Question

x is a normally distributed random variable with mean 47 and standard deviation 24. what is the probability that x is between 10 and 84? write your answer as a decimal rounded to the nearest thousandth.

Explanation:

Step1: Calculate z-score for 10

The formula for z-score is $z = \frac{x - \mu}{\sigma}$, where $\mu$ is the mean and $\sigma$ is the standard deviation. For $x = 10$, $\mu = 47$, $\sigma = 24$.
$z_1=\frac{10 - 47}{24}=\frac{-37}{24}\approx - 1.54$

Step2: Calculate z-score for 84

For $x = 84$,
$z_2=\frac{84 - 47}{24}=\frac{37}{24}\approx1.54$

Step3: Find probabilities from z-table

We need to find $P(-1.54 < Z < 1.54)$, which is $P(Z < 1.54)-P(Z < - 1.54)$. From z - table, $P(Z < 1.54)\approx0.9382$, $P(Z < - 1.54)\approx0.0618$.

Step4: Calculate the probability

$P(-1.54 < Z < 1.54)=0.9382 - 0.0618 = 0.8764$

Answer:

0.876