Applied Statistics and Probability for Engineers

10-3 INFERENCE FOR THE DIFFERENCE IN MEANS OF TWO NORMAL DISTRIBUTIONS, VARIANCES UNKNOWN 345

Minitab will also perform power and sample size calculations for the two-sample t-test (equal

variances). The output from Example 10-7 is as follows:

Power and Sample Size

2-Sample tTest

Testing mean 1mean 2 (versus not)

Calculating power for mean 1mean 2 difference

Alpha 0.05 Sigma 2.7

Sample Target Actual

Difference Size Power Power

4 10 0.8500 0.8793

The results agree fairly closely with the results obtained from the O.C. curve.

10-3.4 Confidence Interval on the Difference in Means

Case 1: ^21 ^22 ^2

To develop the confidence interval for the difference in means  1   2 when both variances

are equal, note that the distribution of the statistic

(10-18)

is the tdistribution with n 1 n 2 2 degrees of freedom. Therefore P(t 2,  2 T

t 2,  2 ) 1  . Now substituting Equation 10-18 for Tand manipulating the quan-

tities inside the probability statement will lead to the 100(1)% confidence interval on

 1   2.

n 1 n 2

n 1 n 2

T

X 1 X 2  1  1  22

Sp

B

1

n 1

1

n 2

If , s^21 and s^22 are the sample means and variances of two random samples of

sizes n 1 and n 2 , respectively, from two independent normal populations with un-

known but equal variances, then a 100(1)% confidence interval on the differ-

ence in means  1  2 is

(10-19)

where is the pooled estimate

of the common population standard deviation, and is the upper  2

percentage point of the tdistribution with n 1 n 2 2 degrees of freedom.

t 2, n 1 n 2  2

sp 231 n 1  12 s^211 n 2  12 s^224
1 n 1 n 2  22

 1  2 x 1 x 2 t 2, n 1 n 2  2 ̨ sp

B

1

n 1

1

n 2

x 1 x 2 t 2, n 1 n 2  2 ̨ sp

B

1

n 1

1

n 2

x 1 , x 2

Definition

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