9.2: What Is a Test of Significance? Statistics Chap 9:Introduction to Inference

Types of Statistical Inference Confidence Intervals: Used to estimate a population parameter. Significance Tests: Used to assess the evidence provided by data about some claim concerning a population.

Example: Shooting free throws The basketball team at State University claims that each of their players makes 80% of his free throws. How do we test the claim? How do we decide whether the claim is true or false? An outcome that is very unlikely if a claim is true is good evidence that the claim is not true.

Example: Is the coffee fresh? It is commonly assumed that people prefer freshly-brewed coffee to instant coffee. A skeptic claims that only half of all coffee drinkers prefer fresh-brewed coffee. (i.e., coffee drinkers are indifferent regarding fresh-brewed vs instant coffee) In an experiment, 50 subjects taste two unmarked cups of coffee and indicate their preference. One cup contains instant coffee; the other, fresh-brewed coffee. The sample statistic is the proportion of subjects who state they prefer fresh-brewed coffee.

Example: Is the coffee fresh?

Hypothesis Testing The purpose of hypothesis testing is not to question the computed value of the sample statistic but to make a judgement about the difference between that sample statistic and a hypothesized population parameter.

Significance Test

Example 9.7: Count Buffon’s coin

Significance Level

Basic Idea: An outcome that would rarely happen if the null hypothesis were true is good evidence that the null hypothesis is not true.

Significance Level The higher the significance level we use for hypothesis testing, the higher the probability of rejecting a null hypothesis when it is true. Type I error:Rejecting a null hypothesis when it is true. Type II error:Accepting a null hypothesis when it is false.

Example A company is evaluating the promotability of its employees. The HR director tells the president that roughly 80% of the employees are “promotable.” The president assembles a special committee to assess the promotability of all employees. This committee conducts in-depth interviews with 150 employees and finds that in its judgment only 70% of the sample are qualified for promotion. The president wants to test at the 0.05 significance level the hypothesis that 0.8 of the employees are promotable.

Example

A member of a public interest group concerned with environmental pollution asserts at a public hearing that “fewer than 60% of the industrial plants in the area are complying with air-pollution standards.” Attending this meeting is an official of the EPA who believes that 60% of the plants are complying with the standards. She decides to test the hypothesis at the 0.02 significance level. The official makes a thorough search of the records in her office. She samples 60 plants from a population of over 10,000 plants and finds that 33 are complying with air- pollution standards.

Example

Exercises p 513: 9.19 p 521: 9.25, 9.27

Exercises p : 9.29,

Activity 9.2: Roll the dice p 510