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ISCAM: Investigating Statistical Concepts, Applications, and Methods

Beth L. Chance, Allan J. Rossman

Section 4.5 Summary of One Proportion \(z\)-Procedures for Proportion of Large Population

Let \(\pi\) be the (unknown) population proportion of successes and given that you have a representative sample from the population of interest.
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Test of \(H_0: \pi = \pi_0\).

Standardized statistic: \(z_0 = \frac{\hat{p} - \pi_0}{\sqrt{\frac{\pi_0(1-\pi_0)}{n}}}\)
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p-value:
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  • If \(H_a: \pi > \pi_0\text{,}\) the p-value is \(P(Z \geq z_0)\text{.}\)
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  • If \(H_a: \pi \lt \pi_0\text{,}\) the p-value is \(P(Z \leq z_0)\text{.}\)
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  • If \(H_a: \pi \neq \pi_0\text{,}\) the p-value is \(2P(Z \geq |z_0|)\)
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Sampling distribution showing test statistic and p-value
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C% Confidence Interval for \(\pi\).

Wald Interval: \(\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\)
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where \(z^*\) is the \(100 \times \frac{1-C}{2}\)th percentile of the standard normal distribution.
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Technical conditions: \(n\hat{p} > 10\) and \(n(1 - \hat{p}) > 10\) (which means there are at least 10 successes and 10 failures in the sample) and population size \(> 20n\text{.}\)
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Adjusted Wald Interval: \(\tilde{p} \pm z^* \sqrt{\frac{\tilde{p}(1-\tilde{p})}{\tilde{n}}}\)
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where \(\tilde{p} = \frac{X + 0.5z^{*2}}{n + z^{*2}}\) and \(\tilde{n} = n + z^{*2}\)
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(For 95% confidence, add two successes and two failures, aka Plus Four Method.)
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Technical conditions: population size \(> 20n\)
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Technology Instructions.

Hint 1. Theory-Based Inference applet: One Proportion instructions
  • Specify the sample size and either the sample count or the sample proportion
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    • Remember to adjust these values if performing the Adjusted Wald interval
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  • Check the Test of Significance box and/or the Confidence Interval box
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  • Specify the value of \(\pi_0\text{,}\) use the \(\lt\) button to change the direction of the alternative
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  • Specify the confidence level
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Hint 2. R instructions
R: iscamonepropztest(observed, n, hypothesized, alternative, conf.level)
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  • Specify the number of successes or \(\hat{p}\text{,}\) \(n\text{,}\) \(\pi_0\text{,}\) alternative, and confidence level (in that order)
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    • Remember to adjust inputs if performing the Adjusted Wald interval
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  • For the alternative, choose "two.sided" or "less" or "greater" (with the quotes)
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  • If no alternative is specified, be sure to label the confidence level (conf.level)
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Hint 3. JMP instructions
JMP: Analyze > Distribution
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  • Use a column of data or Tabled data (column of outcomes, column of counts)
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  • Use the hot spot to select Test Probabilities (specify the hypothesized probability for success) and select the alternative hypothesis (the "Pearson" two-sided p-value matches the normal approximation, the one-sided p-value is the exact binomial) and press Done.
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  • Use the hot spot to select Confidence Interval and select the confidence level for the Score confidence interval
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