Question

In Nebraska, the average ACT score is 21.7 with a standard deviation of 1.1. We collect a random sample of 30 students who took the exam last year.

Part 1: (6 pts)

Check the all necessary conditions in detail (not just yes or no) (1 pt each) and give the sampling model and parameters to 2 decimal places (2 pts).

Part 2: (8 pts)

What is the probability that the average composite ACT score is 22.1 or more? Show your calculations for finding the z-score to three decimal places (4 pts), then find the probability to four decimal places using the appropriate probability notation (2 pts). Write a sentence that gives your solution in context (2 pts).

Answer 1

Let X be a random variable which takes the value of the ACT score of the students.

The average ACT score is = = 21.7 with a standard deviation = = 1.1.

n = sample size = 30

here n = 30 ,

Also population standard deviation of X is known ,

So that by central limit theorem the sampling distribution of the sample mean is approximately normal distribution with mean

and standard deviation =

What is the probability that the average composite ACT score is 22.1 or more?

That is we want to find P( > 22.1 )

The formula of Z score for mean is as follow:

For = 22.1, we get

So P(Z > 2 ) = 1 - P(Z < 2) 1 - 0.9772 = 0.0228 (This is the final answer)

Since 0.0228 is very small ( less than 0.05 ) so the average ACT score is more that 22.1 is unusual.