Estimation of difference among two means, Mathematics

Estimation of difference among two means

We know that the standard error of a sample is given by the value of the standard deviation (σ) divided by the square root of the number of items in the sample (√n).

However, when given two samples, the standard errors is described by

885_Estimation of difference among two means.png

Also note that we do calculate approximately the interval not from the mean although from the difference between the two samples means that is: (x¯A - x¯B)

The appropriate number of confidence level does not change

Thus the confidence interval is described by:

 (x¯A - x¯B) ± Confidence level S(A - x¯B)

=  (x¯A - x¯B)  ± Z  S(A - x¯B)

Illustration

Given two samples A and B of 100 and 400 items respectively, they contain the means x¯1= 7 ad x¯2 = 10 and standard deviations of 2 and 3 respectively. Construct confidence interval at 70 percent confidence level?

Solution

Sample                        A                                  B

                               x¯1 =      7                      x¯2= 10

                               n1 = 100                       n2 = 400

                               S1 = 2                           S2 = 3

The standard error of the samples A and B is described by:

S(A - x¯B) = √{(4/100) + (9/400)}

= ¼ = 0.25

At 70 percent confidence level, then appropriate number is equal to 1.04 or as read from the normal tables

(x¯1 - x¯2)= 7 - 10 = - 3 = 3

We take the absolute value of the difference among the means for illustration, the value of   x¯ = absolute value of X that is a positive value of X.

Therefore Confidence interval is described by:

= 3± 1.04 (0.25 )          From the normal tables a z value of 1.04 provide a value of 0.7.

= 3± 0.26

= 3.26 and 2.974

Hence 2.974 ≤ X ≤ 3.26

Posted Date: 2/19/2013 12:31:50 AM | Location : United States







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