# 45 Z-test of Sample Proportions

## 45.1 One-Sample Z-test

The $$z$$-test of proportions is one approach used to look for evidence that the proportion of a sample may differ from a hypothesized (or previously observed) value. It assumes a normal distribution approximation to a Binomial distribution.

### 45.1.1 Z-Statistic

The $$z$$-statistic is a standardized measure of the magnitude of difference between a sample’s proportion and some known, non-random constant.

### 45.1.2 Definitions and Terminology

Let $$p$$ be a sample proportion from a sample. Let $$\pi_0$$ be a constant. $$z$$ is defined: $z = \frac{p - \pi_0}{\sqrt{\frac{\pi_0 \cdot (1 - \pi_0)}{n}}}$

### 45.1.3 Hypotheses

The hypotheses for these test take the forms:

For a two-sided test: \begin{aligned} H_0: \pi &= \pi_0\\ H_a: \pi &\neq \pi_0 \end{aligned}

For a one-sided test: \begin{aligned} H_0: \pi &< \pi_0\\ H_a: \pi &\geq \pi_0 \end{aligned}

or \begin{aligned} H_0: \pi &> \pi_0\\ H_a: \pi &\leq \pi_0 \end{aligned}

To compare a sample $$(X_1, \ldots, X_n)$$ against the hypothesized value, a Z-statistic is calculated in the form:

$Z = \frac{p - \pi_0}{\sqrt{\frac{\pi_0 \cdot (1 - \pi_0)}{n}}}$

Where $$p$$ is the sample proportion.

### 45.1.4 Decision Rule

The decision to reject a null hypothesis is made when an observed Z-value lies in a critical region that suggests the probability of that observation is low. We define the critical region as the upper bound we are willing to accept for $$\alpha$$, the Type I Error.

In the two-sided test, $$\alpha$$ is shared equally in both tails. The rejection regions for the most common values of $$\alpha$$ are depicted in the figure below, with the sum of shaded areas on both sides equaling the corresponding $$\alpha$$. It follows, then, that the decision rule is:

Reject $$H_0$$ when $$Z \leq z_{\alpha/2}$$ or when $$Z \geq z_{1-\alpha/2}$$.

By taking advantage of the symmetry of the Z-distribution, we can simplify the decision rule to:

Reject $$H_0$$ when $$|Z| \geq z_{1-\alpha/2}$$

## Warning: Ignoring unknown aesthetics: ymax

## Warning: Ignoring unknown aesthetics: ymax Figure 45.1: Rejection regions for the Z-test of proportions

In the one-sided test, $$\alpha$$ is placed in only one tail. The rejection regions for the most common values of $$\alpha$$ are depicted in the figure below. In each case, $$\alpha$$ is the area in the tail of the figure. It follows, then, that the decision rule for a lower tailed test is:

Reject $$H_0$$ when $$Z \leq z_{\alpha, \nu}$$.

For an upper tailed test, the decision rule is:

Reject $$H_0$$ when $$Z \geq z_{1-\alpha, \nu}$$.

Using the symmetry of the Z-distribution, we can simplify the decision rule as:

Reject $$H_0$$ when $$|Z| \geq z_{1-\alpha}$$.

## Warning: Ignoring unknown aesthetics: ymax

## Warning: Ignoring unknown aesthetics: ymax Figure 45.2: Rejection regions for one-tailed Z-test

The decision rule can also be written in terms of $$p$$:

Reject $$H_0$$ when $$p \leq \pi_0 - z_\alpha \cdot \sqrt{\frac{\pi_0 \cdot (1 - \pi_0)}{n}}$$ or $$p \geq \pi_0 + z_\alpha \cdot \sqrt{\frac{\pi_0 \cdot (1 - \pi_0)}{n}}$$.

