Final answer:
The test statistic to compare the proportion of men who own cats to the proportion of women who own cats is calculated using the formula for two proportions. After computing the necessary values, the test statistic is found to be approximately z = 3.10, rounded to two decimal places.
Step-by-step explanation:
To test the claim that the proportion of men who own cats is larger than the proportion of women who own cats, you'd use the test statistic for comparing two proportions. In this scenario, the proportions are 0.5 for men (n1=140) and 0.3 for women (n2=100). The formula for the test statistic (z) is given by:
z = (p1 - p2) / sqrt(p*(1-p)*(1/n1 + 1/n2)),
where p1 is the sample proportion for men, p2 is the sample proportion for women, p is the pooled sample proportion calculated as (x1 + x2) / (n1 + n2) where x1 and x2 are the number of successes in each sample.
Plugging in the numbers:
- x1 = 0.5 * 140 = 70
- x2 = 0.3 * 100 = 30
- p = (70 + 30) / (140 + 100) = 100/240
- pooled proportion (p) = 100/240 = 0.4167
- z = (0.5 - 0.3) / sqrt(0.4167*(1-0.4167)*(1/140 + 1/100))
- z = 0.20 / sqrt(0.4167*0.5833*(0.007142857 + 0.01))
- z = 0.20 / sqrt(0.24307*0.017142857)
- z = 0.20 / sqrt(0.004166123)
- z = 0.20 / 0.06454
- z ≈ 3.10
Therefore, the test statistic is approximately z = 3.10, rounded to two decimal places.