- Check the conditions for creating a confidence interval for population proportion.
- Describe the connection between the confidence level and the confidence interval.
- Calculate a confidence interval for a population proportion.
The [latex]z[/latex] critical value, [latex]z^{*}[/latex]
Let’s find [latex]z^{*}[/latex] value using the Normal Distribution Statistical Tool.
- Select the Find Percentile/Quantile tab.
- The tool defaults to a standard normal distribution with mean [latex]\mu = 0[/latex]and standard deviation [latex]\sigma = 1[/latex].
- Select Two-tailed and enter [latex]95[/latex] for Central Probability, since we want the middle [latex]95\%[/latex].
The [latex]z[/latex] critical value, [latex]z^{*}[/latex], for a [latex]95\%[/latex] confidence level is [latex]1.96[/latex].
Note the tool presents both the [latex]z[/latex] critical value and its negative counterpart, [latex]-1.96[/latex].
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Margin of error
Now that we have the standard error and [latex]z[/latex] critical value, we can calculate the margin of error that we can use to find the confidence interval for a population proportion.
margin of error, [latex]ME[/latex]
Margin of error is the width of the confidence interval.
[latex]ME = z^{*} \cdot (\text{standard error})[/latex]
where:
- standard error = [latex]\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}[/latex]
- [latex]z^{*}[/latex] is the point on the standard normal distribution such that the proportion of area under the curve between [latex]−z^{*}[/latex] and [latex]+z^{*}[/latex] is [latex]C[/latex], the confidence level.