Question

11.92 Delayed Perinatal Stroke. In the article "Prothrombotic Factors in Children With Stroke or Porencephaly" (Pediatrics Journal, Vol. 116, Issue $2$, pp. 447-453), J. Lynch et al. compared differences and similarities in children with arterial ischemic stroke and porencephaly. Three classification categories were used: perinatal stroke, delayed perinatal stroke, and childhood stroke. Of $59$children, $25$were diagnosed with delayed perinatal stroke. At the $5\%$significance level, do the data provide sufficient evidence to conclude that delayed perinatal stroke does not comprise one-third of the cases among the three categories?

Short answer

At the $5\%$ level, the test results are statistically significant. Yes, the data is sufficient to establish that delayed perinatal stroke does not affect one-third of the cases in any of the three groups.

Step-by-step solution

Given information

To find the data provide sufficient evidence to conclude that delayed perinatal stroke does not compromise one-third of the cases among the three categories at the significance level $5\%$.

Explanation

Since, the significance level is$5\%,n=59,p_0=0.3,$and $x=25$
Determine the value of $n{p}_0$ as:
$n{p}_0=\left(59\right)(0.3)$
$=17.7$
Then, find the value of $n\left(1-p_0\right)$ as:
$n(1-p_0)=\left(59\right)(1-0.3)$
Both $n{p}_0$and $n\left(1-p_0\right)$are greater than $5$. So one proportion $z$test is appropriate to use.

The null hypothesis: $H_0:p_0=0.3$
The alternate hypothesis: $H_{\mathrm{a}}:p_0\ne 0.3$

Explanation

Determine the sample proportion as:
$\hat{p}=\frac{x}{n}$

$=\frac{25}{29}$

$=0.465$
For $z$, state the expression as:
$z=\frac{\hat{p}-p_0}{\sqrt{\frac{P_0\left(1-p_0\right)}{n}}}$
$=\frac{0.424-0.3}{\sqrt{{\displaystyle \frac{0.3(1-0.3)}{59}}}}$
$=\frac{0.124}{0.06}$
$=2.074$
Also,
$\alpha =0.05$
The test is two tailed.

Explanation

The critical value of $z$from the standard table for $\alpha =0.025$ as:
$z0.05=\pm 1.96$
In the rejection region, the test statistic falls . Therefore, the hypothesis $H_0$ is rejected.
At the $5\%$level, the test results are statistically significant.
As a result, yes, the data is sufficient to establish that delayed perinatal stroke does not affect one-third of the cases in any of the three groups.