Given two dependent random samples with the following results:
| Population 1 | 30 | 47 | 19 | 49 | 42 | 31 | 24 |
|---|---|---|---|---|---|---|---|
| Population 2 | 45 | 37 | 29 | 44 | 46 | 46 | 34 |
Use this data to find the $90 \%$ confidence interval for the true difference between the population means. Assume that both populations are normally distributed.
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 (1)
Calculate the differences between paired observations.
Calculate the sample mean of the differences: d ˉ = − 5.5714285714 .
Calculate the sample standard deviation of the differences: s d = 9.7785 .
Calculate the margin of error: E = 7.18118 .
The margin of error is 7.181180 .
Explanation
Calculate the differences First, we need to calculate the differences between each pair of observations from Population 1 and Population 2. These differences are calculated as d i = x 1 i − x 2 i .
List the differences The differences are: 30 − 45 = − 15 47 − 37 = 10 19 − 29 = − 10 49 − 44 = 5 42 − 46 = − 4 31 − 46 = − 15 24 − 34 = − 10
Calculate the mean of the differences Next, we calculate the mean of these differences, denoted as d ˉ . This is given by the formula: d ˉ = n ∑ d i where n is the number of pairs (which is 7 in this case).
Calculate the mean d ˉ = 7 − 15 + 10 − 10 + 5 − 4 − 15 − 10 = 7 − 39 = − 5.5714285714
Calculate the standard deviation of the differences Now, we calculate the standard deviation of the differences, denoted as s d . This is given by the formula: s d = n − 1 ∑ ( d i − d ˉ ) 2 where n is the number of pairs (which is 7 in this case).
Calculate the standard deviation First, calculate the squared differences ( d i − d ˉ ) 2 :
( − 15 − ( − 5.5714 ) ) 2 = ( − 9.4286 ) 2 = 88.8980 ( 10 − ( − 5.5714 ) ) 2 = ( 15.5714 ) 2 = 242.4796 ( − 10 − ( − 5.5714 ) ) 2 = ( − 4.4286 ) 2 = 19.6125 ( 5 − ( − 5.5714 ) ) 2 = ( 10.5714 ) 2 = 111.7551 ( − 4 − ( − 5.5714 ) ) 2 = ( 1.5714 ) 2 = 2.4692 ( − 15 − ( − 5.5714 ) ) 2 = ( − 9.4286 ) 2 = 88.8980 ( − 10 − ( − 5.5714 ) ) 2 = ( − 4.4286 ) 2 = 19.6125
Sum of squared differences = 88.8980 + 242.4796 + 19.6125 + 111.7551 + 2.4692 + 88.8980 + 19.6125 = 573.7249
Then, s d = 7 − 1 573.7249 = 6 573.7249 = 95.6208 = 9.7785
Determine the critical t-value We are given a 90% confidence level. The degrees of freedom are n − 1 = 7 − 1 = 6 . The critical t-value ( t α /2 ) for a 90% confidence level with 6 degrees of freedom is 1.943.
Calculate the margin of error Now, we calculate the margin of error (E) using the formula: E = t α /2 ⋅ n s d E = 1.943 ⋅ 7 9.7785 = 1.943 ⋅ 2.64575 9.7785 = 1.943 ⋅ 3.69559 = 7.18118
Final Answer Therefore, the margin of error is approximately 7.18118. Rounding to six decimal places, we get 7.181180.
Example of Usage Imagine you are conducting a study to compare the effectiveness of two different teaching methods. You collect data from two groups of students, where each student in one group is paired with a student in the other group based on similar academic backgrounds. By calculating the confidence interval for the true difference between the means of the two groups, you can determine if there is a statistically significant difference in the effectiveness of the two teaching methods. The margin of error helps you understand the precision of your estimate.
Examples
Imagine you are conducting a study to compare the effectiveness of two different teaching methods. You collect data from two groups of students, where each student in one group is paired with a student in the other group based on similar academic backgrounds. By calculating the confidence interval for the true difference between the means of the two groups, you can determine if there is a statistically significant difference in the effectiveness of the two teaching methods. The margin of error helps you understand the precision of your estimate.
