ELEMENTARY STATISTICS 3 E William Navidi and Barry

ELEMENTARY STATISTICS 3 E William Navidi and Barry Monk ©Mc. Graw-Hill Education. All rights reserved. Authorized only for instructor use in the classroom. No reproduction or further distribution permitted without the prior written consent of Mc. Graw-Hill Education.

Basic Principles of Hypothesis Testing Section 9. 1 ©Mc. Graw-Hill Education.

Objectives 1. Define the null and alternate hypotheses 2. State conclusions to hypothesis tests 3. Distinguish between Type I and Type II errors ©Mc. Graw-Hill Education.

Objective 1 Define the null and alternate hypotheses ©Mc. Graw-Hill Education.

Null and Alternate Hypotheses A study published in the Journal of the Air and Waste Management Association reported that the mean amount of particulate matter (PM) produced by cars and light trucks in an urban setting is 35 milligrams of PM per mile of travel. Suppose that a new engine design is proposed that is intended to reduce the amount of PM in the air. There are two possible outcomes that could happen with the new engine design: either the new design will reduce the level of PM, or it will not. These possibilities are called hypotheses. One of the hypotheses is called the null hypothesis and the other is called the alternate hypothesis. ©Mc. Graw-Hill Education.

Null Hypothesis • ©Mc. Graw-Hill Education.

Alternate Hypothesis • ©Mc. Graw-Hill Education.

Example 1: Type of Test • ©Mc. Graw-Hill Education.

Example 2: Type of Test • ©Mc. Graw-Hill Education.

Example 3: Type of Test • ©Mc. Graw-Hill Education.

Purpose of a Hypothesis Test The purpose of a hypothesis test is to determine how plausible the null hypothesis is. The idea behind a hypothesis test is similar to a criminal trial. At the beginning of a trial, the defendant is assumed to be innocent. Then the evidence is presented. If the evidence strongly indicates that the defendant is guilty, we abandon the assumption of innocence and conclude the defendant is guilty. In a hypothesis test, the null hypothesis is like the defendant in a criminal trial. ©Mc. Graw-Hill Education.

How a Hypothesis Test is Conducted At the start of a hypothesis test, we assume that the null hypothesis is true. Then we look at the evidence, which comes from data that have been collected. If the data strongly indicate that the null hypothesis is false, we abandon our assumption that it is true and believe the alternate hypothesis instead. This is referred to as rejecting the null hypothesis. ©Mc. Graw-Hill Education.

Objective 2 State conclusions to hypothesis tests ©Mc. Graw-Hill Education.

Stating Conclusions • ©Mc. Graw-Hill Education.

Example 1: Stating Conclusions • ©Mc. Graw-Hill Education.

Example 2: Stating Conclusions • ©Mc. Graw-Hill Education.

Objective 3 Distinguish between Type I and Type II errors ©Mc. Graw-Hill Education.

Type I and Type II Errors • ©Mc. Graw-Hill Education.

Example 1: Types of Errors • ©Mc. Graw-Hill Education.

Example 2: Types of Errors • ©Mc. Graw-Hill Education.

Example 3: Types of Errors • ©Mc. Graw-Hill Education.

You Should Know. . . • How to write the null and alternate hypotheses • How to determine whether a hypothesis test is left-tailed, right-tailed, or two-tailed • How to state the conclusion to a hypothesis test • How to distinguish between Type I and Type II errors and correct decisions ©Mc. Graw-Hill Education.

ELEMENTARY STATISTICS 3 E William Navidi and Barry Monk ©Mc. Graw-Hill Education. All rights reserved. Authorized only for instructor use in the classroom. No reproduction or further distribution permitted without the prior written consent of Mc. Graw-Hill Education.

Section 9. 3 ©Mc. Graw-Hill Education.

Objectives 1. Test a hypothesis about a mean using the P-value method 2. Test a hypothesis about a mean using the critical value method ©Mc. Graw-Hill Education.

Objective 1* Test a hypothesis about a mean using the P-value method *(Tables) ©Mc. Graw-Hill Education.

Performing a Hypothesis Test • ©Mc. Graw-Hill Education.

Assumptions • ©Mc. Graw-Hill Education.

Performing a Hypothesis Test (Continued) • • • ©Mc. Graw-Hill Education. •

Computing P-Values • ©Mc. Graw-Hill Education.

Estimating the P-value From a Table • ©Mc. Graw-Hill Education.

P-value From a Table for a Two-tailed Test • ©Mc. Graw-Hill Education.

Example: Hypothesis Test (P-value, Table) • ©Mc. Graw-Hill Education.

Example: Hypothesis Test (P-value, Table, Continued 1) We first check the assumptions. Because the sample is small, the population must be approximately normal. We check this with a dotplot of the data. There is no evidence of strong skewness, and no outliers. We may proceed. • ©Mc. Graw-Hill Education.

