Question:
Suppose the monthly cost of housing for a UNC student is a random variable with mean 690 dollars and standard deviation equal to 113 dollars.
a) We will choose 70 UNC students at random and calculate the average monthly housing cost for the group. What will be the mean and standard deviation of the sampling distribution for the sample mean ? Give your answers to 2 decimal places.
b) How large a sample is required for the standard deviation of the sampling distribution to be below 10? Your answer should be an integer.
Answer:
a) Mean μ = 690.00 dollars
Sample standard deviation ≈ 13.51 dollars
b) For the standard deviation of the sampling distribution to be below 10 the sample size, n ≥ ≈ 128 students
Step-by-step explanation:
Here, based on the central limit theorem, the sampling distribution sample mean is equal to the population mean
The mean, μ of the 70 UNC student monthly housing cost is therefore,
Mean μ = 690 dollars
The Sampling distribution sampling mean standard deviation is given as
, therefore,
Where:
The population standard deviation, σ = 113 dollars and
Sample size, n = 70 we have
Sample standard deviation = 113/√70 = 13.506 dollars ≈ 13.51 dollars
B. For the sample standard deviation to be below 10, we have
113/√n < 10
∴ √n > 113/10
n > 11.3² = 127.69 ≈ 128 students
