Math Problem Statement

Select the correct answer for the blank: If everything else stays the same, the required sample size ____ as the confidence level increases to reach the same margin of error.

Solution

The correct answer is increases.

As the confidence level increases, we require more certainty that the sample statistic (e.g., a sample mean) falls within a given margin of error from the population parameter. To achieve this higher level of confidence, the sample size must increase, assuming that everything else (like the margin of error and variability in the population) remains the same.

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

Here are five related questions to expand on this:

  1. How does the margin of error change with sample size?
  2. What is the relationship between confidence level and margin of error?
  3. How do you calculate the required sample size for a given confidence level and margin of error?
  4. What other factors affect sample size calculations in a study?
  5. How does variability in the population influence the required sample size?

Tip: When determining sample size, always check the trade-off between the confidence level and the precision you need in your results. Higher confidence levels require larger sample sizes.

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Math Problem Analysis

Mathematical Concepts

Statistics
Sample Size
Confidence Levels
Margin of Error

Formulas

Sample size formula: n = (Z^2 * p * (1 - p)) / E^2
Where Z is the z-score for the desired confidence level, p is the population proportion, and E is the margin of error.

Theorems

Central Limit Theorem
Relationship between Confidence Interval and Sample Size

Suitable Grade Level

Grades 10-12 or Introductory College Statistics