Question

About 74% of 100 randomly sampled MCC students believe they can achieve the American dream and about 62% of 100 randomly sampled Ferris State Universtiy students believe they can achieve the American dream. Construct a 95% confidence interval for the difference in the proportions of Montcalm Community College students and Ferris State University students who believe they can achieve the American dream.

If the assumptions are met, we are 95% confident that the difference in the population proportions of MCC and FSU students who believe they can achieve the American dream is between ___ and _____ .
If many groups of 100 randomly selected MCC students and 100 randomly selected FSU students were surveyed, then a different confidence interval would be produced from each group. About ____% of these confidence intervals will contain the true population proportion of the difference in the proportions of MCC students and FSU students who believe they can achieve the American dream about _____% will not contain the true population difference in proportions.

Answer 1

To construct the 95% confidence interval for the difference in proportions of MCC and FSU students who believe in the American dream, calculate the standard error, multiply it by the z-score to get the margin of error, and then apply it to the difference in sample proportions.

Step-by-step explanation:

To construct a 95% confidence interval for the difference in the proportions of Montcalm Community College (MCC) and Ferris State University (FSU) students who believe they can achieve the American dream, we need to follow certain steps. Given that about 74% of MCC students and 62% of FSU students believe in achieving the American dream, the sample proportions (p1 and p2) are 0.74 and 0.62 respectively, both based on samples of 100 students.

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

SE = √[(p1*(1-p1)/n1) + (p2*(1-p2)/n2)]

SE = √[(0.74*(1-0.74)/100) + (0.62*(1-0.62)/100)]

Next, find the z-score for the 95% confidence level, which is typically 1.96 for a two-tailed test.

Then, calculate the margin of error (ME) by multiplying the z-score by the standard error:

ME = z * SE

Now, the confidence interval is found by subtracting and adding the margin of error from and to the difference in sample proportions:

Confidence Interval = (p1 - p2) ± ME

Putting the numbers in,

Confidence Interval = (0.74 - 0.62) ± (1.96 * SE)

After calculating the above, we will get the confidence interval range. If many groups of 100 randomly selected MCC and FSU students were surveyed, about 95% of these confidence intervals will contain the true population proportion of the difference in the proportions, and about 5% will not.