Section 7.1 Summary

• Significance tests and confidence intervals for the mean $\mu$ of a Normal population are based on the sample mean $\overline{x}$ of an SRS. Because of the central limit theorem, the resulting procedures are approximately correct for other population distributions when the sample is large.

• The standard error of the sample mean is

  ${\text{SE}}_{\overline{x}}={\displaystyle \frac{s}{\sqrt{n}}}$

• The standardized sample mean, or one-sample $z$ statistic,

  $z={\displaystyle \frac{\overline{x}-\mu }{\sigma /\sqrt{n}}}$

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  has the $N\left(0,1\right)$ distribution. If the standard deviation $\sigma /\sqrt{n}$ of $\overline{x}$ is replaced by the standard error ${\text{SE}}_{\overline{x}}=s/\sqrt{n}$, the one-sample $t$ statistic

  $t={\displaystyle \frac{\overline{x}-\mu }{s/\sqrt{n}}}$

  has the $t$ distribution with $n-1$ degrees of freedom.

• There is a $t$ distribution for every positive degrees of freedom $k$. All are symmetric distributions similar in shape to Normal distributions. The $t\left(k\right)$ distribution approaches the $N\left(0,1\right)$ distribution as $k$ increases.

• The margin of error for level $C$ confidence is

  $m=t^{*}\times {\text{SE}}_{\overline{x}}=t^{*}{\displaystyle \frac{s}{\sqrt{n}}}$

  where $t^{*}$ is the value for the $t\left(n-1\right)$ density curve with area $C$ between $-t^{*}$ and $t^{*}$.

• A level $C$ confidence interval for the mean $\mu$ of a Normal population is

  $\overline{x}\pm m$

• Significance tests for $H_0:\mu ={\mu }_0$ are based on the one-sample $t$ statistic. $P$-values or fixed significance levels are computed from the $t\left(n-1\right)$ distribution.
• A matched pairs analysis is needed when subjects or experimental units are matched in pairs or when there are two measurements on each individual or experimental unit and the question of interest concerns the difference between the two measurements.
• These one-sample procedures are used to analyze matched pairs data by first taking the differences within the matched pairs to produce a single sample.
• One-sample equivalence testing assesses whether a population mean $\mu$ is practically different from a hypothesized mean ${\mu }_0$. This test requires a threshold $\delta$, which represents the largest difference between $\mu$ and ${\mu }_0$ such that the means are considered equivalent.
• The $t$ procedures are relatively robust against lack of Normality, especially for larger sample sizes. The $t$ procedures are useful for non-Normal data when $n\ge 15$ unless the data show outliers or strong skewness.