Chapter 7: Inference for Means

SECTION 7.2 SUMMARY

• Significance tests and confidence intervals for the difference between the means $μ_{1}$ and $μ_{2}$ of two Normal populations are based on the difference ${\bar{x}}_{1} - {\bar{x}}_{2}$ 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 $n_{1}$ and $n_{2}$ are drawn from two Normal populations with parameters $μ_{1}$ , $σ_{1}$ and $μ_{2}$ , $σ_{2}$ the two-sample z statistic

$z = \frac{({\bar{x}}_{1} - {\bar{x}}_{2}) - (μ_{1} - μ_{2})}{\sqrt{\frac{σ_{1}^{2}}{n_{1}} + \frac{σ_{2}^{2}}{n_{2}}}}$

has the N(0, 1) distribution.

• The two-sample t statistic

$t = \frac{({\bar{x}}_{1} - {\bar{x}}_{2}) - (μ_{1} - μ_{2})}{\sqrt{\frac{s_{1}^{2}}{n_{1}} + \frac{s_{2}^{2}}{n_{2}}}}$

does not have a t distribution. However, good approximations are available.

• Conservative inference procedures for comparing $μ_{1}$ and $μ_{2}$ are obtained from the two-sample t statistic by using the $t (k)$ distribution with degrees of freedom k equal to the smaller of $n_{1} - 1$ and $n_{2} - 1$ .

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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 $μ_{1} - μ_{2}$ is given by

$({\bar{x}}_{1} - {\bar{x}}_{2}) \pm t * \sqrt{\frac{s_{1}^{2}}{n_{1}} + \frac{s_{2}^{2}}{n_{2}}}$

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 $n_{1} - 1$ and $n_{2} - 1$ . The quantity

$t * \sqrt{\frac{s_{1}^{2}}{n_{1}} + \frac{s_{2}^{2}}{n_{2}}}$

is the margin of error.

• Significance tests for $H_{0} : μ_{1} - μ_{2} = Δ_{0}$ use the two-sample t statistic

$t = \frac{({\bar{x}}_{1} - {\bar{x}}_{2}) - Δ_{0}}{\sqrt{\frac{s_{1}^{2}}{n_{1}} + \frac{s_{2}^{2}}{n_{2}}}}$

The P-value is approximated using the $t (k)$ distribution where k is estimated from the data using software or is the smaller of $n_{1} - 1$ and $n_{2} - 1$ .

• 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

$s_{p}^{2} = \frac{(n_{1} - 1) s_{1}^{2} + (n_{2} - 1) s_{2}^{2}}{n_{1} + n_{2} - 2}$

of the unknown common variance and the $t (n_{1} + n_{2} - 2)$ distribution. We do not recommend this procedure for regular use.