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 = 33, d= 2.874, α = 0.05, sd = 0.828
Answer (2)
Calculate the degrees of freedom: df = n − 1 = 33 − 1 = 32 .
Find the critical t-value: t α /2 , df = t 0.025 , 32 ≈ 2.0369 .
Calculate the margin of error: E = t α /2 ⋅ n s d = 2.0369 ⋅ 33 0.828 ≈ 0.2936 .
Calculate the confidence interval: ( d ˉ − E , d ˉ + E ) = ( 2.874 − 0.2936 , 2.874 + 0.2936 ) = ( 2.580 , 3.168 ) .
The 95% confidence interval for the mean of the paired differences is ( 2.580 , 3.168 ) .
Explanation
Understand the problem and provided data We are given the sample size n = 33 , the sample mean of the differences d ˉ = 2.874 , the significance level α = 0.05 , and the sample standard deviation of the differences s d = 0.828 . We want to construct a confidence interval for the mean of the paired differences.
Calculate degrees of freedom First, we need to calculate the degrees of freedom, which is given by df = n − 1 . In this case, df = 33 − 1 = 32 .
Find the critical t-value Next, we need to find the critical t-value, t α /2 , from the t-distribution table with df = 32 degrees of freedom and a significance level of α /2 = 0.05/2 = 0.025 . Using a t-table or a calculator, we find that t 0.025 , 32 ≈ 2.0369 .
Calculate the margin of error Now, we calculate the margin of error E , which is given by the formula: E = t α /2 ⋅ n s d Substituting the values, we get: E = 2.0369 ⋅ 33 0.828 ≈ 2.0369 ⋅ 5.74456 0.828 ≈ 2.0369 ⋅ 0.14414 ≈ 0.2936
Calculate the confidence interval We calculate the lower and upper bounds of the confidence interval using the formulas: Lower Bound = d ˉ − E Upper Bound = d ˉ + E Substituting the values, we get: Lower Bound = 2.874 − 0.2936 = 2.5804 Upper Bound = 2.874 + 0.2936 = 3.1676
State the final answer Therefore, the 95% confidence interval for the mean of the paired differences is approximately ( 2.580 , 3.168 ) , rounding to three decimal places.
Examples
In medical research, paired difference tests are often used to compare the effectiveness of a treatment on the same subject before and after the treatment. For example, suppose we measure a patient's blood pressure before and after taking a new medication. The paired difference is the difference between the blood pressure measurements. A confidence interval for the mean paired difference would provide a range of values within which the true average change in blood pressure due to the medication is likely to fall. This helps researchers determine if the medication has a statistically significant effect.
To construct the confidence interval, we calculated the degrees of freedom, found the critical t-value, determined the margin of error, and then computed the interval endpoints. The resulting 95% confidence interval for the mean of the paired differences is approximately (2.580, 3.168). This interval indicates that we are 95% confident the true mean of the paired differences falls within this range.
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