During 2006, 3.0% of all U.S. households were burglary victims. For a simple random sample of 300 households from a certain region, suppose that 18 households were victimized by burglary during that year. Apply an appropriate hypothesis test and the 0.05 level of significance in determining whether the region should be considered as having a burglary problem greater than that for the nation as a whole.
Solution
Given that $n = 300$, $X= 18$.
The sample proportion is
$$ \begin{aligned} \hat{p}&=\frac{X}{n}=\frac{18}{300}=0.06 \end{aligned} $$
Step 1 Hypothesis Testing Problem
The hypothesis testing problem is
$H_0 : p = 0.03$ against $H_1 : p > 0.03$ ($\text{right-tailed}$)
Step 2 Test Statistic
The test statistic for testing above hypothesis testing problem is
$$ \begin{aligned} Z & = \frac{\hat{p} - p}{\sqrt{\frac{p(1-p)}{n}}} \end{aligned} $$
which follows $N(0,1)$ distribution.
Step 3 Significance Level
The significance level is $\alpha = 0.05$.
Step 4 Critical values
As the alternative hypothesis is $\textit{right-tailed}$, the critical value of $Z$ $\text{ is }$ $\text{1.64}$.
The rejection region (i.e. critical region) for the hypothesis testing problem is $\text{Z > 1.64}$.
Step 5 Computation
The test statistic is
$$ \begin{aligned} Z & = \frac{\hat{p}-p}{\sqrt{\frac{p(1-p)}{n}}}\\ &= \frac{0.06-0.03}{\sqrt{\frac{0.03* (1-0.03)}{300}}}\\ & =3.046 \end{aligned} $$
Step 6 Decision (Traditional approach)
The test statistic is $Z =3.046$ which falls $inside$ the critical region, we $\text{reject}$ the null hypothesis.
OR
Step 6 Decision ($p$-value approach)
This is a $\text{right-tailed}$ test, so the p-value is the area to the left of the test statistic ($Z=3.046$). Thus the $p$-value = $P(Z<3.046)=0.0012$.
The p-value is $0.0012$ which is $\text{less than}$ the significance level of $\alpha = 0.05$, we $\text{reject}$ the null hypothesis.
We conclude that the region should be considered as having a burglary problem greater than that for the nation as a whole.
Further Reading
- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators
