Confidence Intervals

A confidence interval is an interval of plausible values for a population characteristic. A confidence interval is constructed so that we have a chosen level of confidence that the actual value of the population characteristic will be between the lower endpoint and the upper endpoint of the interval. The confidence level is the success rate of the method used to construct a confidence interval.

The general formula for a confidence interval for a population proportion

$p$

when

$\hat{p}$
is the sample proportion from a random sample from the population of interest,
the sample size
$n$
is large
$(n\hat{p} \geq 10 \text{ and } n(1 - \hat{p}) \geq 10)$
, and
if the sample is selected without replacement, the sample size is small relative to the population size (
$n$
is at most 10% of the population size) is

$\hat{p} \pm (z\text{ critical value })\sqrt{\dfrac{\hat{p}(1 - \hat{p})}{n}}$
.

The desired confidence level determines which z critical value is used. The three most commonly used confidence levels, 90%, 95%, and 99%, use z critical values 1.645, 1.96, and 2.58, respectively. The width of the confidence interval is affected by each value in the formula.

Confidence Interval using SALT

In SALT, navigate to the Inferential Statistics tab and select the One Sample Proportion procedure. Enter the number of successes,

$x$

, as an integer. (If a percentage

$p$

is provided, use

$np = x$

to convert the percentage to an integer, where

$n$

is the number of trials.) Then enter the number of trials,

$n$

Select the Confidence Interval option and enter the desired confidence level. Click Generate Results to return a One Sample Proportion Confidence Interval Summary, which displays the lower and upper limits of the confidence interval for

$p$