Question

A study at a college in the west coast reveals that, historically, 45% of their students are minority students. If random samples of size 75 are selected, 80% of the samples will have less than ______% of minority students

Answer 1

80% of samples will have less than 44.944% of minority students.

Step-by-step explanation:

To find the answer to this question, we can use the normal distribution and the standard deviation formula.

First, we need to find the standard deviation, which is the square root of the product of the probability of success and the probability of failure, divided by the sample size.

In this case, the probability of success is 45% and the probability of failure is 55%. So, the standard deviation is sqrt((0.45 * 0.55) / 75) = 0.069.

Then, we can use a Z-table to find the cumulative probability up to a given Z-score.

In this case, we want to find the Z-score that corresponds to 80% of samples having less than a certain percentage of minority students.

We can use the formula Z = (x - mean) / standard deviation, where x is the desired percentage and mean is the average percentage of minority students.

The mean is 45% and the standard deviation is 0.069. So, we can plug these values in and solve for x: 0.8 = (x - 45) / 0.069.

Solving for x, we find x = 45 - (0.069 * 0.8) = 44.944.

So, 80% of samples will have less than 44.944% of minority students.

Answer 2

Approximately 1.12% of random samples of size 75 will have less than 80% minority students.

To find the percentage of random samples of size 75 that will have less than 80% minority students, we can use the binomial probability formula. In this case, we want to find the probability of having fewer than 60 minority students (80% of 75) in a sample.

Using the binomial probability formula:

P(X < 60) = Σ [C(n, x) * p^x * q^(n-x)]

Where:

- n is the sample size (75).

- x is the number of minority students (from 0 to 59).

- p is the probability of success (probability of being a minority student, which is 45% or 0.45).

- q is the probability of failure (1 - p, which is 1 - 0.45 = 0.55).

Calculating the cumulative probability for each value of x from 0 to 59 and summing them up will give us the answer.

P(X < 60) ≈ 0.0112

So, approximately 1.12% of random samples of size 75 will have less than 80% minority students.