Introductory Statistics

# 10.2Two Population Means with Known Standard Deviations

Introductory Statistics10.2 Two Population Means with Known Standard Deviations

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 is:$X ¯ 1 – X ¯ 2 ~N[ μ 1 – μ 2 , ( σ 1 ) 2 n 1 + ( σ 2 ) 2 n 2 ] X ¯ 1 – X ¯ 2 ~N[ μ 1 – μ 2 , ( σ 1 ) 2 n 1 + ( σ 2 ) 2 n 2 ]$
The standard deviation is:
$( σ 1 ) 2 n 1 + ( σ 2 ) 2 n 2 ( σ 1 ) 2 n 1 + ( σ 2 ) 2 n 2$
The test statistic (z-score) is:
$z= ( x ¯ 1 – x ¯ 2 )–( μ 1 – μ 2 ) ( σ 1 ) 2 n 1 + ( σ 2 ) 2 n 2 z= ( x ¯ 1 – x ¯ 2 )–( μ 1 – μ 2 ) ( σ 1 ) 2 n 1 + ( σ 2 ) 2 n 2$

## Example 10.6

Independent groups, population standard deviations known: The mean lasting time of two competing floor waxes is to be compared. Twenty floors are randomly assigned to test each wax. Both populations have a normal distributions. The data are recorded in Table 10.8.

Wax Sample Mean Number of Months Floor Wax Lasts Population Standard Deviation
1 3 0.33
2 2.9 0.36
Table 10.8

### Problem

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

## Try It 10.6

The means of the number of revolutions per minute of two competing engines are to be compared. Thirty engines of each type are randomly assigned to be tested. Both populations have normal distributions. Table 10.9 shows the result. Do the data indicate that Engine 2 has higher RPM than Engine 1? Test at a 5% level of significance.

EngineSample Mean Number of RPMPopulation Standard Deviation
11,50050
21,60060
Table 10.9

## Example 10.7

An interested citizen wanted to know if Democratic U. S. senators are older than Republican U.S. senators, on average. On May 26 2013, the mean age of 30 randomly selected Republican Senators was 61 years 247 days old (61.675 years) with a standard deviation of 10.17 years. The mean age of 30 randomly selected Democratic senators was 61 years 257 days old (61.704 years) with a standard deviation of 9.55 years.

### Problem

Do the data indicate that Democratic senators are older than Republican senators, on average? Test at a 5% level of significance.