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Section 5.5 Summary of One Proportion \(z\) -Procedures for Large Population
Let
\(\pi\) represent the (unknown) population proportion of successes and given that you have a representative sample from the population of interest.
Test of \(H_0\!: \pi = \pi_0\) .
Standardized statistic:
\(z_0 = \frac{\hat{p} - \pi_0}{\sqrt{\frac{\pi_0(1-\pi_0)}{n}}}\)
If
\(H_a\!: \pi > \pi_0\text{,}\) p-value is
\(P(Z \geq z_0)\text{.}\)
If
\(H_a\!: \pi \lt \pi_0\text{,}\) p-value is
\(P(Z \leq z_0)\text{.}\)
If
\(H_a\!: \pi \neq \pi_0\text{,}\) p-value is
\(2P(Z \geq |z_0|)\)
C% Confidence Interval for \(\pi\) .
Wald Interval: \(\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\)
where
\(-z^*\) is the
\(100 \times \frac{1-C}{2}\) th percentile of the standard normal distribution.
Adjusted Wald Interval: \(\tilde{p} \pm z^* \sqrt{\frac{\tilde{p}(1-\tilde{p})}{\tilde{n}}}\)
where
\(\tilde{p} = \frac{X + 0.5z^{*2}}{n + z^{*2}}\) and
\(\tilde{n} = n + z^{*2}\)
(For 95% confidence, add two successes and two failures, aka Plus Four Method.)
Technology Instructions. Hint 1 . Theory-Based Inference applet: One Proportion instructions
Specify the sample size and either the sample count or the sample proportion
Remember to adjust these values if performing the Adjusted Wald interval
Check the Test of Significance box and/or the Confidence Interval box
Specify the value of
\(\pi_0\text{,}\) use the
\(\lt\) button to change the direction of the alternative
Specify the confidence level
Hint 2 . R instructions
R:
iscamonepropztest(observed, n, hypothesized, alternative, conf.level)
Specify the number of successes or
\(\hat{p}\text{,}\) \(n\text{,}\) \(\pi_0\text{,}\) alternative, and confidence level (in that order)
Remember to adjust inputs if performing the Adjusted Wald interval
For the alternative, choose "two.sided" or "less" or "greater" (with the quotes)
If no alternative is specified, be sure to label the confidence level (conf.level)
Hint 3 . JMP instructions
JMP:
Analyze > Distribution
Use a column of data or Tabled data (column of outcomes, column of counts)
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.
Use the hot spot to select Confidence Interval and select the confidence level for the Score confidence interval
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