Question

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.

Let d=(Population 1 entry)−(Population 2 entry)d=(Population 1 entry)−(Population 2 entry). Assume that both populations are normally distributed.

Step 1 of 4 :  Find the mean of the paired differences, x‾d. Round your answer to one decimal place.

Step 2 of 4: Find the critical value that should be used in constructing the confidence interval. Round your answer to three decimal places.

Step 3 of 4: Find the standard deviation of the paired differences to be used in constructing the confidence interval. Round your answer to one decimal place.

Step 4 of 4: Construct the 98% confidence interval. Round your answers to one decimal place.

Answer 1

Population 1	Population 2	Difference
19	24	-5
26	38	-12
26	40	-14
30	38	-8
48	39	9
33	41	-8
31	29	2

1)

∑d = -36

∑d² = 578

n = 7

Mean , x̅d = Ʃd/n = -36/7 = -5.1429 = -5.1

2)

At α = 0.02 and df = n-1 = 6, two tailed critical value, t-crit = T.INV.2T(0.02, 6) = 3.143

3)

Standard deviation, sd = √[(Ʃd² - (Ʃd)²/n)/(n-1)] = √[(578-(-36)²/7)/(7-1)] = 8.0917 = 8.1

4)

98% Confidence interval :

At α = 0.02 and df = n-1 = 6, two tailed critical value, t-crit = T.INV.2T(0.02, 6) = 3.143

Lower Bound = x̅d - t-crit*sd/√n = -5.1429 - 3.143 * 8.0917/√7 = -14.8

Upper Bound = x̅d + t-crit*sd/√n = -5.1429 + 3.143 * 8.0917/√7 = 4.5

-14.8 < µd < 4.5