Question

Given two dependent random samples with the following results:

Population 1	2626	4848	4545	3737	4040	4444	1818
Population 2	3232	3636	3535	3131	3838	3636	2222

Use this data to find the 90%90% 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.

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Step 1 of 4: Find the mean of the paired differences, 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 90% confidence interval. Round your answers to one decimal place.

Answer 1

Step 1 of 4:

Mean of difference calculated using MS Excel is as follows:

The mean of the paired differences d̅ = 4.0

Step 2 of 4:

Level of Confidence = 90%
α = 100% - (Level of Confidence) = 10%
α/2 = 5% = 0.05

Critical value is tα/2

Calculate tα/2 by using t-distribution with degrees of freedom (DF) as n - 1 = 7 - 1 = 6 and α/2 = 0.05 as right-tailed area and left-tailed area.

tα/2 = 1.943

Step 3 of 4:

The standard deviation of the paired differences calculated using MS Excel is as follows:

Step 4 of 4:

Confidence Formula: [d̄ - tα/2•(sd/√n) , d̄ + tα/2•(sd/√n)]

Lower Bound = d̄ - tα/2•(sd/√n) = 4 - (1.943)(6.9282/√7) = -1.1

Upper Bound = d̄ + tα/2•(sd/√n) = 4 + (1.943)(6.9282/√7) = 9.1

The 90% confidence interval is (-1.1, 9.1)