# Assume the random variable x is normally distributed with mean mu = 88 and standard deviation...

## Question:

Assume the random variable {eq}x {/eq} is normally distributed with mean {eq}\mu = 88 {/eq} and standard deviation {eq}\sigma = 4 {/eq}. Find the indicated probability.

{eq}P(x < 85) {/eq}.

## Answer and Explanation:

Given that,

- Population mean, {eq}\mu = 88 {/eq}

- Population standard deviation, {eq}\sigma = 4 {/eq}

Now,

The required probability is {eq}P(X < 85). {/eq}

Now,

{eq}P(X < 85) = P(\frac{X-\mu}{\sigma} < \frac{85-\mu}{\sigma})\\ P(X < 85) = P(Z < \frac{85-88}{4})\\ P(X < 85) = P(Z < -0.75)\\ {/eq}

Excel function for the above probability:

Now,

{eq}\fbox{P(X < 85) = 0.2266} {/eq}

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