Summary of One Proportion <em class="emphasis">z</em>-test

Subsection 3.2 Summary of One Proportion z-test

Summary of One Proportion z-test (Sampling from Binomial Process).

Let \(\pi\) represent the (constant) probability of success for the binomial random process assumed by the null hypothesis.

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To test \(H_0: \pi = \pi_0\)

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We can calculate a p-value based on the normal distribution with mean equal to \(\pi_0\) and standard deviation equal to \(\sqrt{\pi_0(1-\pi_0)/n}\)

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Standardized (Test) Statistic: \(z_0 = \frac{\hat{p} - \pi_0}{\sqrt{\pi_0(1-\pi_0)/n}}\) which follows a \(N(0,1)\) distribution

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p-value:

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if \(H_a: \pi > \pi_0\text{:}\) p-value = \(P(Z > z_0)\)

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if \(H_a: \pi < \pi_0\text{:}\) p-value = \(P(Z < z_0)\)

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if \(H_a: \pi \neq \pi_0\text{:}\) p-value = \(2P(Z > |z_0|)\)

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Diagram showing sample proportions distribution

Validity: This procedure is considered valid as long as the sample size is large relative to the hypothesized probability (\(n \times \pi_0 > 10\) and \(n \times (1 - \pi_0) > 10\)), and you have a representative sample from the process of interest.

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Technology Instructions.

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Hint 1. Theory-Based Inference applet

Theory-Based Inference applet: Specify the hypothesized value, use the button to specify the direction of the alternative (\(<\text{,}\) \(>\text{,}\) or \(\neq\) for not equal), enter the sample size and either the count or the proportion of successes. Press the Calculate button.

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Hint 2. R, ISCAM Workspace

iscamonepropztest(observed, n, hypothesized = π_0, alternative="greater","less", or "two.sided", conf.level). You can enter either the number of successes or the proportion of successes (\(\hat{p}\)) for the "observed" value.

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Hint 3. JMP

Using the ISCAM Journal file, select Hypothesis Test for One Proportion (with Normal Approximation radio button).

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