10.3 One-Proportion z Interval
Assumptions:
- A simple random sample.
- Large sample size: both the number of successes [latex]x[/latex] and the number of failures [latex]n-x[/latex] are at least 5.
Note: Recall that one proportion inferences require [latex]np \geq 5[/latex] and [latex]n(1-p) \geq 5[/latex]. However, [latex]p[/latex] is generally unknown, and estimated with [latex]\hat{p} = \frac{x}{n}[/latex]. Thus, since [latex]n \hat{p} = n \frac{x}{n} = x[/latex] and [latex]n(1 - \hat{p}) = n \left( 1 - \frac{x}{n} \right) = n \left( \frac{n-x}{n} \right) = n-x[/latex], the sample is deemed sufficiently large if [latex]n \hat{p} = x \geq 5[/latex] and [latex]n(1 - \hat{p}) = n -x \geq 5[/latex]. We require at least 5 successes and at least 5 failures in the sample.
A point estimate for the population proportion [latex]p[/latex] is the sample proportion [latex]\hat{p} = \frac{x}{n}[/latex]. Therefore, a [latex](1 – \alpha) \times 100\%[/latex] confidence interval for the population proportion p is
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Two-Tailed
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Upper-Tailed
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Lower-Tailed
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| [latex]\left(\hat{p} - z_{\alpha / 2} \sqrt{ \frac{\hat{p} (1 - \hat{p})}{n}}, \hat{p} + z_{\alpha / 2} \sqrt{ \frac{\hat{p} (1 - \hat{p})}{n}} \right)[/latex] | [latex]\left(\hat{p} - z_{\alpha} \sqrt{ \frac{\hat{p} (1 - \hat{p})}{n}}, 1 \right)[/latex] | [latex]\left(0 , \hat{p} + z_{\alpha} \sqrt{ \frac{\hat{p} (1 - \hat{p})}{n}} \right)[/latex] |
Note: Since the range of proportion is between 0 and 1, the right-end point of the upper-tailed interval is bounded by 1 and the left-end point of the lower-tailed interval is bounded by 0.
Example: One-Proportion Z Interval
A credit card company sent out [latex]n=400[/latex] advertisements, and [latex]x=30[/latex] customers responded. Obtain a 95% confidence interval for the proportion of respondents.
Check the assumptions:
- We have a simple random sample (SRS).
- Both the number of successes [latex]x=30[/latex] and number of failures [latex]n-x = 400 - 30 = 370[/latex] are greater than 5.
The sample proportion is
[latex]\hat{p} = \frac{x}{n} = \frac{30}{400} = 0.075.[/latex]
[latex]1 - \alpha = 0.95 \Longrightarrow \alpha = 0.05 \Longrightarrow z_{\alpha / 2} = z_{0.025} = 1.96.[/latex]
A 95% confidence interval for the proportion of respondents is
[latex]\hat{p} \pm z_{\alpha / 2} \sqrt{\frac{\hat{p} (1 - \hat{p})}{n}} = 0.075 \pm 1.96 \times \sqrt{\frac{0.075 (1 - 0.075)}{400}} = (0.049, 0.101).[/latex]
Interpretation: We are 95% confident that the proportion of respondents is somewhere between 0.049 and 0.101, i.e., we are 95% confident that the percentage of respondents is somewhere between 4.9% and 10.1%.
