Math Problem Statement

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​)

Solution

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

To solve this, we'll standardize the values of xx using the z-score formula:

z=xμσz = \frac{x - \mu}{\sigma}

Step 1: Calculate the z-scores

  • For x=74x = 74:

z1=74874=134=3.25z_1 = \frac{74 - 87}{4} = \frac{-13}{4} = -3.25

  • For x=85x = 85:

z2=85874=24=0.5z_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)P(z < -3.25) corresponds to a very small probability, typically near 0.0006.
  • P(z<0.5)P(z < -0.5) corresponds to a probability of approximately 0.3085.

Step 3: Compute the required probability

The probability P(74<x<85)P(74 < x < 85) is the

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Math Problem Analysis

Mathematical Concepts

Normal Distribution
Z-score

Formulas

Z-score formula: z = (x - μ) / σ

Theorems

Central Limit Theorem

Suitable Grade Level

Advanced High School / College