Given two dependent random samples with the following results:
| Population 1 | 31 | 17 | 38 | 45 | 16 | 21 | 32 |
|---|---|---|---|---|---|---|---|
| Population 2 | 44 | 23 | 43 | 34 | 22 | 33 | 38 |
Use this data to find the [tex]$90 \%$[/tex] confidence interval for the true difference between the population means. Assume that both populations are [tex]$n$[/tex].
Step 4 of 4: Construct the [tex]$90 \%$[/tex] confidence interval. Round your answers to one decimal place.
Answer (1)
Calculate the differences between paired data points.
Find the mean and standard deviation of these differences: d ˉ ≈ 5.2857 , s d ≈ 7.8679 .
Determine the critical t-value for a 90% confidence level with 6 degrees of freedom: t 0.05 , 6 ≈ 1.943 .
Construct the confidence interval: ( − 0.5 , 11.1 ) .
Explanation
Problem Analysis We are given two dependent random samples and asked to find the 90% confidence interval for the true difference between the population means.
Calculate Differences First, we calculate the difference between each pair of data points. These differences are:
d 1 = 44 − 31 = 13 d 2 = 23 − 17 = 6 d 3 = 43 − 38 = 5 d 4 = 34 − 45 = − 11 d 5 = 22 − 16 = 6 d 6 = 33 − 21 = 12 d 7 = 38 − 32 = 6
Calculate Mean of Differences Next, we calculate the mean of these differences: d ˉ = 7 13 + 6 + 5 − 11 + 6 + 12 + 6 = 7 37 ≈ 5.2857
Calculate Standard Deviation of Differences Then, we calculate the standard deviation of the differences:
First, we find the squared differences from the mean: ( 13 − 5.2857 ) 2 ≈ 60.9694 ( 6 − 5.2857 ) 2 ≈ 0.5102 ( 5 − 5.2857 ) 2 ≈ 0.0816 ( − 11 − 5.2857 ) 2 ≈ 265.2857 ( 6 − 5.2857 ) 2 ≈ 0.5102 ( 12 − 5.2857 ) 2 ≈ 45.1531 ( 6 − 5.2857 ) 2 ≈ 0.5102
Sum of squared differences ≈ 60.9694 + 0.5102 + 0.0816 + 265.2857 + 0.5102 + 45.1531 + 0.5102 ≈ 373.0198
The sample standard deviation is: s d = n − 1 ∑ ( d i − d ˉ ) 2 = 7 − 1 373.0198 = 6 373.0198 ≈ 62.1699 ≈ 7.8679
Determine Critical t-Value We are given a 90% confidence level, so α = 1 − 0.90 = 0.10 , and α /2 = 0.05 . With n − 1 = 7 − 1 = 6 degrees of freedom, the t-critical value t 0.05 , 6 ≈ 1.943 .
Calculate Margin of Error Now, we calculate the margin of error: E = t α /2 , n − 1 ⋅ n s d = 1.943 ⋅ 7 7.8679 ≈ 1.943 ⋅ 2.6458 7.8679 ≈ 1.943 ⋅ 2.9737 ≈ 5.7781
Construct Confidence Interval Finally, we construct the 90% confidence interval: ( d ˉ − E , d ˉ + E ) = ( 5.2857 − 5.7781 , 5.2857 + 5.7781 ) ≈ ( − 0.4924 , 11.0638 )
Rounding to one decimal place, the 90% confidence interval is ( − 0.5 , 11.1 ) .
Final Answer The 90% confidence interval for the true difference between the population means is approximately ( − 0.5 , 11.1 ) .
Examples
Understanding the difference between dependent samples is useful in many real-world scenarios. For example, imagine a company wants to test the effectiveness of a new training program on its employees' performance. They measure each employee's performance before and after the training. The 'before' and 'after' scores are dependent because they come from the same employee. By calculating a confidence interval for the difference in means, the company can estimate the true impact of the training program on employee performance. If the interval contains zero, it suggests the training may not have a significant effect. If the interval is entirely positive, it suggests a significant improvement in performance.
