Math Problem Statement

A normal population has mean μ = 34 and standard deviation σ = 10. (a) What proportion of the population is between 10 and 20? (b) What is the probability that a randomly chosen value will be between 28 and 38? Round the answers to at least four decimal places.

Solution

To solve the problems based on the given normal distribution with μ=34\mu = 34 and σ=10\sigma = 10, let’s work through each part.


Part (a): Proportion between 10 and 20

For a normal distribution, we calculate probabilities using the z-score formula:

z=xμσz = \frac{x - \mu}{\sigma}

  1. Compute zz-scores for x=10x = 10 and x=20x = 20:

    • z10=103410=2.4z_{10} = \frac{10 - 34}{10} = -2.4
    • z20=203410=1.4z_{20} = \frac{20 - 34}{10} = -1.4
  2. Use the standard normal distribution table or a calculator to find the cumulative probabilities:

    • P(Z2.4)0.0082P(Z \leq -2.4) \approx 0.0082
    • P(Z1.4)0.0808P(Z \leq -1.4) \approx 0.0808
  3. The proportion between 1010 and 2020 is the difference: P(10X20)=P(Z1.4)P(Z2.4)=0.08080.0082=0.0726P(10 \leq X \leq 20) = P(Z \leq -1.4) - P(Z \leq -2.4) = 0.0808 - 0.0082 = 0.0726


Part (b): Probability between 28 and 38

  1. Compute zz-scores for x=28x = 28 and x=38x = 38:

    • z28=283410=0.6z_{28} = \frac{28 - 34}{10} = -0.6
    • z38=383410=0.4z_{38} = \frac{38 - 34}{10} = 0.4
  2. Use the standard normal distribution table or a calculator to find the cumulative probabilities:

    • P(Z0.6)0.2743P(Z \leq -0.6) \approx 0.2743
    • P(Z0.4)0.6554P(Z \leq 0.4) \approx 0.6554
  3. The probability between 2828 and 3838 is the difference: P(28X38)=P(Z0.4)P(Z0.6)=0.65540.2743=0.3811P(28 \leq X \leq 38) = P(Z \leq 0.4) - P(Z \leq -0.6) = 0.6554 - 0.2743 = 0.3811


Final Answers

  • (a): Proportion between 10 and 20: 0.0726
  • (b): Probability between 28 and 38: 0.3811

Would you like additional explanations or clarifications on how these probabilities were calculated?


Related Questions

  1. How do you interpret the cumulative probabilities in the standard normal distribution table?
  2. What are the steps for using a calculator to determine probabilities for a normal distribution?
  3. How would the probabilities change if the standard deviation were larger or smaller?
  4. How can you calculate probabilities for values outside the range provided?
  5. How can the z-score formula be rearranged to find xx given a probability?

Tip:

When working with normal distributions, always ensure your zz-score calculations are accurate by double-checking the formula and values substituted.

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Math Problem Analysis

Mathematical Concepts

Normal Distribution
Probability
Z-scores

Formulas

Z-score formula: z = (x - μ) / σ
Probability between two values in a normal distribution

Theorems

Properties of the Standard Normal Distribution

Suitable Grade Level

College-level Statistics or Advanced High School