Question

1. The SAT test scores have an average value of 1200 with a standard deviation of 105. A random sample of 35 scores is selected for study.

A) What is the shape, mean(expected value) and standard deviation of the sampling distribution of the sample mean for samples of size 35?

B) What is the probability that the sample mean will be larger than 1235?

C) What is the probability that the sample mean will fall within 25 points of the population mean?

D) What is the probability that the sample mean will be less than 1175?

Answer 1

Let X be the SAT test scores

Given X has a mean of 1200 and a standard deviation of 105

Central limit theorem: If X is a random variable with a mean and standard deviation then the sample mean also follows a normal distribution with a mean and standard deviation when n is large (n>30) where n is the sample size

Given n = 35

a) follows a normal distribution with a mean and standard deviation

has bell-shaped curve

b)

  where Z follows a standard normal distribution

From Z table P(Z > 1.97) = 0.0244

Therefore the probability that the sample mean will be larger than 1235 is 0.0244

c)

  where Z follows a standard normal distribution

From Z table P(Z < -1.41) = 0.0793, P(Z < 1.41) = 0.9207

Therefore the probability that the sample mean will fall within 25 points of the population mean is 0.9207-0.0793 = 0.8414

d)

  where Z follows a standard normal distribution

From Z table P(Z < -1.41) = 0.0793

Therefore the probability that the sample mean will be less than 1175 is 0.0793