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 normal.
Step 4 of 4: Construct the $90 \%$ confidence interval. Round your answers to one decimal place.
Answer (2)
Calculate the differences between each pair of observations.
Calculate the sample mean of the differences: d ˉ ≈ − 1.4 .
Calculate the sample standard deviation of the differences: s d ≈ 9.8 .
Construct the 90% confidence interval: ( − 8.6 , 5.8 ) .
( − 8.6 , 5.8 )
Explanation
Understand the problem and provided data We are given two dependent random samples and asked to find the 90% confidence interval for the true difference between the population means. The data for population 1 is: 35, 34, 45, 30, 49, 25, 34. The data for population 2 is: 44, 24, 40, 42, 38, 31, 43. Since the samples are dependent, we will consider the differences between paired observations.
Calculate the differences First, calculate the differences between each pair of observations: d i = x 1 i − x 2 i . The differences are: -9, 10, 5, -12, 11, -6, -9.
Calculate the mean of the differences Next, calculate the sample mean of the differences: d ˉ = n ∑ d i .
d ˉ = 7 − 9 + 10 + 5 − 12 + 11 − 6 − 9 = 7 − 10 ≈ − 1.42857
Calculate the standard deviation of the differences Then, calculate the sample standard deviation of the differences: s d = n − 1 ∑ ( d i − d ˉ ) 2 .
s d = 7 − 1 ( − 9 − ( − 1.42857 ) ) 2 + ( 10 − ( − 1.42857 ) ) 2 + ( 5 − ( − 1.42857 ) ) 2 + ( − 12 − ( − 1.42857 ) ) 2 + ( 11 − ( − 1.42857 ) ) 2 + ( − 6 − ( − 1.42857 ) ) 2 + ( − 9 − ( − 1.42857 ) ) 2 s d = 6 ( − 7.57143 ) 2 + ( 11.42857 ) 2 + ( 6.42857 ) 2 + ( − 10.57143 ) 2 + ( 12.42857 ) 2 + ( − 4.57143 ) 2 + ( − 7.57143 ) 2 s d = 6 57.32653 + 130.61224 + 41.32653 + 111.75510 + 154.46939 + 20.89796 + 57.32653 s d = 6 573.71428 = 95.61905 ≈ 9.778499
Find the critical t-value Determine the critical t-value, t α /2 , n − 1 , for a 90% confidence interval with n − 1 degrees of freedom. Since the confidence level is 90%, α = 1 − 0.90 = 0.10 , so α /2 = 0.05 . The degrees of freedom are n − 1 = 7 − 1 = 6 . The t-value for 0.05 and 6 degrees of freedom is approximately 1.943.
Calculate the margin of error Calculate the margin of error: E = t α /2 , n − 1 ⋅ n s d .
E = 1.943 ⋅ 7 9.778499 ≈ 1.943 ⋅ 2.64575 9.778499 ≈ 1.943 ⋅ 3.6955 ≈ 7.18118
Construct the confidence interval Construct the confidence interval: ( d ˉ − E , d ˉ + E ) .
Lower bound: − 1.42857 − 7.18118 = − 8.60975 Upper bound: − 1.42857 + 7.18118 = 5.75261 Rounding to one decimal place, the 90% confidence interval is (-8.6, 5.8).
State the final answer The 90% confidence interval for the true difference between the population means is approximately (-8.6, 5.8).
Examples
Understanding the confidence interval for the difference between two population means is very useful in various real-world scenarios. For example, a company might want to compare the effectiveness of two different training programs on employee performance. By collecting data on employee performance before and after each training program, they can calculate the confidence interval for the difference in means. This interval provides a range within which the true difference in effectiveness likely lies, helping the company make informed decisions about which training program to implement. If the interval contains 0, it suggests there might not be a significant difference between the two programs.
To find the 90% confidence interval for the true difference between the population means, we calculated the sample mean and standard deviation of the differences, found the critical t-value, determined the margin of error, and constructed the confidence interval. The resulting interval is approximately (-8.6, 5.8).
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