• Significance tests and confidence intervals for the difference between the means and of two Normal populations are based on the difference between the sample means from two independent SRSs. Because of the central limit theorem, the resulting procedures are approximately correct for other population distributions when the sample sizes are large.
• When independent SRSs of sizes and are drawn from two Normal populations with parameters , and , the two-sample z statistic
has the N(0, 1) distribution.
• The two-sample t statistic
does not have a t distribution. However, good approximations are available.
• Conservative inference procedures for comparing and are obtained from the two-sample t statistic by using the distribution with degrees of freedom k equal to the smaller of and .
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• More accurate probability values can be obtained by estimating the degrees of freedom from the data. This is the usual procedure for statistical software.
• An approximate level C confidence interval for is given by
Here, t* is the value for the t(k) density curve with area C between −t* and t*, where k is computed from the data by software or is the smaller of and . The quantity
is the margin of error.
• Significance tests for use the two-sample t statistic
The P-value is approximated using the distribution where k is estimated from the data using software or is the smaller of and .
• The guidelines for practical use of two-sample t procedures are similar to those for one-sample t procedures. Equal sample sizes are recommended.
• If we can assume that the two populations have equal variances, pooled two-sample t procedures can be used. These are based on the pooled estimator
of the unknown common variance and the distribution. We do not recommend this procedure for regular use.