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Significance Testing for AP Statistics (page 2)

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By — McGraw-Hill Professional
Updated on Feb 4, 2011

z-Procedures versus t-Procedures

In this chapter, we explore inference for means and proportions. When we deal with means, we may use, depending on the conditions, either t-procedures or z-procedures. With proportions, assuming the proper conditions are met, we deal only with large samples—that is, with z-procedures.

When doing inference for a population mean, or for the difference between two population means, we will usually use t-procedures. This is because z-procedures assume that we know the population standard deviation (or deviations in the two-sample situation) which we rarely do. We typically use t-procedures when doing inference for a population mean or for the difference between two population means when:

  1. The sample is a simple random sample from the population
  2. and

  3. The sample size is large (rule of thumb: n ≥ 30) or the population from which the sample is drawn is approximately normally distributed (or, at least, does not depart dramatically from normal)
  4. You can always use z-procedures when doing inference for population means when:

    1. The samples are simple random samples from the population
    2. and

    3. The population(s) from which the sample(s) is (are) drawn is normally distributed (in this case, the sampling distribution of will also be normally distributed)
    4. and

    5. The population standard deviation(s) is (are) known.

Historically, many texts allowed you to use z procedures when doing inference for means if your sample size was large enough to argue, based on the central limit theorem, that the sampling distribution of is approximately normal. The basic assumption is that, for large samples, the sample standard deviation s is a reasonable estimate of the population standard deviation σ. Today, most statisticians would tell you that it's better practice to always use t-procedures when doing inference for a population mean or for the difference between two population means. You can receive credit on the AP exam for doing a large sample problem for means using z-procedures but it's definitely better practice to use t-procedures.

When using t-procedures, it is important to check in step II of the hypothesis test procedure, that the data could plausibly have come from an approximately normal population. A stemplot, boxplot, etc., can be used to show there are no outliers or extreme skewness in the data. t-procedures are robust against these assumptions, which means that the procedures still work reasonably well even with some violation of the condition of normality. Some texts use the following guidelines for sample size when deciding whether of not to use t-procedures:

  • n < 15. Use t-procedures if the data are close to normal (no outliers or skewness).
  • n > 15. Use t-procedures unless there are outliers or marked skewness.
  • n > 40. Use t-procedures for any distribution.

For the two-sample case discussed later, these guidelines can still be used if you replace n with n1 and n2.

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