Question

this year, the act score has an unknown distribution with a mean of 18.1 points and a standard deviation of 5.9 points. let ( x ) be the act score of a randomly selected student and let ( overline{x} ) be the average act score of a random sample of size 13.

1. describe the probability distribution of ( x ) and state its parameters ( mu ) and ( sigma ):

( x sim ) select an answer ( (mu=square,sigma=square) )

1. explain why the central limit theorem can or can not be applied

the central limit theorem for sample means select an answer be applied because
select an answer
select an answer
the sample size is small ( (n < 30) ) and the distribution of the original population is unknownthe original population is normally distributedthe sample size is large ( (n > 30) ) although the distribution of the original population is unknownthe distribution of the original population is unknown

Explanation:

Step1: Determine the distribution of \(X\)

Since \(X\) represents the ACT score of a randomly - selected student and no information about its distribution (other than mean and standard deviation) is given, we just use the given parameters. The mean \(\mu\) of \(X\) is the population mean, and the standard deviation \(\sigma\) is the population standard deviation.
\(\mu = 18.1\), \(\sigma=5.9\)

Step2: Check the Central Limit Theorem (CLT) conditions

The Central Limit Theorem for sample means states that if the sample size \(n\) is large (\(n\geq30\)) or the population is normally distributed, the sampling distribution of \(\overline{X}\) is approximately normal. Here, \(n = 13<30\) and the distribution of the original population (for \(X\)) is unknown.

Answer:

1. \(X\sim\) (with no specific named distribution) \((\mu = 18.1,\sigma = 5.9)\)
2. The Central Limit Theorem for Sample Means cannot be applied because the sample size is small (\(n < 30\)) and the distribution of the original population is unknown.