Math Problem Statement
Given two independent random samples with the following results:
n1=8 xbar1=127 s1=33 n2=12 xbar2=157 s2=27
Use this data to find the 95% confidence interval for the true difference between the population means. Assume that the population variances are not equal and that the two populations are normally distributed.
Step 2 of 3 : Find the margin of error to be used in constructing the confidence interval. Round your answer to six decimal places.
Solution
To find the margin of error for the confidence interval for the difference between the population means, we follow these steps:
Formula for the Margin of Error:
Where:
- : Critical value from the -distribution for a 95% confidence level, based on the degrees of freedom ().
- : Sample standard deviations.
- : Sample sizes.
Step 1: Compute the degrees of freedom ():
The degrees of freedom for unequal variances is calculated as:
Plug in the values:
- ,
- ,
-
Compute variances and their ratios:
-
Sum these terms for the numerator of :
-
Compute the denominator terms:
- For :
- For :
-
Combine to find :
Thus, .
Step 2: Find :
Using and a 95% confidence level, the critical value from the -distribution table is approximately:
Step 3: Calculate the margin of error:
-
Compute the square root term:
-
Multiply by :
Final Answer:
The margin of error is:
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Math Problem Analysis
Mathematical Concepts
Confidence Intervals
Margin of Error
Degrees of Freedom
t-Distribution
Formulas
Margin of Error: ME = t* × √((s1²/n1) + (s2²/n2))
Degrees of Freedom: df = [(s1²/n1 + s2²/n2)²] / [(s1²/n1)²/(n1-1) + (s2²/n2)²/(n2-1)]
Theorems
Central Limit Theorem
t-Distribution Critical Values
Suitable Grade Level
Undergraduate Statistics
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