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 2 of 4 : Calculate the sample standard deviation of the paired differences. Round your answer to six decimal places.

Calculate the differences between each pair of data points.
Calculate the mean of the differences: d ˉ = 7 − 39 ​ ≈ − 5.571429 .
Calculate the sample standard deviation of the differences: s d ​ = 6 573.714286 ​ ​ ≈ 9.778499 .
The sample standard deviation of the paired differences is 9.778499 ​ .

Explanation

Understand the problem and provided data We are given two dependent random samples and asked to calculate the sample standard deviation of the paired differences. The data is as follows:

Population 1: 30, 47, 19, 49, 42, 31, 24 Population 2: 45, 37, 29, 44, 46, 46, 34

Calculate the differences First, we need to calculate the differences between each pair of data points. We subtract the values of Population 2 from Population 1:

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
So, the differences are: -15, 10, -10, 5, -4, -15, -10

Calculate the mean of the differences Next, we calculate the mean of the differences:

d ˉ = n ∑ i = 1 n ​ d i ​ ​ = 7 − 15 + 10 − 10 + 5 − 4 − 15 − 10 ​ = 7 − 39 ​ ≈ − 5.571429

Calculate the squared differences from the mean Now, we calculate the squared differences from the mean:

( d i ​ − d ˉ ) 2
( − 15 − ( − 5.571429 ) ) 2 = ( − 9.428571 ) 2 ≈ 88.897959 ( 10 − ( − 5.571429 ) ) 2 = ( 15.571429 ) 2 ≈ 242.469388 ( − 10 − ( − 5.571429 ) ) 2 = ( − 4.428571 ) 2 ≈ 19.612245 ( 5 − ( − 5.571429 ) ) 2 = ( 10.571429 ) 2 ≈ 111.755102 ( − 4 − ( − 5.571429 ) ) 2 = ( 1.571429 ) 2 ≈ 2.469388 ( − 15 − ( − 5.571429 ) ) 2 = ( − 9.428571 ) 2 ≈ 88.897959 ( − 10 − ( − 5.571429 ) ) 2 = ( − 4.428571 ) 2 ≈ 19.612245

Calculate the sum of squared differences We calculate the sum of squared differences:

∑ i = 1 n ​ ( d i ​ − d ˉ ) 2 ≈ 88.897959 + 242.469388 + 19.612245 + 111.755102 + 2.469388 + 88.897959 + 19.612245 ≈ 573.714286

Calculate the sample standard deviation Finally, we calculate the sample standard deviation of the differences:

s d ​ = n − 1 ∑ i = 1 n ​ ( d i ​ − d ˉ ) 2 ​ ​ = 7 − 1 573.714286 ​ ​ = 6 573.714286 ​ ​ ≈ 95.619048 ​ ≈ 9.778499
Rounding to six decimal places, we get 9.778499.

State the final answer The sample standard deviation of the paired differences, rounded to six decimal places, is 9.778499.

Examples
Understanding the variability between paired data is crucial in many real-world scenarios. For instance, in clinical trials, researchers might measure a patient's blood pressure before and after a new medication. The paired differences help determine if the medication has a consistent effect. Calculating the standard deviation of these differences allows us to quantify the spread of the medication's effect across the patient population. A smaller standard deviation indicates more consistent results, while a larger one suggests greater variability in patient response. This analysis is vital for assessing the reliability and effectiveness of the treatment.

Answered by GinnyAnswer | 2025-07-05