data could plausibly have come from an approximately normal population. A stemplot, boxplot, dotplot,
or normal probability plot 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, provided there is not much skewness. 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 show no outliers and no skewness.
• 15 < n < 40. 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 n (^1)
and n 2 .
Using Confidence Intervals for Two-Sided Alternatives
Consider a two-sided significance test at, say, α = 0.05 and a confidence interval with C = 0.95. A sample
statistic that would result in a significant result at the 0.05 level would also generate a 95% confidence
interval that does not contain the hypothesized value. Confidence intervals for two-sided hypothesis tests
could then be used in place of generating a test statistic and finding a P -value. If the sample value
generates a C -level confidence interval that does not contain the hypothesized value of the parameter,
then a significance test based on the same sample value would reject the null hypothesis at α = 1 – C .
Questions sometimes ask for such a decision to be made based on a confidence interval. Beware,
however. If a question simply asks if there is statistical evidence to support a hypothesis it is asking for a
test. It is best to do what is asked for.
You should never use confidence intervals for hypothesis tests involving one-sided alternative
hypotheses. For the purposes of this book, confidence intervals are considered to be two sided. (One-
sided confidence intervals are not a part of this course.)
Inference for a Single Population Mean
In step II of the hypothesis-testing procedure, we need to identify the test to be used and justify the
conditions needed. The test can be identified by name or by formula. For example, you could say “We
will do a significance test for a mean” to identify the procedure in words. To identify the procedure with
a formula, write the formula for the test statistic which will usually have the following form:
When doing inference for a single mean, the estimator is , the hypothesized value is μ 0 in the nullhypothesis H 0 : μ = μ 0 , and the standard error is the estimate of the standard deviation of , which is
This can be summarized in the following table.