Final answer:
To test the claim that the proportion of men who own cats is smaller than women, we use a left-tailed test. The test statistic is -0.3029 and the critical value is -1.96. Based on this, we reject the null hypothesis.
Step-by-step explanation:
The claim being tested is whether the proportion of men who own cats is smaller than the proportion of women who own cats. The null hypothesis states that the proportions are equal, while the alternative hypothesis states that the proportion of men owning cats is smaller. Since we are testing whether the proportion of men owning cats is smaller, it is a left-tailed test.
To test the claim, we need to calculate the test statistic and the critical value. The test statistic is found using the formula: z = (p1 - p2) / sqrt((p1(1-p1))/n1 + (p2(1-p2))/n2), where p1 is the proportion of men owning cats, p2 is the proportion of women owning cats, n1 is the sample size of men, and n2 is the sample size of women. The critical value is determined based on the significance level and the type of test (left-tailed, two-tailed, or right-tailed). In this case, since it is a left-tailed test at the 0.025 significance level, the critical value is -1.96.
Based on the given information, the test statistic is calculated to be: z = (0.25 - 0.30) / sqrt((0.25(1-0.25))/80 + (0.30(1-0.30))/40) = -0.3029. Since the test statistic falls within the rejection region (less than -1.96), we reject the null hypothesis. This means that there is sufficient evidence to support the claim that the proportion of men who own cats is smaller than the proportion of women who own cats.