Question

x is a normally distributed random variable with mean 27 and standard deviation 15.
what is the probability that x is between 8 and 46?
write your answer as a decimal rounded to the nearest thousandth.

Explanation:

Step1: Calculate z-scores for 8 and 46

The formula for the z - score is $z=\frac{x - \mu}{\sigma}$, where $\mu = 27$ (mean) and $\sigma=15$ (standard deviation).

For $x = 8$:
$z_1=\frac{8 - 27}{15}=\frac{- 19}{15}\approx - 1.27$

For $x = 46$:
$z_2=\frac{46 - 27}{15}=\frac{19}{15}\approx1.27$

Step2: Find probabilities from z - table

We need to find $P(-1.27

From the standard normal table, $P(Z < 1.27)\approx0.89796$ and $P(Z < - 1.27)\approx0.10204$

Step3: Calculate the probability

$P(-1.27

Step4: Round to the nearest thousandth

Rounding $0.79592$ to the nearest thousandth gives $0.796$

Answer:

0.796