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9. Inferences Based on Two Samples
• Now that we’ve learned to make inferences about a single population, we’ll learn how to compare two populations.
• For example, we may wish to compare the mean gas mileages for two models of automobiles, or the mean reaction times of men and women to a visual stimulus.
• In this chapter we’ll see how to decide whether differences exist and how to estimate the differences between population means and proportions.
9.1 Comparing two population means: Independent Sampling
One of the most commonly used significance tests is the comparison of two population means [pic] and [pic].
Two-sample Problems
• The goal of inference is to compare the responses in two groups.
• Each group is considered to be a sample from a distinct population.
• The responses in each group are independent of those in the other group.
A two sample problem can arise from a randomized comparative experiment that randomly divides the subjects into two groups and exposes each group to a different treatment. The two samples may be of different sizes.
|Two-Sample z Statistic |
Suppose that [pic] is the mean of an SRS of size [pic] drawn from N( [pic] , [pic] ) population and that [pic] is the mean of an SRS of size [pic] drawn from N( [pic] , [pic] ) population. Then the two-sample z statistic
[pic]
has the standard normal N(0,1) sampling distribution.
Large Sample Confidence Interval for [pic]
[pic]
Assumptions: The two samples are randomly selected in an independent manner from the two populations. The sample sizes [pic] and [pic] are large enough.
Example for C.I. of [pic]
Let’s look at Example 9.1 in our textbook (page 431).
Example for Test of Significance
Let’s look at Examples 9.2 and 9.3 in our textbook (page 434 and 435).
In the unlikely event that both population standard deviations are known, the two-sample z statistic is the basis for inference about [pic]. Exact z procedures are seldom used because [pic]and [pic] are rarely known.
The two-sample t procedures
Suppose that the population standard deviations [pic] and [pic] are not known. We estimate them by the sample standard deviations [pic] and [pic] from our two samples.
The Pooled two-sample t procedures
The pooled two-sample t procedures are used when we can safely assume that the two populations have equal variances. The modifications in the procedure are the use of the pooled estimator of the common unknown variance
[pic].
This is called the pooled estimator of [pic].
When both populations have variance [pic], the addition rule for variances says that [pic] has variance equal to the sum of the individual variances, which is
[pic]
The standardized difference of means in this equal-variance case is
[pic]
This is a special two-sample z statistic for the case in which the populations have the same [pic]. Replacing the unknown [pic] by the estimates [pic] gives a t statistic. The degrees of freedom are [pic].
|The Pooled Two-Sample t Procedures |
Suppose that an SRS of size [pic] is drawn from a normal population with unknown mean [pic] and that an independent SRS of size [pic] is drawn from another normal population with unknown mean [pic]. Suppose also that the two populations have the same standard deviation. A level C confidence interval [pic] given by
[pic]
Here t* is the value for [pic] density curve with area C between –t* and t*.
To test the hypothesis [pic],compute the pooled two-sample t statistic
[pic]
In terms of a random variable T having the t([pic] distribution, the P-value for a test of [pic] against
[pic] is P([pic])
[pic] is P([pic])
[pic] is 2P([pic])
[pic]
Example Take Group 1 to be the calcium group and Group 2 to be the placebo group. The evidence that calcium lowers blood pressure more than a placebo is assessed by testing
[pic]
[pic]
Here are the summary statistics for the decrease in blood pressure:
|Group |Treatment | [pic] |[pic] |[pic] |
|1 |Calcium |10 |5.000 |8.743 |
|2 |Placebo |11 |-0.273 |5.901 |
The calcium group shows a drop in blood pressure, and the placebo group has a small increase. The sample standard deviations do not rule out equal population standard deviations. A difference this large will often arise by chance in samples this small. We are willing to assume equal population standard deviations. The pooled sample variance is
[pic]
[pic]
[pic]. So that [pic]
The pooled two-sample t statistic is
[pic]
[pic][pic]
The P-value is [pic], where T has t(19) distribution. From Table, we can see that P lies between 0.05 and 0.10. The experiment found no evidence that calcium reduces blood pressure (t=1.634, df=19, 0.05 ................
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