X

 is a normally distributed random variable with mean 92 and standard deviation 11.

What is the probability that 

X

 is between 59 and 70?

*Use* *the 0.68-0.95-0.997 rule and write your answer as a decimal. Round to the nearest thousandth if* *necessary.*

To solve the problem using the 68-95-99.7 rule, we'll follow a similar process as before.

### Step 1: Understand the Problem
- **Mean (μ)**: 92
- **Standard Deviation (σ)**: 11
- We want to find the probability that \( X \) is between 59 and 70.

### Step 2: Calculate the Z-scores
The Z-score for a value \( X \) in a normal distribution is given by:
\[
Z = \frac{X - \mu}{\sigma}
\]
- For \( X = 59 \):
\[
Z_1 = \frac{59 - 92}{11} = \frac{-33}{11} = -3
\]
- For \( X = 70 \):
\[
Z_2 = \frac{70 - 92}{11} = \frac{-22}{11} \approx -2
\]

### Step 3: Apply the Empirical Rule
- **Z = -2**: This corresponds to 2 standard deviations below the mean. The empirical rule tells us that approximately 95% of the data lies within 2 standard deviations of the mean. Therefore, the area to the left of \( Z = -2 \) is \( \frac{1 - 0.95}{2} = 0.025 \) or 2.5%.
  
- **Z = -3**: This corresponds to 3 standard deviations below the mean. According to the empirical rule, 99.7% of the data lies within 3 standard deviations of the mean. Therefore, the area to the left of \( Z = -3 \) is \( \frac{1 - 0.997}{2} = 0.0015 \) or 0.15%.

### Step 4: Find the Probability Between Z = -3 and Z = -2
To find the probability that \( X \) is between 59 and 70, subtract the area to the left of \( Z = -3 \) from the area to the left of \( Z = -2 \):
\[
P(-3 < Z < -2) = P(Z < -2) - P(Z < -3) = 0.025 - 0.0015 = 0.0235
\]

### Final Answer
The probability that \( X \) is between 59 and 70 is approximately **0.024** (rounded to the nearest thousandth).

Would you like more details or have any questions?

Here are 5 related questions for further practice:
1. What is the probability that \( X \) is between 70 and 92?
2. What is the probability that \( X \) is greater than 103?
3. If the standard deviation changes to 8 but the mean remains the same, what is the probability that \( X \) is between 64 and 81?
4. Calculate the Z-score for \( X = 81 \) with the same mean and standard deviation.
5. What is the probability that \( X \) is less than 70?

**Tip:** Remember, Z-scores help you understand how far a value is from the mean in terms of standard deviations. This is very useful in probability calculations involving normal distributions.

Learn how to calculate probabilities using the normal distribution and Z-score. Solve the problem of finding the probability that a normally distributed random variable X, with mean 92 and standard deviation 11, lies between 59 and 70. Detailed steps and calculations included.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation