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Hypothesis tests for a population proportion

The discussion above is an example of a hypothesis test for a population proportion. There are three basic sets of hypotheses that can be tested. In all cases we make use of the central limit theorem for proportions: if the sample size is large, then the sampling distribution of the sample proportion is approximately normal with mean $\pi$ and s.d. $\sqrt{\pi(1-\pi)/n}$ where $\pi$ is the population proportion and n is the sample size.


  1. $H_0:\ \pi \le \pi_0$
    $H_1:\ \pi > \pi_0$
    The burden of proof is to show that $\pi > \pi_0$, where $\pi_0$ is the reference value that is initially assumed to be true. The test statistic for this test is

    \begin{displaymath}
\hat{z} = \frac{\hat{p} - \pi_0}{\sqrt{\pi_0(1-\pi_0)/n}}.
\end{displaymath}

    The p-value is $P(Z > \hat{z})$, the area to the right of $\hat{z}$ under the standard normal density. Note that the s.d. used for the test statistic is not the same as what we used for confidence intervals.

  2. $H_0:\ \pi \ge \pi_0$
    $H_1:\ \pi < \pi_0$
    The burden of proof is to show that $\pi < \pi_0$, where $\pi_0$ is the reference value that is initially assumed to be true. The test statistic for this test is

    \begin{displaymath}
\hat{z} = \frac{\hat{p} - \pi_0}{\sqrt{\pi_0(1-\pi_0)/n}}.
\end{displaymath}

    The p-value is $P(Z < \hat{z})$, the area to the left of $\hat{z}$ under the standard normal density. Note that evidence for the alternative hypothesis would be a sample proportion that is less than $\pi_0$ in which case the test statistic would be negative.

  3. The previous two sets of hypotheses are examples of one-sided tests - we only are interested in detecting the possibility that the population proportion falls on one particular side of the reference value. In the first case, we only are interested in showing that $\pi > \pi_0$. The possibility that $\pi $might be less than $\pi_0$ instead of equal to $\pi_0$ is of no concern and so these two are lumped together into the null hypothesis that $\pi \le \pi_0$. However, there are situations in which three separate actions would be taken depending on whether $\pi < \pi_0$, $\pi > \pi_0$, or $\pi = \pi_0$. In this case we must proceed in two steps. The first is to test two-sided hypotheses,
    $H_0:\ \pi = \pi_0$
    $H_1:\ \pi \ne \pi_0$
    If the null hypothesis is not rejected, then we take the action associated with the hypothesis $\pi = \pi_0$. If the null hypothesis is rejected, then we take the action associated with the hypothesis $\pi < \pi_0$ if $\hat{p} < \pi_0$ and we take the action associated with the hypothesis $\pi > \pi_0$ if $\hat{p} > \pi_0$. The test statistic is $\vert\hat{z}\vert$ and the p-value for this two-sided test includes both tail areas,

    \begin{displaymath}
P(Z > \vert\hat{z}) + P(Z < -\hat{z}).
\end{displaymath}

One important characteristic of a hypothesis test is its power function which represents the probability that the test will reject the null hypothesis expressed as a function of the actual population proportion. Suppose in the example above the level of significance was 5%. In this case the null hypothesis, $\pi > 0.25$, would be rejected if $\hat{z} \ge 1.645$ since $P(Z \ge 1.645) = 0.05$. This is equivalent to

\begin{displaymath}
\hat{p} \ge 0.25 + 1.645\sqrt{(.25)(.75)/750} = 0.276.
\end{displaymath}

If the actual population proportion is $\pi_1$, then the central limit theorem tells us that the samping distribution is approximately normal with mean $\pi_1$ and s.d. $\pi_1(1-\pi_1)/n$. Therefore

\begin{eqnarray*}
Power(\pi_1) &=& P(\hat{p} \ge 0.276) \\
&\approx& P(Z \ge (0.276 - \pi_1)/\sqrt{\pi_1(1-\pi_1)/n}.
\end{eqnarray*}

For example, if $\pi_1 = 0.30$, then the probability this test will reject the null hypothesis is

\begin{displaymath}
Power(0.30) = P(Z \ge -1.434) = 0.924.
\end{displaymath}

That is, if the population proportion is actually $\pi_1 = 0.30$, then there is a 92.4% chance this test will end up rejecting the null hypothesis. Note that the power function evaluated at the null hypothesis value (in this example, 0.25) is equal to the level of significance of the test, $\alpha$. Power functions for the other sets of hypotheses are obtained similarly.


next up previous
Next: Hypothesis tests for a Up: Statistical Decisions Previous: Statistical Decisions
Larry Ammann
2014-12-08