Math Problem Statement
the sample size needed to to procvdie the margin of errorof 8 or less than ith a pprobabiltiy of 0.95when std dev is 11
Solution
To find the sample size needed to achieve a margin of error of 8 or less with a 95% confidence level, given that the standard deviation is 11, we can use the formula for the margin of error (E) in a confidence interval for a population mean:
Where:
- is the margin of error.
- is the Z-score corresponding to the desired confidence level.
- is the population standard deviation.
- is the sample size.
Given:
- Confidence level = 0.95, so the Z-score (for a two-tailed test).
We need to solve for :
Now, let's solve for .
Let me calculate that for you.The calculated sample size is approximately 7.26. Since the sample size must be a whole number, you would need to round up to the next whole number. Therefore, a sample size of 8 is required to achieve a margin of error of 8 or less with a 95% confidence level when the standard deviation is 11.
Would you like any further details or have any other questions?
Here are some related questions you might consider:
- How does changing the confidence level affect the required sample size?
- What happens to the sample size if the margin of error is reduced?
- How does the standard deviation influence the sample size calculation?
- What is the impact of a smaller sample size on the margin of error?
- Can you calculate the sample size for a different margin of error?
Tip: Always round up when calculating sample size because fractional samples aren't possible. This ensures the margin of error requirement is met.
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Math Problem Analysis
Mathematical Concepts
Statistics
Confidence Intervals
Sampling
Formulas
Margin of Error formula in Confidence Intervals
Theorems
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Suitable Grade Level
Advanced High School
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