## $Z$-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.