Math Problem Statement
Construct a confidence interval for the mean of the paired differences for the two populations with the following information. Round the endpoints of the interval to three decimal places, if necessary.
n=31, d¯=2.538, α=0.05, sd=0.556
Solution
To construct a confidence interval for the mean of the paired differences, we can use the formula for the confidence interval of a mean with a t-distribution, since the population standard deviation is unknown and the sample size is small.
The formula for the confidence interval is:
Where:
- is the mean of the paired differences (2.538)
- is the standard deviation of the paired differences (0.556)
- is the sample size (31)
- is the critical value from the t-distribution for a confidence level of with degrees of freedom.
Step 1: Determine the -critical value
For a 95% confidence level (), the -critical value can be found using a t-distribution table or calculator with degrees of freedom. The critical value for is approximately 2.042.
Step 2: Compute the margin of error
Calculate the margin of error using:
Step 3: Construct the confidence interval
Now, calculate the confidence interval as:
This gives the endpoints:
- Lower endpoint:
- Upper endpoint:
Final Confidence Interval
The 95% confidence interval for the mean of the paired differences is:
Would you like further details on the calculations, or do you have any questions?
Here are five related questions to expand your understanding:
- How would the confidence interval change if the sample size were larger?
- What happens to the margin of error if the standard deviation increases?
- How does changing the confidence level (e.g., to 90% or 99%) affect the interval?
- Why do we use the t-distribution instead of the z-distribution here?
- What does the confidence interval suggest about the true mean of the paired differences?
Tip: The width of a confidence interval depends on the sample size, variability in the data, and the confidence level chosen. Increasing the sample size reduces the interval width.
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Math Problem Analysis
Mathematical Concepts
Statistics
Confidence Intervals
Paired Differences
T-Distribution
Formulas
Confidence Interval Formula: d̄ ± t(α/2, n-1) * (sd / √n)
Margin of Error: t(α/2, n-1) * (sd / √n)
Theorems
T-Distribution
Suitable Grade Level
College-level Statistics
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