Assume the random variable x is normally distributed with mean

mu equals 50μ=50

and standard deviation

sigma equals 7σ=7.

Find the indicated probability.

Upper P left parenthesis x greater than 36 right parenthesisP(x>36)

Question content area bottom

Part 1

Upper P left parenthesis x greater than 36 right parenthesisP(x>36)equals=enter your response here

​(Round to four decimal places as​ needed.)

To find the probability \( P(x > 36) \) for a normally distributed random variable \( x \) with mean \( \mu = 50 \) and standard deviation \( \sigma = 7 \), follow these steps:

### Step 1: Standardize the value
We first convert \( x = 36 \) to a z-score using the formula:
\[
z = \frac{x - \mu}{\sigma}
\]
Substitute the values \( \mu = 50 \) and \( \sigma = 7 \):
\[
z = \frac{36 - 50}{7} = \frac{-14}{7} = -2
\]

### Step 2: Use the z-score to find the probability
Now, we need to find \( P(x > 36) = P(z > -2) \). Using standard normal distribution tables or a calculator, we can look up the cumulative probability for \( z = -2 \).

The cumulative probability for \( z = -2 \) is approximately 0.0228. This represents \( P(z \leq -2) \).

### Step 3: Find the desired probability
Since we're looking for \( P(z > -2) \), we subtract this value from 1:
\[
P(z > -2) = 1 - P(z \leq -2) = 1 - 0.0228 = 0.9772
\]

### Final Answer:
\[
P(x > 36) = 0.9772
\]

Would you like more details or have any questions?

Here are five related questions for practice:
1. What is \( P(x < 36) \) for the same distribution?
2. Find \( P(40 < x < 60) \) given \( \mu = 50 \) and \( \sigma = 7 \).
3. How do you calculate \( P(x > 64) \)?
4. What is the z-score for \( x = 55 \)?
5. For \( P(x < 45) \), what value do you get?

**Tip:** When using z-scores, remember that the standard normal distribution is symmetrical around 0, so probabilities for negative and positive z-scores mirror each other.

Assume the random variable x is normally distributed with mean μ = 50 and standard deviation σ = 7. Find the indicated probability P(x > 36).

This page provides a step-by-step solution to find the probability P(x > 36) for a normally distributed random variable with mean μ = 50 and standard deviation σ = 7. Learn how to standardize the value using the z-score formula and calculate the probability using normal distribution tables or calculators. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation