Both confidence intervals and tests of significance for the mean of a Normal population are based on the sample mean , which estimates the unknown . The sampling distribution of depends on the standard deviation . This fact causes no difficulty when is known. When is unknown, we must estimate even though we are primarily interested in .

In this section, we meet the sampling distribution of the standardized mean when we use the sample standard deviation as our estimate of the standard deviation . We then use this sampling distribution in our discussion of both confidence intervals and significance tests for inference about the mean .

distributions

Suppose that we have a simple random sample (SRS) of size from a Normally distributed population with mean and standard deviation . The sample mean is then Normally distributed with mean and standard deviation . When is not known, we estimate it with the sample standard deviation , and then we estimate the standard deviation of by . This quantity is called the standard error of the sample mean , and we denote it by .

The term “standard error” is sometimes used for the actual standard deviation of a statistic. The estimated value is then called the “estimated standard error.” In this book, we use the term “standard error” only when the standard deviation of a statistic is estimated from the data. The term has this meaning in the output of many statistical computer packages and in reports of research in many fields that apply statistical methods.

In the previous chapter, the standardized sample mean, or one-sample statistic,

was used to introduce us to the procedures for inference about . This statistic has the standard Normal distribution . However, when we substitute the standard error for the standard deviation of , this statistic no longer has a Normal distribution. It has a distribution that is new to us, called a distribution.

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A particular distribution is specified by its degrees of freedom. We use to stand for the distribution with degrees of freedom. The degrees of freedom for this statistic come from the sample standard deviation in the denominator of . We saw in Chapter 1 that has degrees of freedom. Thus, there is a different distribution for each sample size. There are also other statistics with different degrees of freedom, some of which we will meet later in this chapter and others we will meet in later chapters.

The distributions were discovered in 1908 by William S. Gosset. Gosset was a statistician employed by the Guinness brewing company, which prohibited its employees from publishing their discoveries that were brewing related. In this case, the company let him publish under the pen name “Student” using an example that did not involve brewing. The distributions are often called “Student’s ” in his honor.

The density curves of the distributions are similar in shape to the standard Normal curve. That is, they are symmetric about 0 and are bell-shaped. Figure 7.1 compares the density curves of the standard Normal distribution and the distributions with 5 and 15 degrees of freedom. The similarity in shape is apparent, as is the fact that the distributions have more probability in the tails and less in the center.

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This greater spread is due to the extra variability caused by substituting the random variable for the fixed parameter . Comparing the two curves, we see that as the degrees of freedom increase, the density curve gets closer to the curve. This reflects the fact that will generally be closer to as the sample size increases.

With the distributions to help us, we can analyze an SRS from a Normal population with unknown or a large sample from a non-Normal population with unknown . Table D in the back of the book gives critical values for the distributions. For convenience, we have labeled the table entries both by the value of needed for significance tests and by the confidence level (in percent) required for confidence intervals. The standard Normal critical values in the bottom row of entries are labeled . This table can be used when you don’t have easy access to computer software.

The one-sample confidence interval

The one-sample confidence interval is similar in both reasoning and computational detail to the procedures of Chapter 6. There, the margin of error for the population mean was . When is unknown, we replace it with its estimate and switch from to . This means that the margin of error for the population mean when we use the data to estimate is .

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In this example, we have given the actual interval (88.3, 130.7) as our answer. Sometimes, we prefer to report the mean and margin of error: the average amount of time is 109.5 minutes with a margin of error of 21.2 minutes.

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The use of the confidence interval in Example 7.1 rests on assumptions that appear reasonable here. First, we assume that our random sample is an SRS from the students at your institution. Second, because our sample size is not large, we assume that the distribution of times is Normal. With only eight observations, this assumption cannot be effectively checked. We can, however, check if the data suggest a severe departure from Normality. Figure 7.2 shows the Normal quantile plot, and we can clearly see there are no outliers or severe skewness. Deciding whether to use the confidence interval for inference about is often a judgment call. We provide some practical guidelines to assist in this decision later in this section.

The one-sample test

Significance tests of the mean using the standard error of are also very similar to the test described in the last chapter. We still carry out the four steps required to do a significance test, but because we use in place of , the distribution we use to find the -value changes from the standard Normal to a distribution. Here are the details.

In this example, we tested the null hypothesis against the two-sided alternative . Because we had suspected that the average time would be smaller, we could have used a one-sided test.

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For these data, our conclusion does not depend on the choice between a one-sided and a two-sided alternative hypothesis. Sometimes, however, this choice will affect the conclusion, and so this choice needs to be made prior to analysis. It is wrong to examine the data first and then decide to do a one-sided test in the direction indicated by the data. If in doubt, always use a two-sided test. This is the alternative hypothesis to use when there is no prior suspicion that the mean is larger or smaller.

Using software

For small data sets, such as the one in Example 7.1 (page 361), it is easy to perform the computations for confidence intervals and significance tests with an ordinary calculator and Table D. For larger data sets, however, software or a statistical calculator eases our work.

