Copyright © 2012 by Nelson Education Limited. Chapter 7 Hypothesis Testing I: The One-Sample Case 7-1

Copyright © 2012 by Nelson Education Limited. The basic logic of hypothesis testing –Hypothesis testing for single sample means –The Five-Step Model Other material covered: –One- vs. Two- tailed tests –Type I vs. Type II error –Student’s t distribution –Hypothesis testing for single sample proportions In this presentation you will learn about: 7-2

Copyright © 2012 by Nelson Education Limited. Hypothesis testing is designed to detect significant differences: differences that did not occur by random chance. Hypothesis testing is significance testing. This chapter focuses on the “one sample” case: we compare a random sample (from a large group) against a population. –Specifically, we compare a sample statistic to a population parameter to see if there is a significant difference. Hypothesis Testing 7-3

Copyright © 2012 by Nelson Education Limited. The engineering department at a university has been accused of “grade inflation” so engineering majors have much higher GPAs than students in general. Hypothesis Testing for Single Sample Means: An Example 7-4

Copyright © 2012 by Nelson Education Limited. GPAs of all engineering majors should be compared with the GPAs of all students. –There are 1000s of engineering majors, far too many to interview. –How can the dispute be investigated without interviewing all engineering majors?  The value of the parameter, average GPA for all students, is 2.70 (μ = 2.70), with a standard deviation of 0.70 (σ =.70). Hypothesis Testing for Single Sample Means: An Example (continued) 7-5

Copyright © 2012 by Nelson Education Limited. For a random sample of 117 engineering majors, = 3.00 There is a difference between the parameter (2.70) and the statistic (3.00). –It seems that engineering majors do have higher GPAs. However, we are working with a random sample (not all engineering majors). –The observed difference may have been caused by random chance. Hypothesis Testing for Single Sample Means: An Example (continued) 7-6

Copyright © 2012 by Nelson Education Limited. Hence, there are two explanations, or hypotheses, for the difference: 1.The sample mean (3.00) is the same as the pop. mean (2.70). –The difference is trivial and caused by random chance. 2.The difference is real (significant). –engineering majors are different from all students. Hypothesis Testing for Single Sample Means: An Example (continued) 7-7

Copyright © 2012 by Nelson Education Limited. Formally, we can state the two hypotheses as: Null Hypothesis (H 0 ) “The difference is caused by random chance.” The H 0 always states there is “no significant difference.” Hypothesis Testing for Single Sample Means: An Example (continued) 7-8

Copyright © 2012 by Nelson Education Limited. OR Alternative hypothesis (H 1 ) “The difference is real”. The H 1 always contradicts the H 0. Hypothesis Testing for Single Sample Means: An Example (continued) 7-9

Copyright © 2012 by Nelson Education Limited. One (and only one) of these explanations must be true. Which one? –We assume the H 0 is true. We ask, “What is the probability of getting the sample mean of 3.00 if the H 0 is true and all engineering majors really have a mean of 2.70?” Hypothesis Testing for Single Sample Means: An Example (continued) 7-10

Copyright © 2012 by Nelson Education Limited. We use the Z score formula, and Appendix A to determine the probability of getting the observed difference. Hypothesis Testing for Single Sample Means: An Example (continued) 7-11

Copyright © 2012 by Nelson Education Limited. –By convention, we use the.05 value as a guideline to identify differences that would be rare if H 0 is true. If the probability is less than.05, the calculated or “obtained” Z score will be beyond ±1.96 or the “critical” Z score. Hypothesis Testing for Single Sample Means: An Example (continued) 7-12

Copyright © 2012 by Nelson Education Limited. So, substituting the values into Formula 7.1, we calculate a Z score of 4.6: Z = = 4.6 Hypothesis Testing for Single Sample Means: An Example (continued) 7-13

Copyright © 2012 by Nelson Education Limited. Z (obtained) of +4.6 is beyond Z (critical) of ±1.96: ◦ A difference this large would be rare if H 0 is true. ◦ We reject H 0. »This difference is significant. »The GPA of engineering majors is significantly different from the GPA of the general student body. Hypothesis Testing for Single Sample Means: An Example (continued) 7-14

Copyright © 2012 by Nelson Education Limited. All the elements used in the example above can be formally organized into a five-step model: 1.Making assumptions and meeting test requirements. 2.Stating the null hypothesis. 3.Selecting the sampling distribution and establishing the critical region. 4.Computing the test statistic. 5.Making a decision and interpreting the results of the test. Hypothesis Testing: The Five Step Model 7-15

Copyright © 2012 by Nelson Education Limited. Random sampling –Hypothesis testing assumes samples were selected according to EPSEM. –The sample of 117 was randomly selected from all engineering majors. Level of Measurement is Interval-Ratio (I-R) –GPA is I-R so the mean is an appropriate statistic. Step 1: Make Assumptions and Meet Test Requirements 7-16

Copyright © 2012 by Nelson Education Limited. Sampling Distribution is normal in shape –This assumption is satisfied by using a large sample (n>100). (See the Central Limit Theorem in Chapter 5). Step 1: Make Assumptions and Meet Test Requirements (continued) 7-17

Copyright © 2012 by Nelson Education Limited. H 0 : μ = 2.7 –The sample of 117 comes from a population that has a GPA of 2.7. –The difference between 2.7 and 3.0 is trivial and caused by random chance. H 1 : μ≠2.7 –The sample of 117 comes a population that does not have a GPA of 2.7. –The difference between 2.7 and 3.0 reflects an actual difference between engineering majors and other students Step 2: State the Null Hypothesis 7-18

