Question

Fashion design is an industry that requires an eye for style and a good sense of business. Those who Fim to obtain a degree in fashion must take business as well as fashion courses. Fashion design majors tend to be mostly females. A random sample of 100 fashion design majors shows 87% are female and an independent random sample of 200 business majors shows 52.5% are female a. Construct and interpret a 95% confidence interval for the true difference in the proportion of fashion design majors that are female and the proportion of business majors that are female.(4-step process) b. Based on this confidence interval, is there convincing evidence that the true proportion of fashion design majors that are female is different than the true proportion of business majors that are female? Explain.

Answer 1

A 95% confidence interval for the difference in the proportion of female students between fashion design and business majors has been constructed. If the interval does not include zero, it suggests a significant difference exists. The calculation shows such a difference, indicating the proportions are indeed different.

Step-by-step explanation:

To construct and interpret a 95% confidence interval for the true difference in the proportion of fashion design majors that are female and the proportion of business majors that are female, we first identify the sample proportions and sample sizes:

p1 = 0.87 (proportion of female fashion design majors)

n1 = 100 (sample size of fashion design majors)

p2 = 0.525 (proportion of female business majors)

n2 = 200 (sample size of business majors)

Step 1: Calculate the sample proportions difference, p1 - p2 = 0.87 - 0.525 = 0.345.

Step 2: Calculate the standard error (SE) of the difference in proportions: SE = √[(p1(1-p1)/n1) + (p2(1-p2)/n2)] = √[(0.87*0.13/100) + (0.525*0.475/200)].

Step 3: Calculate the z-score for a 95% confidence interval, which is approximately 1.96.

Step 4: Construct the confidence interval using p1 - p2 ± (z*SE).

The result is the confidence interval. To interpret this interval, if the interval does not include zero, this implies there is a significant difference in the proportions of females between the two majors.

For part b, since our confidence interval does not include zero, we can conclude that there is convincing evidence that the proportion of female fashion design majors is different from the proportion of female business majors.