This year, the ACT score of a randomly selected student has unknown distribution with a mean of 20.7 points and a standard deviation of 5 points. Let X\displaystyle {X} be the ACT score of a randomly selected student and let X\displaystyle \overline{{{X}}}  be the average ACT score of a random sample of size 45.

1. Describe the probability distribution of X\displaystyle {X} and state its parameters μ\displaystyle \mu and σ\displaystyle \sigma:

 X\displaystyle {X}\sim ( μ=\displaystyle \mu= , σ=\displaystyle \sigma= )

and find the probability that the ACT score of a randomly selected student is less than 32 points.

(Round the answer to 4 decimal places)

2. Use the Central Limit Theorem

to describe the probability distribution of X\displaystyle \overline{{{X}}} and state its parameters μX\displaystyle \mu_{\overline{{{X}}}} and σX\displaystyle \sigma_{\overline{{{X}}}}: (Round the answers to 1 decimal place)

 X\displaystyle \overline{{{X}}}\sim ( μX=\displaystyle \mu_{\overline{{{X}}}}= , σX=\displaystyle \sigma_{\overline{{{X}}}}= )

and find the probability that the average ACT score of a sample of 45 randomly selected students is between 20 and 20 points.

(Round the answer to 4 decimal places)