© Copyright McGraw-Hill 2004 CHAPTER 8 Hypothesis Testing © Copyright McGraw-Hill 2004

© Copyright McGraw-Hill 2004 Objectives Understand the definitions used in hypothesis testing. State the null and alternative hypotheses. Find critical values for the z test. State the five steps used in hypothesis testing. Test means for large samples using the z test. Test means for small samples using the t test. © Copyright McGraw-Hill 2004

© Copyright McGraw-Hill 2004 Objectives (cont’d.) Test proportions using the z test. Test variances or standard deviations using the chi square test. Test hypotheses using confidence intervals. Explain the relationship between type I and type II errors and the power of a test. © Copyright McGraw-Hill 2004

© Copyright McGraw-Hill 2004 Introduction Statistical hypothesis testing is a decision- making process for evaluating claims about a population. In hypothesis testing, the researcher must define the population under study, state the particular hypotheses that will be investigated, give the significance level, select a sample from the population, collect the data, perform the calculations required for the statistical test, and reach a conclusion. © Copyright McGraw-Hill 2004

© Copyright McGraw-Hill 2004 Hypothesis Testing Hypotheses concerning parameters such as means and proportions can be investigated. The z test and the t test are used for hypothesis testing concerning means. © Copyright McGraw-Hill 2004

Methods to Test Hypotheses The three methods used to test hypotheses are: 1. The traditional method. 2. The P-value method. 3. The confidence interval method. © Copyright McGraw-Hill 2004

Statement of a Hypothesis A statistical hypothesis is a conjecture about a population parameter which may or may not be true. There are two types of statistical hypotheses for each situation: the null hypothesis and the alternative hypothesis. ? ? ? ? © Copyright McGraw-Hill 2004

© Copyright McGraw-Hill 2004 Hypotheses The null hypothesis, symbolized by H0, is a statistical hypothesis that states that there is no difference between a parameter and a specific value, or that there is no difference between two parameters. The alternative hypothesis, symbolized by H1, is a statistical hypothesis that states the existence of a difference between a parameter and a specific value, or states that there is a difference between two parameters. © Copyright McGraw-Hill 2004

Hypothesis-Testing Common Phrases © Copyright McGraw-Hill 2004

© Copyright McGraw-Hill 2004 Design of the Study After stating the hypotheses, the researcher’s next step is to design the study. The researcher selects the correct statistical test, chooses an appropriate level of significance, and formulates a plan for conducting the study. © Copyright McGraw-Hill 2004

© Copyright McGraw-Hill 2004 Statistical Test A statistical test uses the data obtained from a sample to make a decision about whether or not the null hypothesis should be rejected. The numerical value obtained from a statistical test is called the test value. © Copyright McGraw-Hill 2004

Possible Outcomes of a Hypothesis Test H0 True H0 False Error Type I Error Type II Correct Decision Reject H0 Do not reject H0 © Copyright McGraw-Hill 2004

Summary of Possible Outcomes A type I error occurs if one rejects the null hypothesis when it is true. A type II error occurs if one does not reject the null hypothesis when it is false. © Copyright McGraw-Hill 2004

© Copyright McGraw-Hill 2004 Error Probabilities The level of significance is the maximum probability of committing a type I error. This probability is symbolized by ; that is , . The probability of a type II error is symbolized by . That is, . © Copyright McGraw-Hill 2004

© Copyright McGraw-Hill 2004  and  Probabilities In most hypothesis testing situations,  cannot easily be computed; however,  and  are related in that decreasing one increases the other. © Copyright McGraw-Hill 2004

© Copyright McGraw-Hill 2004 Hypothesis Testing In a hypothesis testing situation, the researcher decides what level of significance to use. After a significance level is chosen, a critical value is selected from a table for the appropriate test. © Copyright McGraw-Hill 2004

© Copyright McGraw-Hill 2004 Critical Values The critical value(s) separates the critical region from the noncritical region. The symbol for critical value is C.V. © Copyright McGraw-Hill 2004

Critical Values (cont’d) The critical or rejection region is the range of values of the test value that indicates that there is a significant difference and that the null hypothesis should be rejected. The noncritical or nonrejection region is the range of values of the test value that indicates that the difference was probably due to chance and that the null hypothesis should not be rejected. © Copyright McGraw-Hill 2004

© Copyright McGraw-Hill 2004 One-Tailed Test A one-tailed test indicates that the null hypothesis should be rejected when the test value is in the critical region on one side of the mean. A one-tailed test is either right-tailed or left- tailed, depending on the direction of the inequality of the alternative hypothesis. © Copyright McGraw-Hill 2004

© Copyright McGraw-Hill 2004 Left-Tailed Test © Copyright McGraw-Hill 2004

© Copyright McGraw-Hill 2004 Right-Tailed Test © Copyright McGraw-Hill 2004

© Copyright McGraw-Hill 2004 Two-Tailed Test In a two-tailed test, the null hypothesis should be rejected when the test value is in either of the two critical regions. © Copyright McGraw-Hill 2004

