# 5: Inference for Numerical Data

- Page ID
- 295

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

Chapter 4 introduced a framework for statistical inference based on con dence intervals and hypotheses. In this chapter, we encounter several new point estimates and scenarios. In each case, the inference ideas remain the same:

- Determine which point estimate or test statistic is useful.
- Identify an appropriate distribution for the point estimate or test statistic.
- Apply the ideas from Chapter 4 using the distribution from step 2.

Each section in Chapter 5 explores a new situation: the difference of two means (5.1, 5.2); a single mean or difference of means where we relax the minimum sample size condition (5.3, 5.4); and the comparison of means across multiple groups (5.5). Chapter 6 will introduce scenarios that highlight categorical data.

- 5.2: Paired Data
- Two sets of observations are paired if each observation in one set has a special correspondence or connection with exactly one observation in the other data set. To analyze paired data, it is often useful to look at the difference in outcomes of each pair of observations.

- 5.3: Difference of Two Means
- In this section we consider a difference in two population means, μ1−μ2, under the condition that the data are not paired. The methods are similar in theory but different in the details. Just as with a single sample, we identify conditions to ensure a point estimate of the difference is nearly normal. Next we introduce a formula for the standard error, which allows us to apply our general tools discussed previously.

- 5.4: Power Calculations for a Difference of Means (Special Topic)
- It is also useful to be able to compare two means for small samples. In this section we use the t distribution for the difference in sample means. We will again drop the minimum sample size condition and instead impose a strong condition on the distribution of the data.

- 5.5: Comparing many Means with ANOVA (Special Topic)
- In this section, we will learn a new method called analysis of variance (ANOVA) and a new test statistic called F.

- 5.6: Exercises
- Exercises for Chapter 5 of the "OpenIntro Statistics" textmap by Diez, Barr and Çetinkaya-Rundel.