A population of values has a normal distribution with \mu=29.3 and \sigma=65.1. You intend to draw a random sample of size n=142.

Question

A population of values has a normal distribution with $μ = 29.3$ and $σ = 65.1$ . You intend to draw a random sample of size $n = 142$ .
Find the probability that a sample of size n=142 is randomly selected with a mean between 27.7 and 35.3.
$P (27.7 < \overset{―}{X} < 35.3) =$ ?
Write your answers as numbers accurate to 4 decimal places.

Answer 1

Step 1
According to the provided information,
$μ = 29.3$
$σ = 65.1$
$n = 142$
$P (27.7 < X < 35.3) = P (\frac{27.7 - 29.3}{\frac{65.1}{\sqrt{142}}} < \frac{X - μ}{\frac{σ}{\sqrt{n}}} < \frac{35.3 - 29.3}{\frac{65.1}{\sqrt{142}}})$
$= P (- 0.29 < Z < 1.1) (Z = \frac{X - μ}{σ})$
$= 0.4784$ (From standard normal table)
Therefore, the probability that a sample of size $n = 142$ is randomly selected with a mean between 27.7 and 35.3.
$P (27.7 < \overset{―}{X} < 35.3) = 0.4784$