# n 2007, the CDC reported that approximately 6.6 per 1000 (0.66%) children were affected with autism spectrum disorder

#### ByDr. Raju Chaudhari

Feb 24, 2021

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

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.