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$X$ is a normally distributed random variable with mean $18$ and standard deviation $2$ . What is the probability that $X$ is between $17$ and $19$ ? 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

18

and standard deviation

2

. What is the probability that

X

is between

17

and

19

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

Mean and Standard Deviation: We know the mean ( $\mu$ ) is $18$ and the standard deviation ( $\sigma$ ) is $2$ . We need to find $P(17 < X < 19)$ .
Calculate Z-score for $X = 17$ : First, let's find the z-score for $X = 17$ . $Z = \frac{X - \mu}{\sigma} = \frac{17 - 18}{2} = -0.5$ .
Calculate Z-score for $X = 19$ : Now, let's find the z-score for $X = 19$ . $Z = \frac{X - \mu}{\sigma} = \frac{19 - 18}{2} = 0.5$ .
Use $0.68$ - $0.95$ - $0.997$ Rule: Using the $0.68$ - $0.95$ - $0.997$ rule, we know that approximately $68\%$ of the data falls within one standard deviation of the mean. Since our z-scores $-0.5$ and $0.5$ are within one standard deviation, we can use this rule.
Calculate Probability: So, the probability that $X$ is between $17$ and $19$ is approximately $0.68$ or $68\%$ .

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