Chapter 9: Hypothesis Testing with Two Samples
9.1 Two Population Means with Unknown Standard Deviations
Learning Objectives
By the end of this section, the student should be able to:
- Conduct and interpret hypothesis tests for two population means, population standard deviations unknown.
Hypothesis Tests for Two Population Means, Population Standard Deviations Unknown
- The two independent samples are simple random samples from two distinct populations.
- For the two distinct populations:
- if the sample sizes are small, the distributions are important (should be normal)
- if the sample sizes are large, the distributions are not important (need not be normal)
The comparison of two population means is very common. A difference between the two samples depends on both the means and the standard deviations. Very different means can occur by chance if there is great variation among the individual samples. In order to account for the variation, we take the difference of the sample means,
[latex]{\overline{X}}_{1}[/latex] – [latex]{\overline{X}}_{2}[/latex], and divide by the standard error in order to standardize the difference. The result is a t-score test statistic.
Because we do not know the population standard deviations, we estimate them using the two sample standard deviations from our independent samples. For the hypothesis test, we calculate the estimated standard deviation, or standard error, of the difference in sample means, [latex]{\overline{X}}_{1}[/latex] – [latex]{\overline{X}}_{2}[/latex].
The test statistic (t-score) is calculated as follows:
and
[latex]{\overline{x}}_{2}[/latex]
are the sample means. μ1 and μ2 are the population means.
The number of degrees of freedom (df) requires a somewhat complicated calculation. However, a computer or calculator calculates it easily. The df are not always a whole number. The test statistic calculated previously is approximated by the Student’s t-distribution with df as follows:
When both sample sizes n1 and n2 are five or larger, the Student’s t approximation is very good. Notice that the sample variances (s1)2 and (s2)2 are not pooled. (If the question comes up, do not pool the variances.)
Exercises
Example
Independent groups
The average amount of time boys and girls aged seven to 11 spend playing sports each day is believed to be the same. A study is done and data are collected, resulting in the data in [link]. Each population has a normal distribution.
| Sample Size | Average Number of Hours Playing Sports Per Day | Sample Standard Deviation | |
|---|---|---|---|
| Girls | 9 | 2 | [latex]0.866[/latex] |
| Boys | 16 | 3.2 | 1.00 |
Is there a difference in the mean amount of time boys and girls aged seven to 11 play sports each day? Test at the 5% level of significance.
Solution
The population standard deviations are not known. Let g be the subscript for girls and b be the subscript for boys. Then, μg is the population mean for girls and μb is the population mean for boys. This is a test of tw