7.5 Central limit theorem: using the central limit theorem

A study involving stress is done on a college campus among the students. The stress scores follow a uniform distribution with the lowest stress score equal to 1 and the highest equal to 5. Using a sample of 50 students, find:

1. The probability that the mean stress score for the 50 students is less than 2.65.
2. The probability that the total of the 50 stress scores is more than 175.

Let $X$ = one stress score.

Problems 1. asks you to find a probability for a mean . Problems 2. asks you to find a probability for a total or sum . The sample size, $n$ , is equal to 50.

Since the individual stress scores follow a uniform distribution, $X$ ~ $U(1,5)$ where $a=1$ and $b=5$ (See Continuous Random Variables for the uniform).

${\mu }_X=\frac{a+b}{2}=\frac{1+5}{2}=3$

${\sigma }_X=\sqrt{\frac{(b-a{)}^2}{12}}=\sqrt{\frac{(5-1{)}^2}{12}}=1.15$

For problems 1., let $\overline{X}$ = the mean stress score for the 50 students. Then,

$\overline{X}$ ~ $N(3,\frac{1.15}{\sqrt{50}})$ where $\mathrm{n\; =\; 50}$ .

This problem is also worked out in a Two Column Model Step by Step Example. This model is in the next section of this chapter. This is the model that we will use in class, for homework, and for the statistics project.

For problem 2., let $\mathrm{\Sigma X}$ = the sum of the 50 stress scores. Then, $\mathrm{\Sigma X}$ ~ $N\left[\right(\mathrm{50})\cdot (3),\sqrt{50}\cdot 1.15]$