10.1 Two Population Means with Unknown Standard Deviations

HYPOTHESIS TESTING FOR DIFFERENCE BETWEEN MEANS WITH TWO SAMPLES

When testing about claims that compare two groups (populations), we’ll need to obtain samples from each group to test those claims. Typically studies comparing two groups (such as the treatment and control groups in experiments/clinical trials or a company claiming that its product lasts longer than than its competitors, etc.) will require us to obtain sample from each group and use the difference between the two groups to test claim made about those populations.

Independent Samples: Two samples are independent if the selection of sample members from one population does not in any way affects the selection of sample members in the second population.

In this unit, we’ll perform testing using independent samples as well as depending samples. Keep in mind that the tests using independent samples follow a different procedure than that of the dependent samples.

All of hypothesis testing for means with two samples will fall into one of the following scenario:

Note that for scenarios 1 and 2, the null hypothesis may be written with [latex]=[/latex] sign instead of [latex]\ge[/latex] or [latex]\le[/latex].

When conducting a hypothesis test that compares two independent population means, the following characteristics should be present:

1. The two independent samples are simple random samples that are independent from two distinct populations.
2. If the sample sizes are small, the populations should be normally distributed. This requirement can be relaxed for larger sample sizes.

If the standard deviations of the two populations (or variances) are known, then we can conduct a [latex]z-[/latex]test to test the difference between the means. However, more often not, population standard deviations are not known. Therefore we normally do not perform this z-test. For sample sizes that are large, this test may be fine.

For the more realistic scenario when we don’t know the population standard deviations, we can estimate them using the two sample standard deviations from our independent samples. If we happen to know that the population standard deviations are the same (or similar), then we can combine sample information to compute a pooled estimate of the standard deviation and use that to compute the standard error (SE), of the difference in sample means, [latex]\bar X_1-\bar X_2[/latex]: \[s_\text{pooled}=\sqrt{\frac{(n_1-1)s_1^2 +(n_2-1)s_2^2}{n_1+n_2-2}}\]

The standard error, SE: \[\text{SE}=s_\text{pooled}\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}.\] The degrees of freedom, [latex]df[/latex], for the underlying [latex]t-[/latex]distribution is given by: [latex]df=n_1+n_2-2[/latex].

For situations where the population standard deviations are not known to be equal (unpooled), we calculate the estimated standard deviation, or standard error (SE), of the difference in sample means, [latex]\bar X_1-\bar X_2[/latex] as follows: \[\text{SE}=\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}.\] The degrees of freedom, [latex]df[/latex], for the underlying [latex]t-[/latex]distribution is calculated either by using a conservative estimate [latex]df=\text{min}(n_1-1, n_2-1)[/latex] or by a more complicated formula, which our calculators can handle easily. \[df = \frac{\left( \frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}\right)^2}{\frac{1}{n_1-1}\left(\frac{s_1^2 }{n_1} \right)^2+\frac{1}{n_2-1} \left(\frac{s_2^2 }{n_2} \right)^2}\]

You don’t need to compute they gnarly formula by hand, just use a calculator. When reporting the [latex]df[/latex] computed using this formula, it doesn’t always come out to be an whole number so you round down to the nearest whole number if required.

If we’re comparing two populations, we may find that the population means are different. It may be due to a difference in the populations or it may be due to chance. A hypothesis test can help determine if a difference in the estimated means reflects a difference in the population means.

Generally, the null hypothesis states that the two population means are the same. That is [latex]H_0:\mu_1=\mu_2[/latex]. Recall that we start hypothesis testing by assuming the equality in the null hypothesis is true. In this case, we’re assuming that [latex]\mu_1=\mu_2[/latex] which means that the two population means are the same (no difference). With no difference between the population means, we can compute the the standard deviation of the sampling distribution of the difference of the sample means as above (see both pooled and unpooled cases) and the test statistic is: \[ \frac{(\bar x_1-\bar x_2 )-(\mu_1-\mu_2)}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}.\] These formulas for two sample test for the difference of means are shown here so that you can recognize them for what they are but when conducting two sample tests, we’ll be using calculators to speed up our calculations so that you won’t have to remember these formulas.