7.4.2 - Confidence Intervals

Standard Normal Distribution Method Section

The normal distribution can also be used to construct confidence intervals. You used this method when you first learned to construct confidence intervals using the standard error method. Recall the formula you used:

95% Confidence Interval
\(sample\;statistic \pm 2 (standard\;error)\)

The 2 in this formula comes from the normal distribution. According to the 95% Rule, approximately 95% of a normal distribution falls within 2 standard deviations of the mean.

The normal curve showing the empirical rule.
µ−2 σ µ−1 σ µ+1 σ µ−3 σ µ+3 σ µ µ+2 σ 68% 95% 99.7%

Using the normal distribution, we can conduct a confidence interval for any level using the following general formula:

General Form of a Confidence Interval
sample statistic \(\pm\) \(z^*\) (standard error)
\(z^*\) is the multiplier

The \(z^*\) multiplier can be found by constructing a z distribution in Minitab Express.

 

z* Multiplier for a 90% Confidence Interval Section

What z* multiplier should be used to construct a 90% confidence interval?

For a 90% confidence interval, we would find the z scores that separate the middle 90% of the z distribution from the outer 10% of the z distribution:

Minitab Express output: Normal distribution showing the values that separate the outer 10% from the inner 90%
0.05 1.64485 -1.64485 0 0.05 0.0 0.1 0.2 0.3 0.4 Density X DistributionPlot Normal,Mean,StDev=1

For a 90% confidence interval, the \(z^*\) multiplier will be 1.64485.