Chapter 5 - Statistical Inference

formulating the competing hypotheses

1. Identify the relevant population parameter of interest. 2. Determine whether it is a one- or two-tailed test. 3. Include some form of the equality sign in the null hypothesis and use the alternative hypothesis to establish a claim.

parameter

A constant, although its value may be unknown

statistic

A variable whose value depends on the chosen random sample.

Construction of competing hypotheses

As a general guideline, we use the alternative hypothesis as a vehicle to establish something new—that is, contest the status quo. In most applications, the null hypothesis regarding a particular population parameter of interest is specified with one of the following signs: =, ≤, or ≥; the alternative hypothesis is then specified with the corresponding opposite sign: ≠, >, or <.

t distribution

If a random sample of size n is taken from a normal population with a finite variance, then the statistic follows the t df distribution, where df denotes degrees of freedom.

p-value

Is the likelihood of obtaining a sample mean that is at least as extreme as the one derived from the given sample, under the assumption that the null hypothesis is true as an equality

confidence interval / interval estimate

Provides a range of values that, with a certain level of confidence, contains the population parameter of interest.

estimator / point estimator

When a statistic is used to estimate a parameter

null hypothesis vs alternative hypothesis

When constructing a hypothesis test, we define a null hypothesis, denoted H₀, and an alternative hypothesis, denoted HA. We conduct a hypothesis test to determine whether or not sample evidence contradicts H₀.

margin of error

accounts for the standard error of the estimator and the desired confidence level of the interval.

Type II error

do not reject the null hypothesis when the null hypothesis is actually false

estimate

particular value of the estimator

Type I error

reject the null hypothesis when the null hypothesis is actually true

central limit theorem (CLT)

the sum or the average of a large number of independent observations from the same underlying distribution has an approximate normal distribution