Question

experts claim that 5 % of crimes are committed by women. is there enough evidence to reject the claim if in a sample of 50 crimes, 5 were committed by women

Answer 1

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

$$H_0: p=0.05$$

$$H_1: p\not=0.05$$

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

 (2) Rejection Region

If the significance level is not stated use $\alpha=0.05.$

The critical value for a two-tailed test is $z_c=1.96.$

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

(3) Test Statistics

$$\hat{p}={x \over n}={5 \over 50}=0.1$$

The z-statistic is computed as follows:

$$z={\hat{p}-p_0 \over \sqrt{p_0(1-p_0)/n}}={0.1-0.05 \over \sqrt{0.05(1-0.05)/50}}=1.622$$

(4) Decision about the null hypothesis

Since it is observed that $|z|=1.622<1.96=z_c,$ it is then concluded that the null hypothesis is not rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population proportion $p$ is different than $p_0,$ at the $\alpha=0.05$ significance level.

Confidence Interval

The 95% confidence interval for $p$ is: $0.017<p<0.183.$