7.4.2 - Confidence Intervals

7.4.2 - Confidence Intervals

Standard Normal Distribution Method

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

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.


7.4.2.1 - Video Example: 98% CI for Mean Atlanta Commute Time

7.4.2.1 - Video Example: 98% CI for Mean Atlanta Commute Time

Construct a 98% confidence interval to estimate the mean commute time in the population of all Atlanta residents.


This example uses a dataset is built in to StatKey: Confidence Interval for a Mean, Median, Std. The dataset is titled 'Atlanta Commute.'

Video Walkthrough


7.4.2.2 - Video Example: 90% CI for the Correlation between Height and Weight

7.4.2.2 - Video Example: 90% CI for the Correlation between Height and Weight

Construct a 90% confidence interval to estimate the correlation between height and weight in the population of all adult men.


Video Walkthrough


7.4.2.3 - Example: 99% CI for Proportion of Women Students

7.4.2.3 - Example: 99% CI for Proportion of Women Students

Scenario: Data were collected from a representative sample of 501 World Campus STAT 200 students. In that sample, 284 students were women and 217 were not women. Construct a 99% confidence interval to estimate the proportion of all World Campus students who are women. 


StatKey was used to construct a sampling distribution using bootstrapping methods:

StatKey Bootstrap Distribution Plot

Because this distribution is approximately normal, we can approximate the sampling distribution using the z distribution. We will use the standard error, 0.022, from this distribution.

The original sample statistic was \(\widehat p =\frac{284}{501}=0.567\). 

We can find the \(z^*\) multiplier by constructing a z distribution to find the values that separate the middle 99% from the outer 1%:

Minitab Express output: z distribution showing the middle 99% versus the outer 1%

The \(z^*\) multiplier is 2.57583

Recall the general form of a confidence interval: sample statistic \(\pm\) \(z^*\) (standard error) where \(z^*\) is the multiplier. So in this case we have...

\(0.567 \pm 2.57583 (0.022)\)

\(0.567 \pm 0.057\)

\([0.510, 0.624]\)

I am 99% confident that the proportion of all World Campus students who are women is between 0.510 and 0.624


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