Math Problem Statement

Samples of size 40 from population A with proportion 0.25 and samples of size 30 from population B with proportion 0.05.

Mean:  Std. Error: 

Are the sample sizes large enough for the Central Limit Theory to apply? i.. i. Yes     ii.. ii. No

Solution

To determine if the sample sizes are large enough for the Central Limit Theorem (CLT) to apply, we need to check if both sample sizes meet the rule of thumb that states np and n(1-p) should be greater than 5 for both populations.

  1. Population A:

    • Sample size (n) = 40
    • Proportion (p) = 0.25
    • np = 40 * 0.25 = 10
    • n(1-p) = 40 * 0.75 = 30
  2. Population B:

    • Sample size (n) = 30
    • Proportion (p) = 0.05
    • np = 30 * 0.05 = 1.5 (not sufficient)
    • n(1-p) = 30 * 0.95 = 28.5

Since np for Population B is less than 5, the sample size is not large enough for the CLT to apply.

Answer: ii. No

Would you like more details or have any questions? Here are some related questions:

  1. What is the Central Limit Theorem?
  2. How can we calculate the standard error for proportions?
  3. What are the implications of not meeting the CLT conditions?
  4. How can we increase the sample sizes effectively?
  5. What are alternative methods if sample sizes are insufficient?

Tip: Always check the conditions for applying the CLT before performing inference on sample proportions.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Statistics
Sampling Distributions
Proportions

Formulas

np > 5
n(1-p) > 5

Theorems

Central Limit Theorem

Suitable Grade Level

Grades 11-12