Given two dependent random samples with the following results:
| Population 1 | 19 | 26 | 26 | 30 | 48 | 33 | 31 |
|---|---|---|---|---|---|---|---|
| Population 2 | 24 | 38 | 40 | 38 | 39 | 41 | 29 |
Use this data to find the $98 \%$ confidence interval for the true difference between the population means. Assume that both populations.
Step 4 of 4: Construct the $98 \%$ confidence interval. Round your answers to one decimal place.
Answer (1)
Calculate the differences between paired observations: d i = x 1 i − x 2 i .
Calculate the mean of the differences: d ˉ = n ∑ d i ≈ − 5.14 .
Calculate the margin of error: E = t α /2 , n − 1 ⋅ n s d ≈ 9.61 .
Construct the confidence interval: ( d ˉ − E , d ˉ + E ) = ( − 14.8 , 4.5 ) .
Explanation
Problem Analysis We are given two dependent random samples and asked to find the 98% confidence interval for the true difference between the population means. We will calculate the differences between paired observations, then find the mean and standard deviation of these differences. Finally, we'll use the t-distribution to construct the confidence interval.
Calculate Differences First, we calculate the difference d i for each pair of data points:
d 1 = 19 − 24 = − 5 d 2 = 26 − 38 = − 12 d 3 = 26 − 40 = − 14 d 4 = 30 − 38 = − 8 d 5 = 48 − 39 = 9 d 6 = 33 − 41 = − 8 d 7 = 31 − 29 = 2
Calculate Mean of Differences Next, we calculate the sample mean of the differences: d ˉ = n ∑ d i = 7 − 5 − 12 − 14 − 8 + 9 − 8 + 2 = 7 − 36 ≈ − 5.14
Calculate Standard Deviation of Differences Now, we calculate the sample standard deviation of the differences:
First, we find the squared differences from the mean:
( d i − d ˉ ) 2 :
( − 5 − ( − 5.14 ) ) 2 = ( − 5 + 5.14 ) 2 = ( 0.14 ) 2 = 0.0196 ( − 12 − ( − 5.14 ) ) 2 = ( − 12 + 5.14 ) 2 = ( − 6.86 ) 2 = 47.0596 ( − 14 − ( − 5.14 ) ) 2 = ( − 14 + 5.14 ) 2 = ( − 8.86 ) 2 = 78.4996 ( − 8 − ( − 5.14 ) ) 2 = ( − 8 + 5.14 ) 2 = ( − 2.86 ) 2 = 8.1796 ( 9 − ( − 5.14 ) ) 2 = ( 9 + 5.14 ) 2 = ( 14.14 ) 2 = 199.9396 ( − 8 − ( − 5.14 ) ) 2 = ( − 8 + 5.14 ) 2 = ( − 2.86 ) 2 = 8.1796 ( 2 − ( − 5.14 ) ) 2 = ( 2 + 5.14 ) 2 = ( 7.14 ) 2 = 50.9796
Sum of squared differences: 0.0196 + 47.0596 + 78.4996 + 8.1796 + 199.9396 + 8.1796 + 50.9796 = 392.8572
Then, we calculate the sample standard deviation: s d = n − 1 ∑ ( d i − d ˉ ) 2 = 7 − 1 392.8572 = 6 392.8572 = 65.4762 ≈ 8.09
Find Critical t-value We need to find the critical t-value, t α /2 , n − 1 , for a 98% confidence level with n − 1 = 7 − 1 = 6 degrees of freedom. Since the confidence level is 98%, α = 1 − 0.98 = 0.02 , and α /2 = 0.01 . Using a t-table or calculator, we find that t 0.01 , 6 ≈ 3.143 .
Calculate Margin of Error Now, we calculate the margin of error: E = t α /2 , n − 1 ⋅ n s d = 3.143 ⋅ 7 8.09 ≈ 3.143 ⋅ 2.646 8.09 ≈ 3.143 ⋅ 3.057 ≈ 9.61
Construct Confidence Interval Finally, we construct the 98% confidence interval: ( d ˉ − E , d ˉ + E ) = ( − 5.14 − 9.61 , − 5.14 + 9.61 ) = ( − 14.75 , 4.47 )
Rounding to one decimal place, the 98% confidence interval for the true difference between the population means is ( − 14.8 , 4.5 ) .
Final Answer The 98% confidence interval for the true difference between the population means is approximately ( − 14.8 , 4.5 ) .
Examples
In medical research, this type of confidence interval can be used to estimate the true difference in the effectiveness of two treatments on the same patients. For example, if Population 1 represents a patient's blood pressure before taking a medication and Population 2 represents their blood pressure after taking the medication, the confidence interval would estimate the range of the average blood pressure difference across all patients. This helps determine if the medication has a statistically significant effect. The confidence interval provides a range within which the true average difference in blood pressure is likely to fall, giving doctors a measure of the treatment's reliability.