Example: Hypothesis Test (P-value, Table, Continued 2) • ©Mc. Graw-Hill Education. •

Example: Hypothesis Test (P-value, Table, Continued 3) • ©Mc. Graw-Hill Education. •

Objective 1** Test a hypothesis about a mean using the P-value method **(TI-84 PLUS) ©Mc. Graw-Hill Education.

Performing a Hypothesis Test** • ©Mc. Graw-Hill Education.

Assumptions** • ©Mc. Graw-Hill Education.

Performing a Hypothesis Test (Continued)** • • • ©Mc. Graw-Hill Education. •

Hypothesis Testing on the TI-84 PLUS • ©Mc. Graw-Hill Education.

Example 1: Hypothesis Test (P-value, TI-84 PLUS) • ©Mc. Graw-Hill Education.

• Example 2: Hypothesis Test (P-value, TI-84 PLUS) ©Mc. Graw-Hill Education.

Example 2: Hypothesis Test (P-value, TI-84 PLUS, Continued 1) We first check the assumptions. Because the sample is small, the population must be approximately normal. We check this with a dotplot of the data. There is no evidence of strong skewness, and no outliers. We may proceed. • ©Mc. Graw-Hill Education.

Example 2: Hypothesis Test (P-value, TI-84 PLUS, Continued 2) We enter the data (2. 6, 3. 2, 2. 1, 3. 0, 3. 1, 2. 9, 3. 7) into list L 1. Press STAT and highlight the TESTS menu and select T-Test. • • ©Mc. Graw-Hill Education.

Objective 2 Test a hypothesis about a mean using the critical value method ©Mc. Graw-Hill Education.

• ©Mc. Graw-Hill Education.

Example: Hypothesis Test (Critical Value) • ©Mc. Graw-Hill Education.

Example: Hypothesis Test (Critical Value, Continued 1) • ©Mc. Graw-Hill Education.

Example: Hypothesis Test (Critical Value, Continued 2) • We conclude that the mean number of crashes is greater than six per year. ©Mc. Graw-Hill Education.

You Should Know. . . • ©Mc. Graw-Hill Education.

ELEMENTARY STATISTICS 3 E William Navidi and Barry Monk ©Mc. Graw-Hill Education. All rights reserved. Authorized only for instructor use in the classroom. No reproduction or further distribution permitted without the prior written consent of Mc. Graw-Hill Education.

Hypothesis Tests for a Population Proportion Section 9. 4 ©Mc. Graw-Hill Education.

Objectives 1. Test a hypothesis about a proportion using the P-value method 2. Test a hypothesis about a proportion using the critical value method ©Mc. Graw-Hill Education.

Objective 1 Test a hypothesis about a proportion using the P-value method ©Mc. Graw-Hill

Introduction Can virtual reality be used to enhance education? In a 2016 survey of teachers conducted by Samsung, 85% of them stated that using virtual reality in the classroom would have a positive effect on education. One educational specialist believes that the percentage has now increased to more than 90% since virtual reality equipment has become more available. She samples 500 teachers and finds that 471 of them believe that virtual reality would have a positive effect. Can she conclude that the proportion of teachers who believe that virtual reality would have a positive effect is greater than 90%? This is an example of a problem that calls for a hypothesis test about a population proportion. ©Mc. Graw-Hill Education.

P-Value Method • ©Mc. Graw-Hill Education.

Example: Hypothesis Test (P-value) • ©Mc. Graw-Hill Education.

Example: Hypothesis Test (P-value, Continued) • • ©Mc. Graw-Hill Education. •

Objective 1* Test a hypothesis about a proportion using the P-value method *(TI-84 PLUS)

Introduction* Can virtual reality be used to enhance education? In a 2016 survey of teachers conducted by Samsung, 85% of them stated that using virtual reality in the classroom would have a positive effect on education. One educational specialist believes that the percentage has now increased to more than 90% since virtual reality equipment has become more available. She samples 500 teachers and finds that 471 of them believe that virtual reality would have a positive effect. Can she conclude that the proportion of teachers who believe that virtual reality would have a positive effect is greater than 90%? This is an example of a problem that calls for a hypothesis test about a population proportion. ©Mc. Graw-Hill Education.

Example: Hypothesis Test (P-value, TI-84) • ©Mc. Graw-Hill Education.

Example: Hypothesis Test (P-value, TI-84, Continued) • ©Mc. Graw-Hill Education.

Objective 2 Test a hypothesis about a proportion using the critical value method ©Mc.

Critical Values Method • ©Mc. Graw-Hill Education.

Example: Hypothesis Test (Critical Value, Continued) • • ©Mc. Graw-Hill Education. •

You Should Know. . . • The notations used in performing a hypothesis test about a population proportion • The assumptions for performing a hypothesis test about a population proportion • How to perform a hypothesis test about a population proportion using the P-value method • How to perform a hypothesis test about a population proportion using the critical value method ©Mc. Graw-Hill Education.