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$X$ is a normally distributed random variable with mean $35$ and standard deviation $5$ . What is the probability that $X$ is between $25$ and $45$ ? 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

35

and standard deviation

5

. What is the probability that

X

is between

25

and

45

? Use the

0.68-0.95-0.997

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

Mean and Standard Deviation: Mean ( $\mu$ ) is $35$ and standard deviation ( $\sigma$ ) is $5$ . We need to find $P(25 < X < 45)$ .
Calculate Z-score for $X=25$ : For $X = 25$ , calculate the Z-score: $Z = \frac{X - \mu}{\sigma} = \frac{25 - 35}{5} = \frac{-10}{5} = -2$ .
Calculate Z-score for $X=45$ : For $X = 45$ , calculate the Z-score: $Z = \frac{X - \mu}{\sigma} = \frac{45 - 35}{5} = \frac{10}{5} = 2$ .
Use $0.68$ - $0.95$ - $0.997$ Rule: According to the $0.68$ - $0.95$ - $0.997$ rule, the probability that $X$ is within $2$ standard deviations ( $\pm 2\sigma$ ) from the mean is about $0.95$ .
Calculate Probability: So, $P(25 < X < 45)$ is approximately $0.95$ .

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