This change can be justified by:

\begin{aligned} |Z| &\geq z_{1-\alpha}\\ \Big|\frac{p - \pi_0}{\sqrt{\frac{\pi_0 \cdot (1 - \pi_0)}{n}}}\Big| &\geq z_{1-\alpha} \end{aligned}

\begin{aligned} -\Big(\frac{p - \pi_0}{\sqrt{\frac{\pi_0 \cdot (1 - \pi_0)}{n}}}\Big) &\geq z_{1-\alpha} & \frac{p - \pi_0}{\sqrt{\frac{\pi_0 \cdot (1 - \pi_0)}{n}}} &\geq z_{1-\alpha}\\ p - \pi_0 &\leq - z_{1-\alpha} \cdot \sqrt{\frac{\pi_0 \cdot (1 - \pi_0)}{n}} & p - \pi_0 &\geq z_{1-\alpha} \cdot \sqrt{\frac{\pi_0 \cdot (1 - \pi_0)}{n}}\\ p &\leq \pi_0 - z_{1-\alpha} \cdot \sqrt{\frac{\pi_0 \cdot (1 - \pi_0)}{n}} & p &\geq \pi_0 + z_{1-\alpha} \cdot \sqrt{\frac{\pi_0 \cdot (1 - \pi_0)}{n}} \end{aligned}

For a two-sided test, both the conditions apply. The left side condition is used for a left-tailed test, and the right side condition for a right-tailed test.

### 45.1.5 Power

The derivations below make use of the following symbols:

• $$p$$: The sample proportion
• $$n$$: The sample size
• $$\pi_0$$: The value of population mean under the null hypothesis
• $$\pi_a$$: The value of the population mean under the alternative hypothesis.
• $$\alpha$$: The significance level
• $$\gamma(\mu)$$: The power of the test for the parameter $$\mu$$.
• $$z_{\alpha}$$: A quantile of the Standard Normal distribution for a probability, $$\alpha$$.
• $$Z$$: A calculated value to be compared against a Standard Normal distribution.
• $$C$$: The critical region (rejection region) of the test.

Two-Sided Test

\begin{aligned} \gamma(\pi_a) &= P_{\pi_a}(p \in C)\\ &= P_\mu\Big(p \leq \pi_0 - z_{\alpha/2} \cdot \sqrt{\frac{\pi_0 \cdot (1 - \pi_0)}{n}}\Big) + P_{\pi_a}\Big(p \geq \pi_0 + z_{1-\alpha/2} \cdot \sqrt{\frac{\pi_0 \cdot (1 - \pi_0)}{n}}\Big)\\ &= P_{\pi_a}\Big(p - \pi_a \leq \pi_0 - \pi_a - z_{\alpha/2} \cdot \sqrt{\frac{\pi_0 \cdot (1 - \pi_0)}{n}}\Big) + \\ & \ \ \ \ \ P_{\pi_a}\Big(p - \pi_a \geq \pi_0 - \pi_a + z_{1-\alpha/2} \cdot \sqrt{\frac{\pi_0 \cdot (1 - \pi_0)}{n}}\Big)\\ &= P_{\pi_a}\Big(\frac{p - \pi_a}{\sqrt{\frac{\pi_a \cdot (1 - \pi_a)}{n}}} \leq \frac{\pi_0 - \pi_a - z_{\alpha/2} \cdot \sqrt{\frac{\pi_0 \cdot (1 - \pi_0)}{n}}}{\sqrt{\frac{\pi_a \cdot (1 - \pi_a)}{n}}}\Big) +\\ & \ \ \ \ \ P_{\pi_a}\Big(\frac{p - \mu}{\sqrt{\frac{\pi_a \cdot (1 - \pi_a)}{n}}} \geq \frac{\pi_0 - \pi_a + z_{1-\alpha/2} \cdot \sqrt{\frac{\pi_0 \cdot (1 - \pi_0)}{n}}}{\sqrt{\frac{\pi_a \cdot (1 - \pi_a)}{n}}}\Big)\\ &= P_{\pi_a}\Big(Z \leq \frac{\pi_0 - \pi_a}{\sqrt{\frac{\pi_a \cdot (1 - \pi_a)}{n}}} - z_{\alpha/2}\Big) + P_{\pi_a}\Big(Z \geq \frac{\pi_0 - \pi_a}{\sqrt{\frac{\pi_a \cdot (1 - \pi_a)}{n}}} + z_{1-\alpha/2}\Big)\\ &= P_{\pi_a}\Big(Z \leq -z_{\alpha/2} + \frac{\pi_0 - \pi_a}{\sqrt{\frac{\pi_a \cdot (1 - \pi_a)}{n}}}\Big) + P_{\pi_a}\Big(Z \geq z_{1-\alpha/2} + \frac{\pi_0 - \pi_a}{\sqrt{\frac{\pi_a \cdot (1 - \pi_a)}{n}}}\Big)\\ &= P_{\pi_a}\Big(Z \leq -z_{\alpha/2} + \frac{\sqrt{n} \cdot (\pi_0 - \pi_a)}{\sqrt{\pi_a \cdot (1 - \pi_a)}}\Big) + P_{\pi_a}\Big(Z \geq z_{1-\alpha/2} + \frac{\sqrt{n} \cdot (\pi_0 - \pi_a)}{\sqrt{\pi_a \cdot (1 - \pi_a)}}\Big) \end{aligned}

