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A psychologist wants to compare male and female college students’ impressions of personality based on selected Facebook pages. A nutritionist is interested in the effect of increased calcium on blood pressure. A bank wants to know which of two incentive plans will most increase the use of its debit cards. Two-sample problems such as these are among the most common situations encountered in statistical practice.

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. A two-sample problem can also arise when comparing random samples separately selected from two populations. Unlike the matched pairs designs studied earlier, there is no matching of the units in the two samples, and the two samples may be of different sizes. As a result, inference procedures for two-sample data differ from those for matched pairs.

We can present two-sample data graphically by a back-to-back stemplot (for small samples) or by side-by-side boxplots (for larger samples). Now we will apply the ideas of formal inference in this setting. When both population distributions are symmetric, and especially when they are at least approximately Normal, a comparison of the mean responses in the two populations is most often the goal of inference.

We have two independent samples, from two distinct populations (such as subjects given the latest Apple iPhone and those given the latest Samsung Galaxy smartphone). The same response variable—say, battery life—is measured for both samples. We will call the variable x₁ in the first population and x₂ in the second because the variable may have different distributions in the two populations. Here is the notation that we will use to describe the two populations:

We want to compare the two population means, either by giving a confidence interval for μ₁ −μ₂ or by testing the hypothesis of no difference, H₀: μ₁ = μ₂.

Inference is based on two independent SRSs, one from each population. Here is the notation that describes the samples:

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Throughout this section, the subscripts 1 and 2 show the population to which a parameter or a sample statistic refers.

The two-sample z statistic

The natural estimator of the difference μ₁ −μ₂ is the difference between the sample means, . If we are to base inference on this statistic, we must know its sampling distribution. Here are some facts from our study of probability:

We now know the sampling distribution of when both populations are Normally distributed. The mean and variance of this distribution can be expressed in terms of the parameters of the two populations.

Even though the population mean height of 10-year-old girls is greater than the population mean height of 10-year-old boys, the probability that the sample mean of the girls is greater than the sample mean of the boys in our class is only 73%. Large samples are needed to see the effects of small differences.

As Example 7.10 reminds us, any Normal random variable has the N(0, 1) distribution when standardized. We have arrived at a new z statistic.

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In the unlikely event that both population standard deviations are known, the two-sample z statistic is the basis for inference about μ₁ −μ₂. Exact z procedures are seldom used, however, because σ₁ and σ₂ are rarely known. In Chapter 6, we discussed the one-sample z procedures in order to introduce the ideas of inference. Here we move directly to the more useful t procedures.

The two-sample t procedures

Suppose now that the population standard deviations σ₁ and σ₂ are not known. We estimate them by the sample standard deviations s₁ and s₂ from our two samples. Following the pattern of the one-sample case, we substitute the standard errors for the standard deviations used in the two-sample z statistic. The result is the two-sample t statistic:

Unfortunately, this statistic does not have a t distribution. A t distribution replaces the N(0, 1) distribution only when a single standard deviation (σ) in a z statistic is replaced by its sample standard deviation (s). In this case, we replace two standard deviations (σ₁ and σ₂) by their estimates (s₁ and s₂), which does not produce a statistic having a t distribution.

Nonetheless, we can approximate the distribution of the two-sample t statistic by using the t(k) distribution with an approximation for the degrees of freedom kdf approximation. We use these approximations to find approximate values of t* for confidence intervals and to find approximate P-values for significance tests. Here are two approximations:

In practice, the choice of approximation rarely makes a difference in our conclusion. Most statistical software uses the first option to approximate the t(k) distribution for two-sample problems unless the user requests another method. Use of this approximation without software is a bit complicated; we give the details later in this section (see 447).

If you are not using software, the second approximation is preferred. This approximation is appealing because it is conservative.²³ Margins of error for the level C confidence intervals are a bit larger than they need to be, so the true confidence level is larger than C. For significance testing, the P-values are a bit larger; thus, for tests at a fixed significance level, we are a little less likely to reject H₀ when it is true.

The two-sample t confidence interval

We now apply the basic ideas about t procedures to the problem of comparing two means when the standard deviations are unknown. We start with confidence intervals.