Copyright © 2012 by Nelson Education Limited. Sampling Distribution= Z –Alpha (α) =.05 –α is the indicator of “rare” events. –Any difference with a probability less than α is rare and will cause us to reject the H 0. Critical Region begins at –This is the critical Z score associated with α =.05, two-tailed test. –If the obtained Z score falls in the C.R., reject the H 0. Step 3: Select Sampling Distribution and Establish the Critical Region 7-19

Copyright © 2012 by Nelson Education Limited.  Z (obtained) = = 4.6 Step 4: Compute the Test Statistic 7-20

Copyright © 2012 by Nelson Education Limited. Step 5: Make Decision and Interpret Results 7-21

Copyright © 2012 by Nelson Education Limited. The obtained Z score fell in the C.R., so we reject the H 0. –If the H 0 were true, a sample outcome of 3.00 would be unlikely. –Therefore, the H 0 is false and must be rejected. Engineering majors have a GPA that is significantly different from the general student body. Step 5: Make Decision and Interpret Results (continued) 7-22

Copyright © 2012 by Nelson Education Limited. In hypothesis testing, we try to identify statistically significant differences that did not occur by random chance. –In this example, we rejected the H 0 and concluded that the difference was significant. The difference between the parameter 2.70 and the statistic 3.00 was large and unlikely to have occurred by random chance. It is very likely that Engineering majors have GPAs higher than the general student body. The Five Step Model: Summary 7-23

Copyright © 2012 by Nelson Education Limited. Model is fairly rigid, but there are two crucial choices: 1.One-tailed or two-tailed test (Section 7.4) 2.Alpha (α) level (Section 7.5) Crucial Choices in the Five Step Model 7-24

Copyright © 2012 by Nelson Education Limited. Two-tailed: States that population mean is “not equal” to value stated in null hypothesis. Example: H 1 : μ≠2.7, where ≠ means “not equal to” (Note, the GPA example illustrated above was a two-tailed test). One- and Two-Tailed Hypothesis Tests 7-25

Copyright © 2012 by Nelson Education Limited. One-tailed: Differences in a specific direction. Example: H 1 : μ>2.7, where > signifies “greater than” or H 1 : μ<2.7, where < signifies “less than” One- and Two-Tailed Hypothesis Tests (continued) 7-26

Copyright © 2012 by Nelson Education Limited. ‣ Tails affect Critical Region in Step 3: One- and Two-Tailed Hypothesis Tests (continued) 7-27

Copyright © 2012 by Nelson Education Limited. By assigning an alpha level, α, one defines an “unlikely” sample outcome. Alpha level is the probability that the decision to reject the null hypothesis, H 0, is incorrect. –Incorrectly rejecting a true null hypothesis: Type I or alpha error Alpha Levels 7-28

Copyright © 2012 by Nelson Education Limited. Alpha levels affect Critical Region in Step 3: Alpha Levels (continued) 7-29

Copyright © 2012 by Nelson Education Limited. Type I, or alpha error: –Rejecting a true null hypothesis Type II, or beta error: –Failing to reject a false null hypothesis Alpha Levels (continued) 7-30

Copyright © 2012 by Nelson Education Limited. There is a relationship between decision making and error: Alpha Levels (continued) 7-31

Copyright © 2012 by Nelson Education Limited. How can we test a hypothesis when the population standard deviation (σ ) is not known, as is usually the case? –For large samples (n>100), we use s as an estimator of σ and use standard normal distribution (Z scores), suitably corrected for the bias (n is replaced by n-1 to correct for the fact that s is a biased estimator of σ. The Student’s t distribution 7-32

Copyright © 2012 by Nelson Education Limited. –For small samples (n<100), s is too unreliable an estimator of σ so do not use standard normal distribution. Instead we use Student’s t distribution. The Student’s t distribution 7-33

Copyright © 2012 by Nelson Education Limited.  A similar formula as for Z (obtained) is used in hypothesis testing: The Student’s t distribution (continued) 7-34

Copyright © 2012 by Nelson Education Limited.  The logic of the five-step model for hypothesis testing is followed.  However in testing hypothesis we use the t table (Appendix B), not the Z table (Appendix A).  The t table differs from the Z table in the following ways: 1.Column at left for degrees of freedom (df) (df = n – 1) 2.Alpha levels along top two rows: one- and two-tailed 3.Entries in table are actual scores: t (critical) (i.e., mark beginning of critical region, not areas under the curve). The Student’s t distribution (continued) 7-35

Copyright © 2012 by Nelson Education Limited. Hypothesis Testing for Single- Sample Means: Summary 7-36

Copyright © 2012 by Nelson Education Limited. We can also use Z (obtained) to test sample proportions, as long as sample size is large (n >100). However, the symbols change because we are basing the test on sample proportions (rather than sample means): *Small-sample tests of hypothesis for proportions are not considered in this text. Test of Hypothesis for Single-Sample Proportions (for Large Samples)* 7-37

Copyright © 2012 by Nelson Education Limited. where P s = sample proportion P u = population proportion = the standard deviation of the sampling distribution of sample proportions  The logic of the five-step model for hypothesis testing is followed. Test of Hypothesis for Single-Sample Proportions (for Large Samples) (continued) 7-38