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X is a normally distributed random variable with mean 52 and standard deviation 7.What is the probability that X is between 45 and 59?Use the 0.68-0.95-0.997 rule and write your answer as a decimal.

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We know that


\begin{gathered} \mu=52 \\ \sigma=7 \end{gathered}

If X is a normally distributed random variable with mean 52 and standard desviation 7, the probability that X is between 45 and 59 is represented as


P(45Then, we must write 45 and 59 as the mean 52 plus or minus a multiple k of the standard desviation 7[tex]\begin{gathered} 45=\mu+\sigma k \\ 45=52+7k \\ 7k=45-52 \\ k=-1 \end{gathered}

So,


45=\mu-\sigma

By a similar calculation


59=\mu+\sigma

This means

[tex]P(45taking into account that X is normally distributed and using the 0.68-0.95-0.997 rule[tex]P(\mu-\sigmaSo, P (45 < X < 59) = 0.68

User Venkat Peri
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