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The design of the study in Example 7.11 is not ideal. Random assignment of students was not possible in a school environment, so existing third-grade classes were used. The effect of the reading programs is, therefore, confounded with any other differences between the two classes. The classes were chosen to be as similar as possible—for example, in terms of the social and economic status of the students. Extensive pretesting showed that the two classes were, on the average, quite similar in reading ability at the beginning of the experiment. To avoid the effect of two different teachers, the researcher herself taught reading in both classes during the eight-week period of the experiment. Therefore, we can be somewhat confident that the two-sample test is detecting the effect of the treatment and not some other difference between the classes. This example is typical of many situations in which an experiment is carried out but randomization is not possible.

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The two-sample t significance test

The same ideas that we used for the two-sample t confidence interval also apply to two-sample t significance tests. We can use either software or the conservative approach with Table D to approximate the P-value.

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Most statistical software requires the raw data for analysis. A few, like Minitab, will also perform a t test on data in summarized form (such as the summary statistics table in Example 7.12, pages 438–439). It is always preferable to work with the raw data because one can also examine the data through plots such as the back-to-back stemplot and those in Figure 7.12.

In order to get a confidence interval as part of the Minitab output, the two-sided alternative was considered. If your software gives you the P-value for only the two-sided alternative, 2P(T ≥ |t|), you need to divide the reported value by 2 after checking that the means differ in the direction specified by the alternative hypothesis.

Robustness of the two-sample procedures

The two-sample t procedures are more robust than the one-sample t methods. When the sizes of the two samples are equal and the distributions of the two populations being compared have similar shapes, probability values from the t table are quite accurate for a broad range of distributions when the sample sizes are as small as n₁ = n₂ = 5.²⁵ When the two population distributions have different shapes, larger samples are needed.

The guidelines for the use of one-sample t procedures can be adapted to two-sample procedures by replacing “sample size” with the “sum of the sample sizes” n₁ + n₂. Specifically,

These guidelines are rather conservative, especially when the two samples are of equal size. In planning a two-sample study, choose equal sample sizes if you can. The two-sample t procedures are most robust against non-Normality in this case, and the conservative probability values are most accurate.

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Here is an example with large sample sizes that are almost equal. Even if the distributions are not Normal, we are confident that the sample means will be approximately Normal. The two-sample t test is very robust in this case.

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In this example the exact P-value is very small because t = 3.71 says that the observed difference in means is over 3.5 standard errors above the hypothesized difference of zero (μ₁ = μ₂). In this study, the researchers also compared energy intake and energy expenditure between late and early eaters. Despite the observed weight loss difference of 2.2 kg, no significant differences in these variables were found.

In this and other examples, we can choose which population to label 1 and which to label 2. After inspecting the data, we chose early eaters as Population 1 because this choice makes the t statistic a positive number. This avoids any possible confusion from reporting a negative value for t. Choosing the population labels is not the same as choosing a one-sided alternative after looking at the data. Choosing hypotheses after seeing a result in the data is a violation of sound statistical practice.

Inference for small samples

Small samples require special care. We do not have enough observations to examine the distribution shapes, and only extreme outliers stand out. The power of significance tests tends to be low, and the margins of error of confidence intervals tend to be large. Despite these difficulties, we can often draw important conclusions from studies with small sample sizes. If the size of an effect is very large, it should still be evident even if the n’s are small.

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Figure 7.14 gives outputs for this analysis from several software packages. Although the formats differ, the basic information is the same. All report the sample sizes, the sample means and standard deviations (or variances), the t statistic, and its P-value. All agree that the P-value is small, though some give more detail than others. Software often labels the groups in alphabetical order. Always check the means first and report the statistic (you may need to change the sign) in an appropriate way. Be sure to also mention the size of the effect you observed, such as “The mean weight loss for the early eaters was 6.44 kg higher than for the late eaters.”

