Math Problem Statement
Solution
To solve this problem, we need to determine the minimum sample size, , required to estimate the mean completion time with a 99% confidence level and a margin of error of minutes. Given information includes:
- Population standard deviation, minutes
- Confidence level,
- Margin of error, minutes
Step-by-Step Solution
-
Identify the Z-score for a 99% confidence level. For a 99% confidence level, the Z-score (critical value) associated with this level is approximately .
-
Apply the sample size formula for estimating a population mean. The formula to determine the sample size when estimating a mean is:
Plugging in the known values:
-
Calculate the value inside the parentheses first.
-
Square the result.
-
Round up to the nearest whole number. Since we need the minimum sample size, we round up to the next whole number.
Answer
The minimum sample size needed is 177.
Would you like further details on this calculation, or do you have any questions?
Here are 5 related questions for practice:
- How would the required sample size change if the confidence level was reduced to 95%?
- What is the impact of increasing the margin of error on the sample size?
- How is the sample size calculation affected if the population standard deviation is not known?
- If the margin of error is reduced to 1 minute, what would the new sample size be?
- How does changing the sample size affect the confidence interval width?
Tip: Remember, increasing confidence levels or reducing margin of error both increase required sample sizes, reflecting more stringent conditions for precision.
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 Determination
Formulas
n = (z * σ / E)^2
Theorems
Central Limit Theorem
Suitable Grade Level
Grades 11-12, College Level
Related Recommendation
Sample Size Calculation for 95% Confidence in Estimating Mean Assembly Line Time
Minimum Sample Size Calculation for 95% Confidence Interval
Minimum Sample Size for 90% Confidence Interval in Mean Shopping Time
Calculate Minimum Sample Size for Confidence Interval - Statistics Problem
Determine Minimum Sample Size for 95% Confidence with Population Standard Deviation 12.6