}{
\vspace*{1ex}
{\bf z=1.96 ver...
...l.\\
\hspace*{2em}Using $z=2.00$\space in gives 95.4\% confidence interval.
} }


next up previous contents index
Next: t-based Confidence Interval for Up: Confidence Intervals Previous: Confidence Intervals

2003-09-08