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There are two other things to notice in the outputs. First, SAS and SPSS only give results for the two-sided alternative. To get the P-value for the one-sided alternative, we must first check the mean difference to make sure it is in the proper direction. If it is, we divide the given P-value by 2. Also, SAS and SPSS report the results of two t procedures: a special procedure that assumes that the two population variances are equal and the general two-sample procedure that we have just studied. We don’t recommend the “equal-variances” procedures, but we describe them later, in the section on pooled two-sample t procedures.

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Software approximation for the degrees of freedom

We noted earlier that the two-sample t statistic does not have a t distribution. Moreover, the distribution changes as the unknown population standard deviations and change. However, the distribution can be approximated by a t distribution with degrees of freedom given by

This is the approximation used by most statistical software. It is quite accurate when both sample sizes and are 5 or larger.

The number df given by the preceding approximation is always at least as large as the smaller of n₁ − 1 and n₂ − 1. On the other hand, the number df is never larger than the sum n₁ + n₂ − 2 of the two individual degrees of freedom. The number df is generally not a whole number. There is a t distribution with any positive degrees of freedom, even though Table D contains entries only for whole-number degrees of freedom. When the number df is small and is not a whole number, interpolation between entries in Table D may be needed to obtain an accurate critical value or P-value. Because of this and the need to calculate df, we do not recommend regular use of this approximation if a computer is not doing the arithmetic. With a computer, however, the more accurate procedures are painless.

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The pooled two-sample t procedures

There is one situation in which a t statistic for comparing two means has exactly a t distribution. This is when the two Normal population distributions have the same standard deviation. As we’ve done with other t statistics, we will first develop the z statistic and then, from it, the t statistic. In this case, notice that we need to substitute only a single standard error when we go from the z to the t statistic. This is why the resulting t statistic has a t distribution.

Call the common—and still unknown—standard deviation of both populations σ. Both sample variances and estimate . The best way to combine these two estimates is to average them with weights equal to their degrees of freedom. This gives more weight to the sample variance from the larger sample, which is reasonable. The resulting estimator of is

This is called the pooled estimator of σ²pooled estimator of σ² because it combines the information in both samples.

When both populations have variance σ², the addition rule for variances says that has variance equal to the sum of the individual variances, which is

The standardized difference between means in this equal-variance case is, therefore,

This is a special two-sample z statistic for the case in which the populations have the same σ. Replacing the unknown σ by the estimate gives a t statistic. The degrees of freedom are , the sum of the degrees of freedom of the two sample variances. This t statistic is the basis of the pooled two-sample t inference procedures.

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As usual, we first examine the data. To compare the effects of the two treatments, take the response variable to be the amount of the decrease in blood pressure. Inspection of the data reveals that there are no outliers. Side-by-side boxplots and Normal quantile plots (Figure 7.15 and 7.16) give a more detailed picture. The calcium group has a somewhat short left tail, but there are no severe departures from Normality that will prevent use of t procedures. To examine the question of the researchers who collected these data, we perform a significance test.

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Sample size strongly influences the P-value of a test. An effect that fails to be significant at a specified level α in a small sample can be significant in a larger sample. In the light of the rather small samples in Example 7.19, the evidence for some effect of calcium on blood pressure is rather good. The published account of the study combined these results for blacks with the results for whites and adjusted for pretest differences among the subjects. Using this more detailed analysis, the researchers were able to report a P-value of 0.008.

Of course, a P-value is almost never the last part of a statistical analysis. To make a judgment regarding the size of the effect of calcium on blood pressure, we need a confidence interval.

The pooled two-sample t procedures are anchored in statistical theory and so have long been the standard version of the two-sample t in textbooks. But they require the assumption that the two unknown population standard deviations are equal. We discuss methods to assess this condition in Chapter 12.

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The pooled t procedures are therefore a bit risky. They are reasonably robust against both non-Normality and unequal standard deviations when the sample sizes are nearly the same. When the samples are quite different in size, the pooled t procedures become sensitive to unequal standard deviations and should be used with caution unless the samples are large. Unequal standard deviations are quite common. In particular, it is not unusual for the spread of data to increase when the center of the data increases. We recommend regular use of the unpooled t procedures because most software automates the Satterthwaite approximation.