Given two dependent random samples with the following results:
Population 1: 44, 39, 42, 31, 40, 36, 42
Population 2: 32, 32,, 27, 18, 38, 30, 37
Use this data to find the 95% confidence interval for the true difference between the population means. Let d=(Population 1 entry)−(Population 2 entry) Assume that both populations are normally distributed.
Find the mean of the paired differences, find the critical value that should be used in constructing the confidence interval, and find the standard deviation of the paired differences, so that you may construct the 95% confidence interval.
Confidence Interval for the paired t test
From the Given Data: Xd = 8.571, Sd = 4.791
The t critical for alpha = 0.05, df = n - 1 = 7-1 = 6 is 2.4469
The Confidence Interval is Given by
Lower Limit = Xd - tcritical * Sd / Sqrt(n) = 8.571 - 2.4461 * 4.791/sqrt(7) = 8.571 - 4.431 = 4.140
Upper Limit = Xd + tcritical * Sd / Sqrt(n) = 8.571 + 2.4461 * 4.791/sqrt(7) = 8.571 + 4.431 = 13.002
The 95% confidence interval is 4.140 13.002
__________________________________________
Calculations Of Mean and SD
Mean = Sum of Observations/n
Standard Deviation = Sqrt(Variance)
Variance = SS/n-1, where SS = SUM (X - Mean)2.
# | Difference | Mean | (X-Mean)2 |
1 | 12 | 8.571 | 11.758 |
2 | 7 | 8.571 | 2.468 |
3 | 15 | 8.571 | 41.332 |
4 | 13 | 8.571 | 19.616 |
5 | 2 | 8.571 | 43.178 |
6 | 6 | 8.571 | 6.61 |
7 | 5 | 8.571 | 12.752 |
8 | 0 | 0 | |
Total | 60 | 137.714 |
n | 7 |
Sum | 60 |
Mean | 8.571 |
SS | 137.714 |
Variance | 22.9523 |
Std Dev | 4.791 |
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