Question

According to a study conducted for Gateway Computers, 59% of men and 70% of women say that weight is an extremely/very important factor in purchasing a laptop computer. Suppose this survey was conducted using 374 men and 481 women. Do these data show enough evidence to declare that a significantly higher proportion of women than men believe that weight is an extremely/very important factor in purchasing a laptop computer? Use alpha= 0.05.

Write the correct R commands for solving this problem. What is the p value? What is the statistical decision?

Use R command to construct a 90% confidence interval to estimate the difference in proportion of women and men believe that weight is an extremely/very important factor in purchasing a laptop computer.

Answer 1

Solution:-

a)

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: P_Men> P_Women
Alternative hypothesis: P_Men < P_Women

Note that these hypotheses constitute a one-tailed test.

Formulate an analysis plan. For this analysis, the significance level is 0.05. The test method is a two-proportion z-test.

Analyze sample data. Using sample data, we calculate the pooled sample proportion (p) and the standard error (SE). Using those measures, we compute the z-score test statistic (z).

p = (p₁ * n₁ + p₂ * n₂) / (n₁ + n₂)

p = 0.65188

SE = sqrt{ p * ( 1 - p ) * [ (1/n₁) + (1/n₂) ] }
SE = 0.03284
z = (p₁ - p₂) / SE

z = - 3.35

where p₁ is the sample proportion in sample 1, where p₂ is the sample proportion in sample 2, n₁ is the size of sample 1, and n₂ is the size of sample 2.

Since we have a one-tailed test, the P-value is the probability that the z-score is less than -3.35.

Thus, the P-value = less than 0.0001.

Interpret results. Since the P-value (almost 0) is less than the significance level (0.05), we have to reject the null hypothesis.

From the above test we have sufficient evidence in the favor of the claim that that a significantly higher proportion of women than men believe that weight is an extremely/very important factor in purchasing a laptop computer.

b) 90% confidence interval to estimate the difference in proportion of women and men believe that weight is an extremely/very important factor in purchasing a laptop computer is C.I = (- 0.1640, - 0.05598).

C.I = (0.59 - 0.70) + 1.645 × 0.03284

C.I = - 0.11 + 0.05402

C.I = (- 0.1640, - 0.05598)