The hypothesis test in Example 7.4 leads us to conclude that the mean return on the client’s account differs from that of the stock index. Now let’s assess the return on the client’s account with a confidence interval.

The confidence interval suggests that the broker’s management of this account had a long-term mean somewhere between a loss of 3.04% and a gain of 0.84% per month. We are interested not in the actual mean but in the difference between the broker’s process and the diversified S&P 500 index.

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Matched pairs procedures

The smartphone use problem of Case 7.1 concerns only a single population. We know that comparative studies are usually preferred to single-sample investigations because of the protection they offer against confounding. For that reason, inference about a parameter of a single distribution is less common than comparative inference.

One common comparative design, however, makes use of single-sample procedures. In a matched pairs study, subjects are matched in pairs and the outcomes are compared within each matched pair. For example, an experiment to compare two marketing campaigns might use pairs of subjects that are the same age, sex, and income level. The experimenter could toss a coin to assign the two campaigns to the two subjects in each pair. The idea is that matched subjects are more similar than unmatched subjects, so comparing outcomes within each pair is more efficient (that is, reduces the standard deviation of the estimated difference of treatment means). Matched pairs are also common when randomization is not possible. For example, before-and-after observations on the same subjects call for a matched pairs analysis.

This result, however, does not fully address the goal of this study. A lack of statistical significance does not prove the null hypothesis is true. If that were the case, we would simply design poor experiments whenever we wanted to prove the null hypothesis. The more appropriate method of inference in this setting is to consider equivalence testing. With this approach, we try to prove that the mean difference is within some acceptable region around 0. We can actually perform this test using a confidence interval.

If the resulting 90% confidence interval would have been outside the stated region or contained values both within and outside the stated region, we would not have been able to conclude that the means are equivalent.

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Robustness of the one-sample procedures

The matched pairs procedures use one-sample confidence intervals and significance tests for differences. They are, therefore, based on an assumption that the population of differences has a Normal distribution. For the histogram of the 76 differences in Example 7.7 shown in Figure 7.8 (page 369), the data appear to be slightly skewed. Does this slight non-Normality suggest that we should not use the procedures for these data?

All inference procedures are based on some conditions, such as Normality. Procedures that are not strongly affected by violations of a condition are called robust. Robust procedures are very useful in statistical practice because they can be used over a wide range of conditions with good performance.

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The condition that the population be Normal rules out outliers, so the presence of outliers shows that this condition is not fulfilled. The procedures are not robust against outliers, because and are not resistant to outliers.

Fortunately, the procedures are quite robust against non-Normality of the population, particularly when the sample size is large. The procedures rely only on the Normality of the sample mean . This condition is satisfied when the population is Normal, but the central limit theorem tells us that a mean from a large sample follows a Normal distribution closely even when individual observations are not Normally distributed.

To convince yourself of this fact, use the Statistic applet to study the sampling distribution of the one-sample statistic. From one of three population distributions, 10,000 SRSs of a user-specified sample size are generated, and a histogram of the statistics is constructed. You can then compare this estimated sampling distribution with the distribution. When the population distribution is Normal, the sampling distribution is always distributed. For the other two distributions, you should see that as increases, the histogram looks more like the distribution.

To assess whether the procedures can be used in practice, Normal quantile plots, stemplots, histograms, and boxplots are all good tools for checking for skewness and outliers. For most purposes, the one-sample procedures can be safely used when unless an outlier or clearly marked skewness is present. In fact, the condition that the data are an SRS from the population of interest is the more crucial assumption, except in the case of small samples. Here are practical guidelines, based on the sample size and plots of the data, for inference on a single mean:⁵

For the measurement study in Example 7.7 (pages368–370), there is only slight skewness and no outliers. With observations, we should feel comfortable that the procedures give approximately correct results.

BEYOND THE BASICS: The Bootstrap

Confidence intervals and significance tests are based on sampling distributions. In this section, we have used the fact that the sampling distribution of is when the data are an SRS from an population. If the data are not Normal, the central limit theorem tells us that this sampling distribution is still a reasonable approximation as long as the distribution of the data is not strongly skewed and there are no outliers. Even a fair amount of skewness can be tolerated when the sample size is large.

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What if the population does not appear to be Normal and we have only a small sample? Then we do not know what the sampling distribution of looks like. The bootstrap is a procedure for approximating sampling distributions when theory cannot tell us their shape.⁶

The basic idea is to act as if our sample were the population. We take many samples from it. Each of these is called a resample. We calculate the mean for each resample. We get different results from different resamples because we sample with replacement. Thus, an observation in the original sample can appear more than once in a resample. We treat the resulting distribution of ’s as if it were the sampling distribution and use it to perform inference. If we want a 95% confidence interval, for example, we could use the middle 95% of this distribution.

The bootstrap is practical only when you can use a computer to take a large number of samples quickly. It is an example of how the use of fast and easy computing is changing the way we do statistics.