Even though this situation is not likely (knowing the population standard deviations is very unlikely), the following example illustrates hypothesis testing for independent means with known population standard deviations. The sampling distribution for the difference between the means is normal in accordance with the central limit theorem. The random variable is $\stackrel{\u2013}{{X}_{1}}\u2013\stackrel{\u2013}{{X}_{2}}$. The normal distribution has the following format:

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 10.3.

Wax

Sample mean number of months floor wax lasts

Population standard deviation

1

3

0.33

2

2.9

0.36

Table 10.3

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

Solution 1

This is a test of two independent groups, two population means, population standard deviations known.

Random Variable: ${\stackrel{\u2013}{X}}_{1}\text{\u2013}{\stackrel{\u2013}{X}}_{2}$ = difference in the mean number of months the competing floor waxes last.

${H}_{0}:{\mu}_{1}\le {\mu}_{2}$

${H}_{a}:{\mu}_{1}>{\mu}_{2}$

The words "is more effective" says that wax 1 lasts longer than wax 2, on average. "Longer" is a “>” symbol and goes into H_{a}. Therefore, this is a right-tailed test.

Distribution for the test: The population standard deviations are known so the distribution is normal. Using the formula for the test statistic we find the calculated value for the problem.

The estimated difference between he two means is : ${\stackrel{\u2013}{X}}_{1}$ – ${\stackrel{\u2013}{X}}_{2}$ = 3 – 2.9 = 0.1

Compare calculated value and critical value and Z_{α}: We mark the calculated value on the graph and find the calculated value is not in the tail therefore we cannot reject the null hypothesis.

Make a decision: the calculated value of the test statistic is not in the tail, therefore you cannot reject H_{0}.

Conclusion: At the 5% level of significance, from the sample data, there is not
sufficient evidence to conclude that the mean time wax 1 lasts is longer (wax 1 is more effective) than the mean time wax 2 lasts.

Try It 10.7

The means of the number of revolutions per minute of two competing engines are to be compared. Thirty engines are randomly assigned to be tested. Both populations have normal distributions. Table 10.4 shows the result. Do the data indicate that Engine 2 has higher RPM than Engine 1? Test at a 5% level of significance.

Engine

Sample mean number of RPM

Population standard deviation

1

1,500

50

2

1,600

60

Table 10.4

Example 10.8

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.

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

Solution 1

This is a test of two independent groups, two population means. The population standard deviations are unknown, but the sum of the sample sizes is 30 + 30 = 60, which is greater than 30, so we can use the normal approximation to the Student’s-t distribution. Subscripts: 1: Democratic senators 2: Republican senators

Random variable:${\stackrel{\u2013}{X}}_{1}\text{\u2013}{\stackrel{\u2013}{X}}_{2}$ = difference in the mean age of Democratic and Republican U.S. senators.

The words "older than" translates as a “>” symbol and goes into H_{a}. Therefore, this is a right-tailed test.

Figure 10.8

Make a decision: The p-value is larger than 5%, therefore we cannot reject the null hypothesis. By calculating the test statistic we would find that the test statistic does not fall in the tail, therefore we cannot reject the null hypothesis. We reach the same conclusion using either method of a making this statistical decision.

Conclusion: At the 5% level of significance, from the sample data, there is not sufficient evidence to conclude that the mean age of Democratic senators is greater than the mean age of the Republican senators.