Given two dependent random samples with the following results:
| Population 1 | 24 | 31 | 44 | 31 | 26 | 22 | 25 |
|---|---|---|---|---|---|---|---|
| Population 2 | 26 | 40 | 48 | 39 | 29 | 18 | 16 |
Use this data to find the $80 \%$ confidence interval for the true difference between the population means. Assume that both populations are $n$
Step 3 of 4 : Calculate the margin of error to be used in constructing the confidence interval. Round your answer to six decimal places.
Answer (2)
Calculate the differences between paired observations.
Determine the mean and standard deviation of these differences: d ˉ ≈ − 1.857143 , s d ≈ 6.414270 .
Find the critical t-value for an 80% confidence interval with 6 degrees of freedom: t 0.10 , 6 = 1.440 .
Calculate the margin of error: E = t α /2 ⋅ n s d ≈ 3.491084 .
Explanation
Calculate the Differences First, we need to calculate the differences between each pair of observations in Population 1 and Population 2. These differences are calculated as d i = x 1 i − x 2 i .
List the Differences The differences are: 24 − 26 = − 2 31 − 40 = − 9 44 − 48 = − 4 31 − 39 = − 8 26 − 29 = − 3 22 − 18 = 4 25 − 16 = 9
Calculate the Mean of the Differences Next, we calculate the mean of these differences, denoted as d ˉ . This is done by summing the differences and dividing by the number of pairs, n = 7 .
d ˉ = n ∑ d i = 7 − 2 − 9 − 4 − 8 − 3 + 4 + 9 = 7 − 13 ≈ − 1.857143
Calculate the Standard Deviation of the Differences Now, we calculate the standard deviation of the differences, denoted as s d . This measures the spread of the differences around the mean difference. s d = n − 1 ∑ ( d i − d ˉ ) 2 First, we calculate the squared differences from the mean: ( − 2 − ( − 1.857143 ) ) 2 = ( − 0.142857 ) 2 ≈ 0.020408 ( − 9 − ( − 1.857143 ) ) 2 = ( − 7.142857 ) 2 ≈ 51.020408 ( − 4 − ( − 1.857143 ) ) 2 = ( − 2.142857 ) 2 ≈ 4.591837 ( − 8 − ( − 1.857143 ) ) 2 = ( − 6.142857 ) 2 ≈ 37.734694 ( − 3 − ( − 1.857143 ) ) 2 = ( − 1.142857 ) 2 ≈ 1.306122 ( 4 − ( − 1.857143 ) ) 2 = ( 5.857143 ) 2 ≈ 34.306122 ( 9 − ( − 1.857143 ) ) 2 = ( 10.857143 ) 2 ≈ 117.877551 Sum of squared differences: 0.020408 + 51.020408 + 4.591837 + 37.734694 + 1.306122 + 34.306122 + 117.877551 = 246.857142 Then, we divide by n − 1 = 7 − 1 = 6 :
6 246.857142 ≈ 41.142857 Finally, we take the square root: s d = 41.142857 ≈ 6.414270
Determine the Critical t-Value We are constructing an 80% confidence interval, so α = 1 − 0.80 = 0.20 . Thus, α /2 = 0.10 . We need to find the t-value t α /2 with n − 1 = 6 degrees of freedom. From a t-table, we find that t 0.10 , 6 = 1.440 .
Calculate the Margin of Error Now we calculate the margin of error E using the formula: E = t α /2 ⋅ n s d = 1.440 ⋅ 7 6.414270 ≈ 1.440 ⋅ 2.645751 6.414270 ≈ 1.440 ⋅ 2.424253 ≈ 3.491084 Rounding to six decimal places, the margin of error is 3.491084 .
Final Answer The margin of error for the 80% confidence interval is approximately 3.491084 .
Examples
Understanding confidence intervals is crucial in various fields. For instance, in medical research, it helps determine the reliability of the difference in effectiveness between two drugs. In marketing, it can assess the range of potential customer responses to a new advertising campaign. By calculating the margin of error, we gain insight into the precision of our estimates, enabling more informed decisions based on sample data. The margin of error, in this case, helps us understand how much the true difference in population means could vary.
The margin of error for the 80% confidence interval is approximately 3.491084. This is calculated by determining the mean and standard deviation of the differences between two dependent samples, then using the critical t-value to compute the margin of error. Following these steps ensures we accurately represent the uncertainty in the estimate of the true difference between population means.
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