Question

In a survey of 500 infants chosen at random, it is found that 240 are girls. Are boy and girl births equally likely according to this survey (use α = 0.05)

Answer 1

$$\hat{p}=\dfrac{240}{500}=0.48$$

The following null and alternative hypotheses for the population proportion needs to be tested:

$H_0:p=0.5$

$H_1:p\not=0.5$

This corresponds to a two-tailed test, for which a z-test for one population proportion will be used.

Based on the information provided, the significance level is $\alpha=0.05,$ and the critical value for a two-tailed test is $z_c=1.96.$

The rejection region for this two-tailed test is $R=\{z:|z|>1.96\}.$

The z-statistic is computed as follows:

$$z=\dfrac{\hat{p}-p_0}{\sqrt{\dfrac{p_0(1-p_0)}{n}}}=\dfrac{0.48-0.5}{\sqrt{\dfrac{0.5(1-0.5)}{500}}}=-0.8944$$

Since it is observed that $|z|=0.8944<1.96=z_c,$ it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is $p=2P(Z<-0.8944)=0.3711,$ and since $p=0.3711>0.05=\alpha,$ it is concluded that the null hypothesis is not rejected.

Therefore, there is not enough evidence to claim that the population proportion $p$ is different than $0.5,$ at the $\alpha=0.05$ significance level.

Therefore, there is enough evidence to claim that boy and girl births are equally likely, at the $\alpha=0.05$ significance level.