## Figure 3.1: A two-tail hypothesis testing for in the example

Marx analyzes the *how* of money in the theory of the value-form and the *why* of money in the theory of the fetish-character, whereas in the theory of the exchange process he examines the question of *through what*. Near the end of Chapter 2, as the final consideration of money prior to Chapter 3 (where he presents the theory of money proper), Marx writes: “The difficulty lies, not in comprehending that money is a commodity, but in discovering how, why, and through what [*wie*, *warum*, *wodurch*] a commodity is money” (Marx, 1976a, p. 186). Marx’s indication of these three difficulties clearly suggests that he managed to brilliantly overcome them, but no hint is provided regarding where this is carried out. My view is that Marx answered the questions *how*, *why*, and *through what *in Section 3 and 4 of Chapter 1 and in Chapter 2, respectively. In other words, the three problems are listed in the order that he solves them in *Capital*.

It is impossible to give an exhaustive list ofsuch testing functionality, but we hope not only to provide severalexamples but also to elucidate some of the logic of statistical hypothesistests with these examples.**Datasets and other files used in this tutorial:****CRAN contributed packages used in this tutorial:**

In the previoustutorial, we used exploratory techniques to identify 92 stars from thedata set that are associated with the Hyades.

## The right-tail test shown in Figure gives the -value .

A left-tail test is not different from the right-tail one, but of course the rejection region is to the left of . For example, if we are interested to determine if is less than 15, we place this conjecture in the alternative hypothesis: . The novelty here is how we use the function to calculate : instead of , we need to use . Figure illustrates this example, where the rejection region is, remember, to the left of .

## The P Value Approach to hypothesis Testing

Here we look at some examples of calculating the power of a test. Theexamples are for both normal and t distributions. We assume that youcan enter data and know the commands associated with basicprobability. All of the examples here are for a two sided test, andyou can adjust them accordingly for a one sided test.

## Calculating p-value for 1-tailed test in a linear model

Figure shows the predicted levels of the dependent variable and its 95% confidence band for the sample values of the variable . In more complex functional forms, the function plots the partial effect of a variable for given levels of the other independent variables in the model. The simplest possible call of this function requires, as arguments, the name of the term for which we wish the partial effect (in our example ), and the object (model) in question. If not otherwise specified, the confidence intervals (band) are determined for a 95% confidence level and the other variables at their means. A simple use of the package is presented in Figure , which plots the partial effects of all variables in the basic model. (Function plots only one graph for one variable, while plots all variables in the model.)

## p-value <- pt(t, df) Right-tail test ((H_{A ..

When you perform a hypothesis test in statistics, a -value helps you determine the significance of your results. are used to test the validity of a claim that is made about a population. This claim that’s on trial, in essence, is called the

## Power is the probability of rejecting the null hypothesis

**Definition: **Assuming that the null hypothesis is true, the p value isthe probability of obtaining a sample mean as extreme or more extreme than thesample mean actually obtained.