in the spring of 2001, a university had 40,000 registered students. to estimate the percentage of all students expecting to graduate in 4 years, a simple random sample of 225 students was drawn, 144 of the sample planned to graduate in 4 years. (a) were the draws made with or without replacement or do we need more information? (b) give your estimate of population percentage (c) briefly but completely describe your box model. what is marked on each ticket? how many tickets in the box? (d) attatch a SE estimate to (b) for the population. Explain your method. Is his SE exact or why not? (e) Find a 98% - CI for the % of all 40,000 students planning to graduate in 4 years
(a) To determine whether the draws were made with or without replacement, we need to know if the 225 students were put back into the population after being selected or not. If the selected students were put back into the population, then the draws were made with replacement. If the selected students were not put back into the population, then the draws were made without replacement.
(b) To estimate the population percentage of all students expecting to graduate in 4 years, we can use the proportion of the sample that planned to graduate in 4 years. In this case, the proportion is given as 144 out of 225 students.
Population Percentage Estimate = (Number of students in the sample planning to graduate in 4 years) / (Total number of students in the sample) = 144 / 225
(c) The box model represents the population of students. In this case, each ticket in the box represents a student, and the value on each ticket indicates whether the student plans to graduate in 4 years or not. There are 40,000 tickets in the box, representing the 40,000 registered students at the university.
(d) The standard error (SE) estimate can be calculated to determine the precision of our population percentage estimate. The formula for SE in this case is:
SE = sqrt((p_hat*(1-p_hat))/n)
Where p_hat represents the proportion in the sample planning to graduate in 4 years, and n is the sample size. In this case, p_hat is 144/225 and n is 225. Substitute these values into the formula to calculate the SE.
(e) To find a 98% Confidence Interval (CI) for the percentage of all 40,000 students planning to graduate in 4 years, we can use the sample proportion and standard error.
The formula for the Confidence Interval is:
CI = p_hat ± z*(SE)
Where p_hat is the sample proportion, z is the critical value for the desired confidence level (98% in this case), and SE is the standard error.
To find the critical value, we look up the z-score corresponding to the 98% confidence level in the z-table. Once we have the critical value, we can calculate the upper and lower bounds of the Confidence Interval by substituting the values into the formula.