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$X$ is a normally distributed random variable with mean $34$ and standard deviation $6$ . What is the probability that $X$ is between $16$ and $52$ ? Use the $0.68-0.95-0.997$ rule and write your answer as a decimal. Round to the nearest thousandth if necessary.

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Find probabilities using the normal distribution I

Full solution

Q.

X

is a normally distributed random variable with mean

34

and standard deviation

6

. What is the probability that

X

is between

16

and

52

? Use the

0.68-0.95-0.997

rule and write your answer as a decimal. Round to the nearest thousandth if necessary.

Calculate Z-score for $X=16$ : Calculate the Z-score for $X = 16$ . $Z = \frac{X - \mu}{\sigma}$ $Z = \frac{16 - 34}{6}$ $Z = \frac{-18}{6}$ $Z = -3$
Calculate Z-score for $X=52$ : Calculate the Z-score for $X = 52$ . $Z = \frac{X - \mu}{\sigma}$ $Z = \frac{52 - 34}{6}$ $Z = \frac{18}{6}$ $Z = 3$
Use probability rule: Use the $0.68$ - $0.95$ - $0.997$ rule to find the probability. $\newline$ The Z-scores $-3$ and $3$ correspond to $\mu - 3\sigma$ and $\mu + 3\sigma$ , respectively. $\newline$ According to the rule, the probability that $X$ is within $3$ standard deviations of the mean ( $\mu \pm 3\sigma$ ) is approximately $0.997$ .
Find probability between $16$ and $52$ : The probability that $X$ is between $16$ and $52$ is approximately $0.997$ .

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