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$X$ is a normally distributed random variable with mean $38$ and standard deviation $4$ . What is the probability that $X$ is between $26$ and $50$ ? 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

38

and standard deviation

4

. What is the probability that

X

is between

26

and

50

? 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 $X=26$ : Mean ( $\mu$ ) is $38$ and standard deviation ( $\sigma$ ) is $4$ . Calculate the z-score for $X=26$ . $\newline$ $Z = \frac{X - \mu}{\sigma} = \frac{26 - 38}{4} = \frac{-12}{4} = -3$ .
Calculate z-score for $X=50$ : Calculate the z-score for $X=50$ . $Z = \frac{X - \mu}{\sigma} = \frac{50 - 38}{4} = \frac{12}{4} = 3$ .
Identify probability range for z-scores: Using the $0.68$ - $0.95$ - $0.997$ rule, identify the probability range for the z-scores. $\newline$ A z-score of $-3$ to $3$ covers almost all the data, which corresponds to approximately $0.997$ of the data.
Calculate probability for $X$ between $26$ and $50$ : The probability that $X$ is between $26$ and $50$ is about $0.997$ .

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