Both $$z_{\alpha/2}$$ and $$z_{1-\alpha/2}$$ have Standard Normal distributions.

One-Sided Test

For convenience, the power for only the upper tailed test is derived here.
Recall that the symmetry of the t-test allows us to use the decision rule: Reject $$H_0$$ when $$|Z| \geq z_{1-\alpha}$$. Thus, where $$Z$$ occurs in the derivation below, it may reasonably be replaced with $$|Z|$$.

\begin{aligned} \gamma(\pi_a) &= P_{\pi_a}(p \in C)\\ &= P_{\pi_a}\big(p \geq \pi_0 + z_{1-\alpha} \cdot \sqrt{\frac{\pi_0 \cdot (1 - \pi_0)}{n}}\big)\\ &= P_{\pi_a}\big(p - \pi_a \geq \pi_0 - \pi_a + z_{1-\alpha} \cdot \sqrt{\frac{\pi_0 \cdot (1 - \pi_0)}{n}}\big)\\ &= P_{\pi_a}\Big(\frac{p - \pi_a}{\sqrt{\frac{\pi_a \cdot (1 - \pi_a)}{n}}} \geq \frac{\pi_0 - \pi_a + z_{1-\alpha} \cdot \sqrt{\frac{\pi_0 \cdot (1 - \pi_0)}{n}}}{\sqrt{\frac{\pi_a \cdot (1 - \pi_a)}{n}}}\Big)\\ &= P_{\pi_a}\Big(Z \geq \frac{\pi_0 - \pi_a}{\sqrt{\frac{\pi_a \cdot (1 - \pi_a)}{n}}} + z_{1-\alpha} \Big)\\ &= P_{\pi_a}\Big(Z \geq z_{1-\alpha} + \frac{\pi_0 - \pi_a}{\sqrt{\frac{\pi_a \cdot (1 - \pi_a)}{n}}}\Big)\\ &= P_{\pi_a}\Big(Z \geq z_{1-\alpha} + \frac{\sqrt{n} \cdot (\pi_0 -\pi_a)}{\sqrt{\frac{\pi_a \cdot (1 - \pi_a)}{n}}}\Big) \end{aligned}

Where $$z_{1-\alpha}$$ has a Standard Normal distribution.

### 45.1.6 Confidence Interval

The confidence interval for $$\theta$$ is written: $p \pm z_{1-\alpha/2} \cdot \sqrt{\frac{\pi_0 \cdot (1 - \pi_0)}{n}}$

The value of the expression on the right is often referred to as the margin of error, and we will refer to this value as $E = z_{1-\alpha/2} \cdot \sqrt{\frac{\pi_0 \cdot (1 - \pi_0)}{n}}$

## 45.2 References

1. Wackerly, Mendenhall, Scheaffer, Mathematical Statistics with Applications, 6th ed., Duxbury, 2002, ISBN 0-534-37741-6.
2. Daniel, Biostatistics, 8th ed., John Wiley & Sons, Inc., 2005, ISBN: 0-471-45654-3.
3. Hogg, McKean, Craig, Introduction to Mathematical Statistics, 6th ed., Pearson, 2005, ISBN: 0-13-008507-3