Question

11.94 Drowning Deaths. In the article "Drowning Deaths of Zero to Five Year Old Children in Victorian Dams, 1989-2001" (Australian Journal of Rural Health, Vol. 13, Issue 5, pp. 300-308), L. Bugeja and R. Franklin examined drowning deaths of young children in Victorian dams to identify common contributing factors and develop strategies for future prevention. Of $11$young children who drowned in Victorian dams located on farms, $5$ were girls, At the $5\%$ significance level, do the data provide sufficient evidence to conclude that, of all young children drowning in Victorian dams located on farms, less than half are girls?

Short answer

At the $5\%$ level, the test results are not statistically significant. No, the evidence does not support the conclusion that less than half of all young children drowning in Victorian dams on farms are girls.

Step-by-step solution

Given information

To find that the data provide sufficient evidence to conclude that, of all young children drowning in Victorian dams located on farms, less than half are girls at the significance level of $5\%$.

Explanation

Let, the significance level is $5\%,n=11,p_0=0.5,$and $x=5$.

Determine the value of $n{p}_0$ as:
$n{p}_0=\left(11\right)(0.5)$

$=5.5$

Then, find the value of $n\left(1-p_0\right)$ as:
$n(1-p_0)=\left(11\right)(1-0.5)$

$=5.5$
The values of $n{p}_0$and $n(1-p_0)$are both larger than $5$.
As a result, only one proportion $z$ test should be used.
The null hypothesis: $H_0:p_0=0.5$

The alternate hypothesis: $H_{\mathrm{a}}:p_0<0.5$

Explanation

Determine the sample proportion as:
$\hat{p}=\frac{x}{n}$
$=\frac{5}{11}$

$=0.455$

Find the value for $z$ as:
$z=\frac{\hat{p}-p_0}{\sqrt{\frac{p_0\left(1-p_0\right)}{n}}}$
$=\frac{0.455-0.5}{\sqrt{{\displaystyle \frac{0.5(1-0.5}{11}}}}$
$=\frac{-0.045}{0.151}$
$=-0.302$
Then, $\alpha =0.05$
The test is left tailed.

Explanation

From the standard table, the critical value of $z$for $\alpha =0.05$, as:
$z_{0.05}=-1.645$
The test statistic is within acceptable bounds.
As a result, the hypothesis $H_0$ is not rejected.
At the $5\%$ level, the test results are not statistically significant.
As a result, the evidence does not support the conclusion that less than half of all young children drowning in Victorian dams on farms are girls.