Math Problem Statement
A random sample of nn measurements was selected from a population with standard deviation σ=10.5σ=10.5 and unknown mean μμ. Calculate a 9999 % confidence interval for μμ for each of the following situations:
(a) n=35, x⎯⎯⎯=86.7n=35, x¯=86.7
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≤μ≤≤μ≤
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(b) n=55, x⎯⎯⎯=86.7n=55, x¯=86.7
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≤μ≤≤μ≤
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(c) n=70, x⎯⎯⎯=86.7n=70, x¯=86.7
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≤μ≤≤μ≤
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(d) In general, we can say that for the same confidence level, increasing the sample size
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the margin of error (width) of the confidence interval. (Enter: ''DECREASES'', ''DOES NOT CHANGE'' or ''INCREASES'', without the quotes.)
Solution
To calculate the confidence intervals for the population mean , we use the formula for the confidence interval when the population standard deviation is known:
where:
- is the sample mean,
- is the z-score corresponding to the desired confidence level (which is based on the standard normal distribution),
- is the population standard deviation,
- is the sample size.
For a 99.99% confidence level, the critical z-value is approximately 3.890 (from standard normal distribution tables).
The population standard deviation , and the sample means are provided as . We will calculate the confidence interval for each scenario:
(a) When :
So the confidence interval is:
(b) When :
So the confidence interval is:
(c) When :
So the confidence interval is:
(d) In general, when the sample size increases, the margin of error DECREASES.
This happens because the margin of error is inversely proportional to the square root of the sample size (), meaning that as the sample size increases, the width of the confidence interval shrinks, resulting in a more precise estimate.
Would you like more details or explanations on any part? Here are some questions that expand on this topic:
- How does the z-score change if we switch to a different confidence level, like 95%?
- Why is the sample size important in determining the width of the confidence interval?
- How does this method change if the population standard deviation is unknown?
- What are the implications of a very small or very large confidence interval in practical terms?
- Can you derive the formula for confidence intervals if the sample follows a t-distribution instead of a normal distribution?
Tip: When increasing the sample size, the reduction in margin of error follows a square root relationship, so doubling the sample size won't halve the margin of error.
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Math Problem Analysis
Mathematical Concepts
Statistics
Confidence Intervals
Normal Distribution
Formulas
Confidence Interval Formula: x̄ ± zα/2 (σ/√n)
Margin of Error: zα/2 (σ/√n)
Theorems
Central Limit Theorem
Standard Normal Distribution
Suitable Grade Level
College/Advanced High School
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