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.
-
Population A:
- Sample size (n) = 40
- Proportion (p) = 0.25
- np = 40 * 0.25 = 10
- n(1-p) = 40 * 0.75 = 30
-
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:
- What is the Central Limit Theorem?
- How can we calculate the standard error for proportions?
- What are the implications of not meeting the CLT conditions?
- How can we increase the sample sizes effectively?
- What are alternative methods if sample sizes are insufficient?
Tip: Always check the conditions for applying the CLT before performing inference on sample proportions.
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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
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