Learning Outcomes
- Conduct a hypothesis test for a difference in two population means with known standard deviations and interpret the conclusion in context
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 [latex]\displaystyle\overline{{X}}_{{1}}-\overline{{X}}_{{2}}[/latex]. The normal distribution has the following format:
Normal distribution is: [latex]\displaystyle\overline{{X}}_{{1}}-\overline{{X}}_{{2}}\sim{N}\Bigg[{\mu_{{1}}-\mu_{{2}},\sqrt{{\dfrac{{(\sigma_{{1}})}^{{2}}}{{n}_{{1}}}+\dfrac{{(\sigma_{{2}})}^{{2}}}{{n}_{{2}}}}}\Bigg]}[/latex]
The standard deviation is: [latex]\displaystyle\sqrt{\dfrac{(\sigma_1)^2}{n_1}+\dfrac{(\sigma_2)^2}{n_2}}[/latex]
The test statistic (z-score) is: [latex]\displaystyle{z}=\dfrac{(\overline{x}_1-\overline{x}_2)-(\mu_1-\mu_2)}{\sqrt{\dfrac{(\sigma_1)^2}{n_1}+\dfrac{(\sigma_2)^2}{n_2}}}[/latex]
Recall: ORDER OF OPERATIONS
| Please | Excuse | My | Dear | Aunt | Sally |
| parentheses | exponents | multiplication | division | addition | subtraction |
| [latex]( \ )[/latex] | [latex]x^2[/latex] | [latex]\times \ \mathrm{or} \ \div[/latex] | [latex]+ \ \mathrm{or} \ -[/latex] | ||
To calculate the test statistic (z-score) follow these steps:
First, find the numerator. Calculate the difference between the two sample means [latex](\overline{x}_1- \overline{x}_2)[/latex] and the two population means [latex](\mu _1- \mu _2)[/latex], then subtract them.
Second, find the denominator. You will end up taking the square root of the entire bottom; a square root can be understood as parentheses.
Step 1: Calculate [latex]\frac{(\sigma _1)^2}{n_1}[/latex] by squaring the population standard deviation of the first population and then dividing by the sample size taken from the first population.
Step 2: Calculate [latex]\frac{(\sigma _2)^2}{n_2}[/latex] by squaring the population standard deviation of the second population and then dividing by the sample size taken from the second population.
Step 3: Add the value you got in Step 1 and Step 2.
Step 4: Find the square root of the value you found in Step 3.
Third, take the numerator and divide by the denominator.
Example 1
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 distribution. The data are recorded in the table.
| Wax | Sample Mean Number of Months Floor Wax Lasts | Population Standard Deviation |
|---|---|---|
| 1 | 3 | 0.33 |
| 2 | 2.9 | 0.36 |
Does the data indicate that wax 1 is more effective than wax 2? Test at a 5% level of significance.
try it 1
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. The table below shows the result. Do the data indicate that Engine 2 has a 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 |
Example 2
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.