Two-Tailed Test (cont’d.) © Copyright McGraw-Hill 2004

Hypothesis-Testing (Traditional Method) Step 1 State the hypothesis, and identify the claim. Step 2 Find the critical value from the appropriate table. Step 3 Compute the test value. Step 4 Make the decision to reject or not reject the null hypothesis. Step 5 Summarize the results. © Copyright McGraw-Hill 2004

© Copyright McGraw-Hill 2004 The z Test The z test is a statistical test for the mean of a population. It can be used when , or when the population is normally distributed and  is known. © Copyright McGraw-Hill 2004

© Copyright McGraw-Hill 2004 The z Test Formula The formula for the z test is: © Copyright McGraw-Hill 2004

The z Test When  is Unknown The central limit theorem states that when the population standard deviation  is unknown, the sample standard deviation s can be used in the formula as long as the sample size is 30 or more. © Copyright McGraw-Hill 2004

© Copyright McGraw-Hill 2004 The P-value The P-value (or probability value) is the probability of getting a sample statistic (such as the mean) or a more extreme sample statistic in the direction of the alternative hypothesis when the null hypothesis is true. © Copyright McGraw-Hill 2004

© Copyright McGraw-Hill 2004 The P-value (cont’d.) The P-value is the actual area under the standard normal distribution curve (or other curve depending on what statistical test is being used) representing the probability of a particular sample statistic or a more extreme sample statistic occurring if the null hypothesis is true. © Copyright McGraw-Hill 2004

Decision Rule When Using a P-Value © Copyright McGraw-Hill 2004

Statistical vs. Practical Significance The researcher should distinguish between statistical significance and practical significance. When the null hypothesis is rejected at a specific significance level, it can be concluded that the difference is probably not due to chance and thus is statistically significant. However, the results may or may not have any practical significance. © Copyright McGraw-Hill 2004

© Copyright McGraw-Hill 2004 The t Test The t test is a statistical test of the mean of a population and is used when the population is normally or approximately normally distributed,  is unknown, and the sample size is less than 30. The formula for the t test is: The degrees of freedom are d.f. = n–1. © Copyright McGraw-Hill 2004

© Copyright McGraw-Hill 2004 z Test for a Proportion A hypothesis test involving a population proportion can be considered as a binomial experiment when there are only two outcomes and the probability of a success does not change from trial to trial. © Copyright McGraw-Hill 2004

Formula for the z Test for Proportions © Copyright McGraw-Hill 2004

Chi-Square Test for Single Variance The formula is: © Copyright McGraw-Hill 2004

Assumptions for Chi-Square Test for Single Variance The sample must be randomly selected from the population. The population must be normally distributed for the variable under study. The observations must be independent of each other. © Copyright McGraw-Hill 2004

Confidence Intervals & Hypothesis Testing There is a relationship between confidence intervals and hypothesis testing. When the null hypothesis is rejected in a hypothesis testing situation, the confidence interval for the mean using the same level of significance will not contain the hypothesized mean. Likewise, when the null hypothesis is not rejected, the confidence interval computed will contain the hypothesized mean. © Copyright McGraw-Hill 2004

Power of a Statistical Test The power of a test measures the sensitivity of the test to detect a real difference in parameters if one actually exists. The higher the power, the more sensitive the test is to detecting a real difference between parameters if there is a difference. © Copyright McGraw-Hill 2004

© Copyright McGraw-Hill 2004 Summary A statistical hypothesis is a conjecture about a population. There are two types of statistical hypotheses: the null hypothesis states that there is no difference, and the alternative hypothesis specifies a difference. © Copyright McGraw-Hill 2004

© Copyright McGraw-Hill 2004 Summary (cont’d.) The z test is used when the population standard deviation is known and the variable is normally distributed or when  is not known and the sample size is greater than or equal to 30. When the population standard deviation is not known and the variable is normally distributed, the sample standard deviation is used, but a t test should be conducted if the sample size is less than 30. © Copyright McGraw-Hill 2004

© Copyright McGraw-Hill 2004 Summary (cont’d.) Researchers compute a test value from the sample data in order to decide whether the null hypothesis should or should not be rejected. Statistical tests can be one-tailed or two- tailed, depending on the hypotheses. © Copyright McGraw-Hill 2004

© Copyright McGraw-Hill 2004 Summary (cont’d.) The null hypothesis is rejected when the difference between the population parameter and the sample statistic is said to be significant. The difference is significant when the test value falls in the critical region of the distribution. The critical region is determined by , the level of significance of the test. © Copyright McGraw-Hill 2004

© Copyright McGraw-Hill 2004 Summary (cont’d.) The significance level of a test is the probability of committing a type I error. A type I error occurs when the null hypothesis is rejected when it is true. The type II error can occur when the null hypothesis is not rejected when it is false. One can test a single variance by using a chi- square test. © Copyright McGraw-Hill 2004

© Copyright McGraw-Hill 2004 Conclusions Researchers are interested in answering many types of questions. For example: “Will a new drug lower blood pressure?” “Will seat belts reduce the severity of injuries caused by accidents?” These types of questions can be addressed through statistical hypothesis testing, which is a decision-making process for evaluating claims about a population. © Copyright McGraw-Hill 2004

© Copyright McGraw-Hill 2004 Homework Page 449 1,2,3,4,5,6,7 © Copyright McGraw-Hill 2004