Question

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.

Answer 1

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