The Roper Organization conducted identical surveys 5 years apart. One question asked of women was Are most men basically kind, gentle, and thoughtful? The earlier survey revealed that, of the 3,000 women surveyed, 2,010 said that they were. The later revealed 1,530 of the 3,000 women surveyed thought that men were kind, gentle, and thoughtful. At the .05 level, can we conclude that women think men are less kind, gentle, and thoughtful in the later survey compared with the earlier one? Hint: For the calculations, assume the latter survey as the first sample.
The decision rule is to reject Ho if z is ________? The test statistic is z = _______? Please show calculations.
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To determine if women's perception of men being kind, gentle, and thoughtful has changed over time, we can conduct a hypothesis test comparing the proportions of women who answered positively in the earlier and later surveys.
Let's set up the null and alternative hypotheses:
Null Hypothesis (Ho): There is no difference in the proportion of women who think men are kind, gentle, and thoughtful between the earlier and later surveys.
Alternative Hypothesis (Ha): The proportion of women who think men are kind, gentle, and thoughtful is lower in the later survey compared to the earlier one.
To test the hypotheses, we can use a two-proportion z-test. The formula for the test statistic is:
z = (p1 - p2) / √(p * (1 - p) * (1/n1 + 1/n2))
Where:
p1 = proportion of women who think men are kind, gentle, and thoughtful in the earlier survey
p2 = proportion of women who think men are kind, gentle, and thoughtful in the later survey
n1 = sample size of the earlier survey
n2 = sample size of the later survey
p = pooled proportion
First, let's calculate the necessary values:
p1 = 2,010 / 3,000 = 0.67 (proportion of women in the earlier survey)
p2 = 1,530 / 3,000 = 0.51 (proportion of women in the later survey)
n1 = n2 = 3,000 (sample size)
p = (p1 * n1 + p2 * n2) / (n1 + n2) = (0.67 * 3000 + 0.51 * 3000) / (3000 + 3000) = 0.59 (pooled proportion)
Next, let's calculate the test statistic z:
z = (p1 - p2) / √(p * (1 - p) * (1/n1 + 1/n2))
= (0.67 - 0.51) / √(0.59 * (1 - 0.59) * (1/3000 + 1/3000))
= 0.16 / √(0.59 * 0.41 * 0.00067)
≈ 0.16 / √(0.00989123)
≈ 0.16 / 0.099447
≈ 1.6091
Now, to determine the decision rule at the 0.05 level, we need to find the critical z-value from the standard normal distribution. A two-tailed z-test with a 0.05 significance level requires a critical z-value of ±1.96.
If the calculated z-value falls outside the range of ±1.96, we will reject the null hypothesis.
In this case, the calculated z-value of 1.6091 does not fall outside the range of ±1.96. Therefore, we fail to reject the null hypothesis.
In summary, at the 0.05 level of significance, we do not have sufficient evidence to conclude that women think men are less kind, gentle, and thoughtful in the later survey compared to the earlier one.