$Z$-Test for Proportion
Let $X$ be the observed number of individuals possessing certain attributes (say, number of successes) in a random sample of size $n$ from a large population, then $\hat{p}=\frac{X}{n}$ be the observed proportion of successes. Let $p$ be the population proportion of successes and $q = 1- p$ be the population proportion of failures.
Assumptions
Assumptions for testing a proportion are as follows:
a. The sample is a random sample.
b. The conditions for binomial experiments are satisfied.
c. $n$ is sufficiently large ($n>20$), $np\geq 5$ and $nq\geq 5$.
Procedure
We wish to test the null hypothesis $H_0 : p = p_0$, where $p_0$ is the specified value of the population proportion.
The standard error of $p$ is
$$ \begin{aligned} SE(\hat{p}) &= \sqrt{\frac{p(1-p)}{n}} \end{aligned} $$
The test statistic for testing $H_0$ is
$$ \begin{aligned} Z & = \frac{\hat{p}-p}{SE(\hat{p})} \end{aligned} $$
which follows standard normal distribution $N(0,1)$.
Step 1 State the hypothesis testing problem
The hypothesis testing problem can be structured in any one of the three situations as follows:
| Situation | Hypothesis Testing Problem |
|---|---|
| Situation A : | $H_0: p=p_0$ against $H_a : p < p_0$ (Left-tailed) |
| Situation B : | $H_0: p=p_0$ against $H_a : p > p_0$ (Right-tailed) |
| Situation C : | $H_0: p=p_0$ against $H_a : p \neq p_0$ (Two-tailed) |
Step 2 Define the test statistic
The test statistic for testing above hypothesis is
$$ \begin{eqnarray*} Z & = & \frac{\hat{p}-p}{SE(\hat{p})}\\ & = &\frac{\hat{p}-p}{\sqrt{\frac{p*(1-p)}{n}}} \end{eqnarray*} $$
The test statistic $Z$ follows standard normal distribution $N(0,1)$.
Step 3 Specify the level of significance $\alpha$
Step 4 Determine the critical values
For the specified value of $\alpha$ determine the critical region depending upon the alternative hypothesis.
- For left-tailed alternative hypothesis: Find the $Z$-critical value using
$$ \begin{aligned} P(Z<-Z_\alpha) &= \alpha. \end{aligned} $$
- For right-tailed alternative hypothesis: $Z_\alpha$.
$$ \begin{aligned} P(Z>Z_\alpha) &= \alpha. \end{aligned} $$
- For two-tailed alternative hypothesis: $Z_{\alpha/2}$.
$$ \begin{aligned} P(|Z|> Z_{\alpha/2}) &= \alpha. \end{aligned} $$
Step 5 Computation
Compute the test statistic under the null hypothesis $H_0$ using equation
$$ \begin{aligned} Z_{obs} &= \frac{\hat{p}-p_0}{\sqrt{\frac{p_0*(1-p_0)}{n}}} \end{aligned} $$
Step 6 Decision (Traditional Approach)
Traditional approach is based on the critical value(s).
- For left-tailed alternative hypothesis: Reject $H_0$ if
$Z_{obs}\leq -Z_\alpha$. - For right-tailed alternative hypothesis: Reject $H_0$ if
$Z_{obs}\geq Z_\alpha$. - For two-tailed alternative hypothesis: Reject $H_0$ if
$|Z_{obs}|\geq Z_{\alpha/2}$.
OR
Step 6 Decision ($p$-value Approach)
$p$-value approach is based on the $p$-value of the test.
| Alternative Hypothesis | Type of Hypothesis | $p$-value |
|---|---|---|
| $H_a: p < p_0$ | Left-tailed | $p$-value $= P(Z\leq Z_{obs})$ |
| $H_a: p > p_0$ | Right-tailed | $p$-value $= P(Z\geq Z_{obs})$ |
| $H_a: p \neq p_0$ | Two-tailed | $p$-value $= 2P(Z\geq |Z_{obs}|)$ |
If $p$-value is less than $\alpha$, then reject the null hypothesis $H_0$ at $\alpha$ level of significance, otherwise fail to reject $H_0$ at $\alpha$ level of significance.
