Assume the random variable x is normally distributed with mean

muμequals=8787

and standard deviation

sigmaσequals=44.

Find the indicated probability.

​P(7474less than<xless than<8585​)

Given that \( x \) is a normally distributed random variable with mean \( \mu = 87 \) and standard deviation \( \sigma = 4 \), we need to find the probability \( P(74 < x < 85) \).

To solve this, we'll standardize the values of \( x \) using the z-score formula:

\[
z = \frac{x - \mu}{\sigma}
\]

### Step 1: Calculate the z-scores
- For \( x = 74 \):

\[
z_1 = \frac{74 - 87}{4} = \frac{-13}{4} = -3.25
\]

- For \( x = 85 \):

\[
z_2 = \frac{85 - 87}{4} = \frac{-2}{4} = -0.5
\]

### Step 2: Find the corresponding probabilities
Using standard normal distribution tables or a calculator:

- \( P(z < -3.25) \) corresponds to a very small probability, typically near 0.0006.
- \( P(z < -0.5) \) corresponds to a probability of approximately 0.3085.

### Step 3: Compute the required probability
The probability \( P(74 < x < 85) \) is the

Learn how to calculate the probability P(74 < x < 85) for a normally distributed random variable with mean μ = 87 and standard deviation σ = 4. This problem involves applying the z-score formula and understanding the properties of the normal distribution.

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