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Comparing Two Independent Population Means with Known Population Standard Deviations

Module by: Susan Dean, Barbara Illowsky, Ph.D.. E-mail the authors

Summary: This module provides an overview of hypothesis testing in situations where there are both two independent population means and known population standard deviations in statistics.

Even though this situation is not likely (knowing the population standard deviations is not likely), the following example illustrates hypothesis testing for independent means, known population standard deviations. The sampling distribution for the difference between the means is normal and both populations must be normal. The random variable is X 1 ¯ - X 2 ¯ X 1 - X 2 . The normal distribution has the following format:

Normal distribution

X 1 ¯ − X 2 ¯ ~ N [ u 1 − u 2 , ( σ 1 ) 2 n 1 + ( σ 2 ) 2 n 2 ] X 1 ¯ − X 2 ¯ ~N[ u 1 − u 2 , ( σ 1 ) 2 n 1 + ( σ 2 ) 2 n 2 ]
(1)

The standard deviation is:

( σ 1 ) 2 n 1 + ( σ 2 ) 2 n 2 ( σ 1 ) 2 n 1 + ( σ 2 ) 2 n 2
(2)

The test statistic (z-score) is:

z = ( x 1 ¯ − x 2 ¯ ) − ( μ 1 − μ 2 ) ( σ 1 ) 2 n 1 + ( σ 2 ) 2 n 2 z= ( x 1 ¯ − x 2 ¯ ) − ( μ 1 − μ 2 ) ( σ 1 ) 2 n 1 + ( σ 2 ) 2 n 2
(3)

Example 1

independent groups, population standard deviations known: The mean lasting time of 2 competing floor waxes is to be compared. Twenty floors are randomly assigned to test each wax. Both populations have a normal distribution. The following table is the result.

Table 1
Wax Sample Mean Number of Months Floor Wax Last Population Standard Deviation
1 3 0.33
2 2.9 0.36

Problem 1

Does the data indicate that wax 1 is more effective than wax 2? Test at a 5% level of significance.

[ Show Solution ]

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