A population of values has a normal distribution with population standard mean of 26.2 and population standard deviations of85.3. You intend to draw a random sample of size n=226.

a.) Find the probability that a **single** randomly selected value is greater than 15.4.
*P*(*X* > 15.4) =  (Round to 4 decimal places)

To find the probability that a single randomly selected value from a normally distributed population is greater than 15.4, we need to standardize the value using the Z-score formula and then use the standard normal distribution to find the corresponding probability.

### Step-by-Step Solution:

1. **Calculate the Z-score for \( X = 15.4 \):**

\[ Z = \frac{X - \mu}{\sigma} \]

where:
- \( \mu = 26.2 \) (population mean)
- \( \sigma = 85.3 \) (population standard deviation)
- \( X = 15.4 \) (value of interest)

\[ Z = \frac{15.4 - 26.2}{85.3} \]

\[ Z = \frac{-10.8}{85.3} \]

\[ Z \approx -0.1265 \]

2. **Find the probability corresponding to \( Z \leq -0.1265 \) using the standard normal distribution table or a calculator.**

Using a Z-table or a standard normal distribution calculator, we find the cumulative probability for \( Z \leq -0.1265 \):

\[ P(Z \leq -0.1265) \approx 0.450 \]

3. **Find the probability \( P(X > 15.4) \):**

Since the total area under the standard normal curve is 1, the probability of \( Z > -0.1265 \) is:

\[ P(Z > -0.1265) = 1 - P(Z \leq -0.1265) \]

\[ P(Z > -0.1265) = 1 - 0.450 \]

\[ P(Z > -0.1265) \approx 0.5500 \]

Thus, the probability that a single randomly selected value is greater than 15.4 is approximately 0.5500.

Would you like more details or have any questions about this solution?

Here are 5 numbered questions for you to ask:
1. How do I interpret the Z-score in a standard normal distribution?
2. Can you explain how to use the Z-table to find probabilities?
3. What steps should I take if the population distribution is not normal?
4. How do I find probabilities for other ranges of values?
5. Can you show an example of calculating the Z-score for a different value?

**Tip:** Always double-check your Z-score calculations and ensure you're using the correct cumulative probability from the Z-table or standard normal distribution calculator.

Explore how to calculate the probability that a randomly selected value from a normal distribution with mean μ = 26.2 and standard deviation σ = 85.3 is greater than 15.4. Learn about the Z-score formula and its application in probability calculations. Suitable for college-level students and beyond.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation