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 $90 \%$ confidence interval for the true difference between the population means. Assume that both populations are norm
Step 2 of 4: Calculate the sample standard deviation of the paired differences. Round your answer to six decimal places.
Answer (1)
Calculate the differences between each pair of data points.
Calculate the mean of the differences: d ˉ = − 1.4285714285714286 .
Calculate the sample standard deviation of the differences: s d = 9.778499251881529 .
Round the sample standard deviation to six decimal places: 9.778499 .
Explanation
Calculate the Differences First, we need to calculate the differences between each pair of data points. We subtract the values of Population 2 from the corresponding values of Population 1.
List the Differences The differences are calculated as follows:
d 1 = 35 − 44 = − 9 d 2 = 34 − 24 = 10 d 3 = 45 − 40 = 5 d 4 = 30 − 42 = − 12 d 5 = 49 − 38 = 11 d 6 = 25 − 31 = − 6 d 7 = 34 − 43 = − 9
So, the differences are: -9, 10, 5, -12, 11, -6, -9.
Calculate the Mean of the Differences Next, we calculate the mean of these differences: d ˉ = n ∑ d i = 7 − 9 + 10 + 5 − 12 + 11 − 6 − 9 = 7 − 10 = − 1.4285714285714286 Rounding to six decimal places is not necessary at this step, but we keep the full precision for further calculations.
Calculate the Squared Differences Now, we calculate the squared differences ( d i − d ˉ ) 2 :
( d 1 − d ˉ ) 2 = ( − 9 − ( − 1.4285714285714286 ) ) 2 = ( − 7.5714285714285715 ) 2 = 57.326530612244895 ( d 2 − d ˉ ) 2 = ( 10 − ( − 1.4285714285714286 ) ) 2 = ( 11.428571428571429 ) 2 = 130.6122448979592 ( d 3 − d ˉ ) 2 = ( 5 − ( − 1.4285714285714286 ) ) 2 = ( 6.428571428571429 ) 2 = 41.3265306122449 ( d 4 − d ˉ ) 2 = ( − 12 − ( − 1.4285714285714286 ) ) 2 = ( − 10.571428571428571 ) 2 = 111.75510204081633 ( d 5 − d ˉ ) 2 = ( 11 − ( − 1.4285714285714286 ) ) 2 = ( 12.428571428571429 ) 2 = 154.46938775510205 ( d 6 − d ˉ ) 2 = ( − 6 − ( − 1.4285714285714286 ) ) 2 = ( − 4.571428571428571 ) 2 = 20.897959183673468 ( d 7 − d ˉ ) 2 = ( − 9 − ( − 1.4285714285714286 ) ) 2 = ( − 7.5714285714285715 ) 2 = 57.326530612244895
Sum the Squared Differences We sum the squared differences: ∑ ( d i − d ˉ ) 2 = 57.326530612244895 + 130.6122448979592 + 41.3265306122449 + 111.75510204081633 + 154.46938775510205 + 20.897959183673468 + 57.326530612244895 = 573.7142857142857
Calculate the Sample Variance of the Differences Now, we calculate the sample variance of the differences: s d 2 = n − 1 ∑ ( d i − d ˉ ) 2 = 7 − 1 573.7142857142857 = 6 573.7142857142857 = 95.6190476190476
Calculate the Sample Standard Deviation of the Differences Finally, we calculate the sample standard deviation of the differences: s d = s d 2 = 95.6190476190476 = 9.778499251881529 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 in paired data is crucial in many fields. For instance, in medical research, you might want to compare the effectiveness of a new drug by measuring patients' symptoms before and after treatment. The standard deviation of the differences helps quantify the consistency of the drug's effect across the patient population. Similarly, in manufacturing, you could compare the performance of two machines by measuring the output of each machine on the same set of tasks. The standard deviation of the differences would indicate how consistently one machine outperforms the other.
