Given two dependent random samples with the following results:
| Population 1 | 35 | 34 | 45 | 30 | 49 | 25 | 34 |
|---|---|---|---|---|---|---|---|
| Population 2 | 44 | 24 | 40 | 42 | 38 | 31 | 43 |
Use this data to find the [tex]$90 \%$[/tex] confidence interval for the true difference between the population means. Assume that both populations are normal.
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 each pair of data points.
Calculate the mean of the differences: d ˉ = − 1.428571 .
Calculate the standard deviation of the differences: s d = 9.778499 .
Calculate the margin of error: E = t α /2 , n − 1 ⋅ n s d = 7.181183 . The margin of error is 7.181183 .
Explanation
Calculate the Differences First, we need to calculate the differences between each pair of data points. These differences are obtained by subtracting the values of Population 2 from the corresponding values of Population 1.
List of Differences The differences are: 35 − 44 = − 9 , 34 − 24 = 10 , 45 − 40 = 5 , 30 − 42 = − 12 , 49 − 38 = 11 , 25 − 31 = − 6 , 34 − 43 = − 9 . So the differences are: − 9 , 10 , 5 , − 12 , 11 , − 6 , − 9 .
Calculate the Mean of Differences Next, we calculate the mean of these differences. The mean is the sum of the differences divided by the number of differences.
Mean of Differences Calculation The mean of the differences, denoted as d ˉ , is calculated as follows: d ˉ = 7 − 9 + 10 + 5 − 12 + 11 − 6 − 9 = 7 − 10 = − 1.4285714285714286 So, d ˉ ≈ − 1.428571 .
Calculate the Standard Deviation of Differences Now, we calculate the standard deviation of the differences. This measures the spread of the differences around the mean.
Standard Deviation Calculation The standard deviation of the differences, denoted as s d , is calculated as follows: First, find the squared differences from the mean: ( − 9 − ( − 1.428571 ) ) 2 = ( − 7.571429 ) 2 ≈ 57.327 ( 10 − ( − 1.428571 ) ) 2 = ( 11.428571 ) 2 ≈ 130.612 ( 5 − ( − 1.428571 ) ) 2 = ( 6.428571 ) 2 ≈ 41.327 ( − 12 − ( − 1.428571 ) ) 2 = ( − 10.571429 ) 2 ≈ 111.755 ( 11 − ( − 1.428571 ) ) 2 = ( 12.428571 ) 2 ≈ 154.469 ( − 6 − ( − 1.428571 ) ) 2 = ( − 4.571429 ) 2 ≈ 20.898 ( − 9 − ( − 1.428571 ) ) 2 = ( − 7.571429 ) 2 ≈ 57.327
Sum of squared differences ≈ 57.327 + 130.612 + 41.327 + 111.755 + 154.469 + 20.898 + 57.327 = 573.715
Then, s d = n − 1 ∑ ( d i − d ˉ ) 2 = 7 − 1 573.715 = 6 573.715 = 95.619 ≈ 9.778499 .
Determine the Critical t-Value We need to find the critical t-value for a 90% confidence interval with n − 1 = 7 − 1 = 6 degrees of freedom. For a 90% confidence interval, α = 1 − 0.90 = 0.10 , so α /2 = 0.05 . The t-critical value for a two-tailed test with 6 degrees of freedom and α /2 = 0.05 is approximately 1.943.
Calculate the Margin of Error Now, we calculate the margin of error (E) using the formula: E = t α /2 , n − 1 ⋅ n s d .
Margin of Error Calculation Plugging in the values, we get: E = 1.943 ⋅ 7 9.778499 = 1.943 ⋅ 2.645751 9.778499 ≈ 1.943 ⋅ 3.695561 ≈ 7.181183 .
Final Answer Rounding the margin of error to six decimal places, we get E ≈ 7.181183 .
Conclusion Therefore, the margin of error for the 90% confidence interval is approximately 7.181183 .
Examples
In medical research, when comparing the effectiveness of two treatments on dependent samples (e.g., measuring a patient's condition before and after a treatment), calculating the confidence interval for the true difference between the means helps determine if the treatment has a statistically significant impact. The margin of error provides a range within which the true difference likely falls, aiding in assessing the reliability of the findings. For instance, if a new drug is tested on a group of patients, and their pain levels are recorded before and after the treatment, the margin of error helps determine the precision of the estimated pain reduction.
The margin of error for the 90% confidence interval is calculated to be approximately 7.181183 based on the differences between two dependent random samples. This involves calculating the mean and standard deviation of the differences and applying the t-distribution. The final margin of error indicates the range within which the true difference between population means is likely to fall.
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