Inference for Population Mean

Inference for the Mean of a Population

• $t$ procedures can be used when data comes from a random sample, the population has a Normal distribution, and the population is much larger than the sample.
• When the population standard deviation ($\sigma$) is unknown, the sample standard deviation ($s_x$) estimates it.
• This leads to the sampling distribution having a t distribution, not a Normal distribution.
• Degrees of freedom (df) specify a t distribution, calculated as $df = n - 1$.
• The t density curve is bell-shaped and symmetric around 0, but has a greater spread than the standard Normal curve, with more probability in the tails and less in the center.
• As degrees of freedom increase, the t density curve approaches the standard Normal curve.
• The standard error of the sample mean ($\bar{x}$) is $SE_{\bar{x}} = \frac{s_x}{\sqrt{n}}$.
• The one-sample t statistic is $t = \frac{\bar{x} - \mu}{\frac{s_x}{\sqrt{n}}}$, with $df = n - 1$.
• The one-sample t confidence interval for $\mu$ is $\bar{x} \pm t^* \frac{s_x}{\sqrt{n}}$, where $t^*$ is the critical value for the t distribution.
• The one-sample t test tests $H_0: \mu = \mu_0$ using the test statistic $t = \frac{\bar{x} - \mu_0}{\frac{s_x}{\sqrt{n}}}$, with $df = n - 1$.
• t procedures are sensitive to outliers.
• If $n \ge 30$, the t procedures can be used even for clearly skewed distributions.
• With $n < 15$, use t procedures only if the data appear close to Normal; if outliers are present or if the data are clearly non-Normal, do not use t.
• With $n \ge 15$, the t procedures can be used unless outliers or strong skewness are present.

Matched Pairs t Procedures

• Use one-sample t procedures to observed differences to compare responses to two treatments in a matched pairs design.

Comparing Two Means

• Two-sample t procedures can be used if data comes from two independent random samples, both populations have Normal distributions, and both populations are much larger than the samples.
• The two-sample t statistic is $t = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$.
• It has approximately a t distribution with degrees of freedom given by software or the smaller of $n_1 - 1$ and $n_2 - 1$.
• The two-sample t confidence interval for $\mu_1 - \mu_2$ is $(\bar{x}_1 - \bar{x}_2) \pm t^* \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$.
• Here, $t^*$ is the critical value for the t distribution with degrees of freedom given by software or the smaller of $n_1 - 1$ and $n_2 - 1$.
• The two-sample t test tests $H_0: \mu_1 = \mu_2$ using the test statistic $t = \frac{(\bar{x}_1 - \bar{x}_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$.
• Here, $t$ has approximately a t distribution with degrees of freedom given by software or the smaller of $n_1 - 1$ and $n_2 - 1$.
• Statistical software can be used to calculate the degrees of freedom.
• Two-sample t procedures are more robust than one-sample t procedures.
• Choose equal sample sizes when planning a two-sample study.
• The two-sample t procedures are not robust against outliers.
• If $n_1 + n_2 < 15$, use t procedures only if the data appear close to Normal; if outliers are present or if the data are clearly non-Normal, do not use t.
• If $n_1 + n_2 \ge 15$, the t procedures can be used except in the presence of outliers or strong skewness.
• Differing sample means and sample medians indicate non-Normality.

Inference for Population Standard Deviation

• When $n$ is small, the distribution of $s^2$ is strongly non-Normal.
• The statistic to test hypotheses about $\sigma^2$ is $\chi^2 = \frac{(n - 1)s^2}{\sigma^2}$.
• When the Normal condition is met, the statistic has a $\chi^2$ distribution with $n - 1$ degrees of freedom.
• The chi-square test tests $H_0: \sigma = \sigma_0$ using the test statistic $\chi^2 = \frac{(n - 1)s^2}{\sigma_0^2}$.
• Here, $\chi^2$ has a $\chi^2$ distribution with $df = n - 1$.
• Inference about $\sigma$ is much less robust than inference about $\mu$.
• The chi-square test is extremely sensitive to non-Normal distributions.
• Always plot the data to verify the Normality condition before using the chi-square test.
• A Normal probability plot can show how far the data are from Normal.