GfK Custom Research North America conducted identical surveys five 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?

This is a test of 2 population proportions. Let 1 and 2 be the subscript for the women that responded positively at the earlier and later survey. The population proportion of women's positive responses at the earlier and later survey would be p1 and p2 respectively.

p1 - p2 = difference in the proportion of women's positive responses at the earlier and later survey.

The null hypothesis is

H0 : p1 = p2

p1 - p2 = 0

The alternative hypothesis is

Ha : p1 > p2

p1 - p2 > 0

it is a right tailed test

Sample proportion = x/n

Where

x represents number of success

n represents number of samples

For earlier survey,

x1 = 2010

n1 = 3000

p1 = 2010/3000 = 0.67

For later survey ,

x2 = 1530

n2 = 3000

p2 = 1530/3000 = 0.51

The pooled proportion, pc is

pc = (x1 + x2)/(n1 + n2)

pc = (2010 + 1530)/(3000 + 3000) = 0.59

1 - pc = 1 - 0.59 = 0.41

z = (p1 - p2)/√pc(1 - pc)(1/n1 + 1/n2)

z = (0.67 - 0.51)/√(0.59)(0.41)(1/3000 + 1/3000)

z = 12.6

From the normal distribution table, the probability value corresponding to the z score is less than 0.00001

Therefore,

p < 0.00001

By using the p value,

Since 0.05 > the p value, we would reject the null hypothesis. Therefore, at

the .05 level, can we can conclude that women think men are less kind, gentle, and thoughtful in the later survey compared with the earlier one