Given two dependent random samples with the following results:
| Population 1 | 24 | 31 | 44 | 31 | 26 | 22 | 25 |
|---|---|---|---|---|---|---|---|
| Population 2 | 26 | 40 | 48 | 39 | 29 | 18 | 16 |
Use this data to find the $80 \%$ confidence interval for the true difference between the population means. Assume that both populations are
Step 4 of 4 : Construct the $80 \%$ confidence interval. Round your answers to one decimal place.
Answer (2)
Calculate the differences between paired data points.
Compute the mean and standard deviation of these differences: d ˉ ≈ − 1.9 , s d ≈ 6.4 .
Find the critical t-value for an 80% confidence level with 6 degrees of freedom: t 0.10 , 6 ≈ 1.440 .
Calculate the confidence interval: ( − 5.3 , 1.6 ) .
Explanation
Problem Analysis We are given two dependent random samples and asked to find the 80% confidence interval for the true difference between the population means. We will follow the solution plan to calculate this interval.
Calculate Differences First, we calculate the difference d i for each pair of data points by subtracting the value in Population 2 from the corresponding value in Population 1. This gives us the following differences:
24 − 26 = − 2 31 − 40 = − 9 44 − 48 = − 4 31 − 39 = − 8 26 − 29 = − 3 22 − 18 = 4 25 − 16 = 9
Calculate Mean of Differences Next, we calculate the mean of these differences ( d ˉ ).
d ˉ = 7 − 2 + ( − 9 ) + ( − 4 ) + ( − 8 ) + ( − 3 ) + 4 + 9 = 7 − 13 ≈ − 1.9
Calculate Standard Deviation of Differences Now, we calculate the standard deviation of the differences ( s d ).
First, we find the squared differences from the mean:
( − 2 − ( − 1.9 ) ) 2 = ( − 0.1 ) 2 = 0.01 ( − 9 − ( − 1.9 ) ) 2 = ( − 7.1 ) 2 = 50.41 ( − 4 − ( − 1.9 ) ) 2 = ( − 2.1 ) 2 = 4.41 ( − 8 − ( − 1.9 ) ) 2 = ( − 6.1 ) 2 = 37.21 ( − 3 − ( − 1.9 ) ) 2 = ( − 1.1 ) 2 = 1.21 ( 4 − ( − 1.9 ) ) 2 = ( 5.9 ) 2 = 34.81 ( 9 − ( − 1.9 ) ) 2 = ( 10.9 ) 2 = 118.81
Then, we sum these squared differences:
∑ ( d i − d ˉ ) 2 = 0.01 + 50.41 + 4.41 + 37.21 + 1.21 + 34.81 + 118.81 = 246.87
Now, we calculate the standard deviation:
s d = n − 1 ∑ ( d i − d ˉ ) 2 = 7 − 1 246.87 = 6 246.87 ≈ 41.145 ≈ 6.4
Find the Critical t-Value We need to find the critical t-value for an 80% confidence interval with n − 1 = 7 − 1 = 6 degrees of freedom. For an 80% confidence level, α = 1 − 0.80 = 0.20 , so α /2 = 0.10 . Looking up t 0.10 , 6 in a t-table, we find that t 0.10 , 6 ≈ 1.440 .
Calculate Margin of Error Now, we calculate the margin of error ( E ).
E = t α /2 , n − 1 ⋅ n s d = 1.440 ⋅ 7 6.4 ≈ 1.440 ⋅ 2.646 6.4 ≈ 1.440 ⋅ 2.419 ≈ 3.5
Construct Confidence Interval Finally, we construct the 80% confidence interval.
( d ˉ − E , d ˉ + E ) = ( − 1.9 − 3.5 , − 1.9 + 3.5 ) = ( − 5.4 , 1.6 )
Rounding to one decimal place, the 80% confidence interval for the true difference between the population means is (-5.3, 1.6).
Final Answer The 80% confidence interval for the true difference between the population means is approximately (-5.3, 1.6).
Examples
Consider a scenario where you want to compare the effectiveness of two different teaching methods on student test scores. You collect paired data from students who were taught using both methods. The confidence interval helps you estimate the true difference in the average test scores between the two methods. For instance, if the interval is (-5.3, 1.6), it suggests that the first method might result in scores that are, on average, 5.3 points lower to 1.6 points higher than the second method, with 80% confidence. This information is valuable for making informed decisions about which teaching method to implement.
The 80% confidence interval for the true difference between the population means is calculated as approximately (-5.3, 1.6) after calculating the mean and standard deviation of the differences, determining the critical t-value, and computing the margin of error.
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