Question

According to the institute for college access and success, 69% of students who graduated from college in 2013 had student loan debt. you want to know if the proportion is the same in wisconsin as in the us as a whole, so you survey a random sample of 100 people who graduated from wisconsin colleges in 2013. a. assuming that the proportion of people with student loan debt is the same in wisconsin as in the whole us, what is the probability that between 65 and 75% of your sample has student loan debt? use a normal approximation for .

Answer 1

By using a normal approximation to the binomial distribution, we calculate z-scores for 65 and 75%, and then find the corresponding probabilities in the standard normal distribution table to determine the likelihood that the sample proportion falls within that range.

Step-by-step explanation:

To determine the probability that between 65 and 75% of a sample of 100 Wisconsin college graduates has student loan debt, given the national average is 69%, we employ the normal approximation to the binomial distribution.

First, we establish the binomial distribution parameters:

- Sample size (№) = 100

- Sample proportion (P) = 0.69

- Therefore, the mean (μ) for the sample is № * P = 100 * 0.69 = 69 students

- The standard deviation (σ) for the sample is (√(№ * P * (1 - P))) = √(100 * 0.69 * (1 - 0.69))

To use normal approximation:

- We need to find the z-scores for 65% (65 students) and 75% (75 students).

- Z = (x - μ) / σ

- For x = 65: Z1 = (65 - 69) / σ

- For x = 75: Z2 = (75 - 69) / σ

After finding the z-scores, we look up the corresponding probabilities in the standard normal distribution table. Finally, the probability that the sample proportion will be between 65% and 75% is the difference between the probabilities associated with Z2 and Z1.