Question

"A survey of randomly selected university students found that 94 of the 106 first-year students and 96 of the 141 second-year students surveyed had purchased used textbooks in the past year.

(a) Construct a 95% confidence interval for the difference in the proportions of first-year and second-year university students who purchased used textbooks. (Round your answers to 4 decimal places, if needed.)

(b) If a 98% confidence interval for the difference in proportion of first-year and second-year university students who purchased used textbooks is (0.0899, 0.322), give an interpretation of this confidence interval."

Answer 1

A 95% confidence interval for the difference in proportions is calculated by finding the sample proportions, computing the standard error, and applying the z-score for the confidence level. For a 98% interval, the interpretation is that we're 98% confident the true difference in proportions is within the given range.

Step-by-step explanation:

To answer your question regarding the construction of a 95% confidence interval for the difference in the proportions of first-year and second-year university students who purchased used textbooks, we would use the formula for the confidence interval of the difference between two proportions. Here, p1 represents the proportion of first-year students who purchased used textbooks, and p2 represents the proportion of second-year students who did the same.

Firstly, we calculate the respective sample proportions:

• p1 = 94/106
• p2 = 96/141

Next, we calculate the standard error (SE) of the difference in sample proportions:

SE = √[(p1(1-p1)/n1) + (p2(1-p2)/n2)]

Where n1 and n2 are the sample sizes of the first and second-year students, respectively.

Then we use the z-score for a 95% confidence interval, which is approximately 1.96, to calculate the interval:

Confidence interval = (p1 - p2) ± z * SE

Plugging in the respective values and rounding to four decimal places, you would get the 95% confidence interval for the difference in proportions.

Regarding part (b), the interpretation of the 98% confidence interval would be: We are 98% confident that the true difference in the proportion of first-year and second-year university students who purchased used textbooks lies between 0.0899 and 0.322. This means that if we were to take many samples and build a confidence interval in this way, 98% of those intervals would contain the true difference in proportions.