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$X$ is a normally distributed random variable with mean $10$ and standard deviation $3$ . What is the probability that $X$ is between $9$ and $11$ ? Write your answer as a decimal rounded to the nearest thousandth.

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Find probabilities using the normal distribution I

Full solution

Q.

X

is a normally distributed random variable with mean

10

and standard deviation

3

. What is the probability that

X

is between

9

and

11

? Write your answer as a decimal rounded to the nearest thousandth.

Calculate z-score for $X=9$ : Mean ( $\mu$ ) is $10$ , standard deviation ( $\sigma$ ) is $3$ . Calculate the z-score for $X=9$ . $\newline$ $Z = \frac{X - \mu}{\sigma} = \frac{9 - 10}{3} = -\frac{1}{3}$ .
Calculate z-score for $X=11$ : Calculate the z-score for $X=11$ . $Z = \frac{X - \mu}{\sigma} = \frac{11 - 10}{3} = \frac{1}{3}$ .
Probability within $1$ standard deviation: Using the $0.68$ - $0.95$ - $0.997$ rule, the probability that $X$ is within $1$ standard deviation ( $\mu \pm \sigma$ ) is about $0.68$ .
Probability between $9$ and $11$ : Since $9$ and $11$ are within $1$ standard deviation from the mean, the probability that $X$ is between $9$ and $11$ is less than $0.68$ .
Estimate exact probability: The exact probability for z-scores $-\frac{1}{3}$ and $\frac{1}{3}$ is not given by the $0.68$ - $0.95$ - $0.997$ rule, but we can estimate it to be roughly half of $0.68$ , because the interval from $9$ to $11$ is about half the interval from $\mu - \sigma$ to $\mu + \sigma$ .

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