Math Problem Statement
Solution
To solve this problem, we need to find the sample size required to achieve a margin of error of 0.1 with a 95% confidence level, and a population proportion .
Formula for sample size:
Where:
- is the Z-score corresponding to the 95% confidence level.
- is the estimated population proportion.
- is the margin of error.
Step-by-step solution:
-
Z-score for 95% confidence: From standard normal distribution tables, the Z-score for 95% confidence is approximately .
-
Plug values into the formula: Simplifying inside the square root:
Let's compute this value to find the sample size.The required sample size is , rounded up to the nearest whole number.
Would you like any further details or explanations? Here are a few related questions to deepen your understanding:
- How is the Z-score determined for different confidence levels?
- Why is the margin of error important in determining the sample size?
- What happens to the sample size if the margin of error is decreased?
- Can the sample size formula be applied when the population proportion is unknown?
- How does the confidence level affect the sample size calculation?
Tip: A smaller margin of error or a higher confidence level always requires a larger sample size to maintain precision in estimates.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Statistics
Confidence Intervals
Sample Size Calculation
Formulas
n = (Z * sqrt(p*(1 - p)) / E)^2
Theorems
Central Limit Theorem
Suitable Grade Level
Undergraduate Statistics or Advanced High School
Related Recommendation
Sample Size Calculation for a 90% Confidence Interval with 8% Margin of Error
Sample Size Calculation for 95% Confidence Interval with Margin of Error 0.03
Sample Size Calculation for a 95% Confidence Interval with p* = 0.25 and Margin of Error 0.1
Estimating Sample Size for Margin of Error in Confidence Intervals
Sample Size Estimation for Proportion with 99% Confidence and 0.03 Margin of Error