In 2007, the CDC reported that approximately 6.6 per 1000 (0.66%) children were affected with autism spectrum disorder. A sample of 900 children from Boston was tested and 7 were diagnosed with autism spectrum disorder. Is the proportion of children affected with autism spectrum disorder higher in Boston as compared to the national estimate? Run the appropriate test at a 5% level of significance.
Solution
Given that $n = 900$, $X= 7$.
The sample proportion is
$$\hat{p}=\frac{X}{n}=\frac{7}{900}=0.008$$.
Hypothesis Testing Problem
The hypothesis testing problem is
$H_0 : p = 0.0066$ against $H_1 : p > 0.0066$ ($\text{right-tailed}$)
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.
Significance Level
The significance level is $\alpha = 0.05$.
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 $\text{Z > 1.64}$.
Computation
The test statistic is
$$ \begin{aligned} Z & = \frac{\hat{p}-p}{\sqrt{\frac{p(1-p)}{n}}}\\ &= \frac{0.008-0.0066}{\sqrt{\frac{0.0066* (1-0.0066)}{900}}}\\ & =0.436 \end{aligned} $$
Decision
Traditional approach:
The test statistic is $Z =0.436$ which falls $outside$ the critical region, we $\text{fail to reject}$ the null hypothesis.
$p$-value approach:
This is a $\text{right-tailed}$ test, so the p-value is the
area to the left of the test statistic ($Z=0.436$). Thus the $p$-value = $P(Z < 0.436) =0.3313$.
The p-value is $0.3313$ which is $\text{greater than}$ the significance level of $\alpha = 0.05$, we $\text{fail to reject}$ the null hypothesis.
We conclude that at 0.05 level of significance the proportion of children affected with autism spectrum disorder is not higher in Boston as compared to the national estimate.
Further Reading
- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators
