Question

Following complaints about the working conditions in some apparel factories both in the United States and abroad, a join government and industry commission recommended in 1998 that companies that monitor and enforce proper standards be allowed to display a “No Sweat” label on their products. Does the presence of these labels influence consumer behavior? A survey of U.S. residents aged 18 or older asked a series of questions about how likel they would be to purchase a gament under various conditions. For some conditions, it was stated that the garment had a “No Sweat” label; for others, there was no mention of such a label. On the basis of the responses, each person was classified as a “label use” or a “label nonuser” there were 296 women surveyed. Of these, 63 were label users. On the other hand, 27 of 251 men were classified as users.

(a) Give a 95% confidence interval for the difference in the proportions.

(b) You would like to compare the women with the men. Set up appropriate hypotheses, and find the test staitistic and the P-value. What do you conclude?

Answer 1

a) = 63/296 = 0.2128

   = 27/251 = 0.1076

The pooled sample proportion(P) = * n1 + * n2)/(n1 + n2)

                                                     = (0.2128 * 296 + 0.1076 * 251)/(296 + 251)

                                                     = 0.1645

SE = sqrt(P(1 - P)(1/n1 + 1/n2))

       = sqrt(0.1645 * (1 - 0.1645) * (1/296 + 1/251))

       = 0.0318

At 95% confidence interval the critical value is z^* = 1.96

The 95% confidence interval for difference in the population proportion is

() +/- z^* * SE

= (0.2128 - 0.1076) +/- 1.96 * 0.0318

= 0.1052 +/- 0.0623

= 0.0429, 0.1675

b) H₀: P1 = P2

    H₁: P1 P2

The test statistic z = ()/SE

                              = (0.2128 - 0.1076)/0.0318

                              = 3.31

P-value = 2 * P(Z > 3.31)

             = 2 * (1 - P(Z < 3.31))

             = 2 * (1 - 0.9995)

             = 0.001

Since the P-value is less than the significance level(0.001 < 0.05), so we should reject the null hypothesis.

At 5% significance level, we can conclude that there is a significant difference in the proportion of women and men label users.