Question

Using a college cost estimator, the average of campus rental at George Town is approximately $547 per month. Suppose these rates are roughly normal with a standard deviation of $100. Say you take a random sample of 25 students living off campus.

1. What would be the mean and standard error for the distribution of the sample mean?

2. What is the probability that the sample mean would be at least $600?

3. What is the probability that the rent for a randomly selected student would be at least $600?

4. Explain the reason for discrepancy between the two probabilities you calculated in parts (B) and  (C).

Answer 1

Let X be the rental amount.

It is given that X is normally distributed with mean $547 and the standard deviation is $100.

The distribution of X is given as, X~N(µ=547, σ=100).

Here, the sample size, n is 25.

1).

It is known that the mean of the sampling distribution of the sample mean is nothing but the population mean.

The standard error of the sampling distribution of the sample mean is given as follows:

Hence, the sampling distribution of rental amounts is given as, X-bar~N(µ=547, σ=20).

2).

The probability that the sample mean is at least 600 is calculated by using the Excel formula “=1-NORM.DIST(600,547,20,TRUE)”.

3).

The probability that the rent of the randomly selected student’s rent is at least 600 is calculated by using the Excel formula “=1-NORM.DIST(600,547,100,TRUE)”.

4).

The probability in part (b) is obtained using the sampling distribution of sample mean.

The probability that the mean of the selected 25 students is at least 600 is obtained.

The probability in part (c) is the probability that the rent of the randomly selected student’s rent is at least 600.