Question

x is a normally distributed random variable with mean 32 and standard deviation 21.
what is the probability that x is between 5 and 59?
write your answer as a decimal rounded to the nearest thousandth.

Explanation:

Step1: Calculate z-scores for 5 and 59

The formula for the z - score is $z=\frac{x - \mu}{\sigma}$, where $\mu = 32$ (mean), $\sigma=21$ (standard deviation).

For $x = 5$:
$z_1=\frac{5 - 32}{21}=\frac{- 27}{21}\approx - 1.286$

For $x = 59$:
$z_2=\frac{59 - 32}{21}=\frac{27}{21}\approx1.286$

Step2: Find the probabilities corresponding to the z - scores

We use the standard normal distribution table (or a calculator with normal distribution functions). The probability that $Z

From the standard normal table, $P(Z < 1.286)\approx0.9009$ and $P(Z < - 1.286)=1 - P(Z < 1.286)\approx1 - 0.9009 = 0.0991$

Step3: Calculate the probability between the two z - scores

The probability that $X$ is between 5 and 59 is $P(5

$P(5

Answer:

0.802