1. Copyright © 2014 Pearson Education. All rights reserved. 9.1-1 9.1 Fundamentals of Hypothesis Testing LEARNING GOAL Understand the goal of hypothesis testing and the basic structure of a hypothesis test, including how to set up the null and alternative hypotheses, how to determine the possible outcomes of a hypothesis test, and how to decide between these possible outcomes.

2. Copyright © 2014 Pearson Education. All rights reserved. 9.1-2 Slide 9.1- 2 A company called ProCare Industries, Ltd. once claimed that its product, called Gender Choice, could increase a woman’s chance of giving birth to a baby girl. How could we test whether the Gender Choice claim is true? One way would be to study a random sample of, say, 100 babies born to women who used the Gender Choice product. If the product does not work, we would expect about half of these babies to be girls. If it does work, we would expect significantly more than half of the babies to be girls. The key question, then, is what constitutes “significantly more.”

3. Copyright © 2014 Pearson Education. All rights reserved. 9.1-3 Slide 9.1- 3 In statistics, we answer such questions through hypothesis testing. Consider a sample in which there are 64 girls among 100 babies born to women who used the product. We begin by assuming that Gender Choice does not work; that is, it does not increase the percentage of girls. If this is true, then we should expect about 50% girls among the population of all births to women using the product. That is, the proportion of girls is equal to 0.50. We now use our sample (with 64 girls among 100 births) to test the above assumption. We conduct this test by calculating the likelihood of drawing a random sample of 100 births in which 64% (or more) of the babies are girls (from a population in which the overall proportion of girls is only 50%).

4. Copyright © 2014 Pearson Education. All rights reserved. 9.1-4 Slide 9.1- 4 If we find that a random sample of births is fairly likely to have 64% (or more) girls, then we do not have evidence that Gender Choice works. However, if we find that a random sample of births is unlikely to have at least 64% girls, then we conclude that the result in the Gender Choice sample is probably due to something other than chance—meaning that the product may be effective. Note, however, that even in this case we will not have proved that the product is effective, as there could still be other explanations for the result (such as that we happened to choose an unusual sample or that there were confounding variables we did not account for).

5. Copyright © 2014 Pearson Education. All rights reserved. 9.1-5 Slide 9.1- 5 TIME OUT TO THINK Suppose Gender Choice is effective and you select a new random sample of 1,000 babies born to women who used the product. Based on the first sample (with 64 girls among 100 births), how many baby girls would you expect in the new sample? Next, suppose Gender Choice is not effective; how many baby girls would you expect in a random sample of 1,000 babies born to women who used the product?

6. Copyright © 2014 Pearson Education. All rights reserved. 9.1-6 Slide 9.1- 6 Formulating the Hypotheses Definitions A hypothesis is a claim about a population parameter, such as a population proportion (p) or population mean (  ). A hypothesis test is a standard procedure for testing a claim about a population parameter.

7. Copyright © 2014 Pearson Education. All rights reserved. 9.1-7 Slide 9.1- 7 There are always at least two hypotheses in any hypothesis test. In the Gender Choice case, the two hypotheses are essentially: (1) that Gender Choice works in raising the population proportion of girls from the 50% that we would normally expect and (2) that Gender Choice does not work in raising the population proportion of girls.

8. Copyright © 2014 Pearson Education. All rights reserved. 9.1-8 Slide 9.1- 8 The null hypothesis is the claim that Gender Choice does not work, in which case the population proportion of girls born to women who use the product should be 50%, or 0.50. Using p to represent the population proportion, we write this null hypothesis as The alternative hypothesis is the claim that Gender Choice does work, in which case the population proportion of girls born to women who use the product should be greater than 0.50. We write this alternative hypothesis as H 0 (null hypothesis): p = 0.50 H a (alternative hypothesis): p > 0.50

9. Copyright © 2014 Pearson Education. All rights reserved. 9.1-9 Slide 9.1- 9 In this chapter, the null hypothesis will always include the condition of equality, just as the null hypothesis for the Gender Choice example is the equality p = 0.50. However, the alternative hypothesis for the Gender Choice case (p > 0.50) is only one of three general forms of alternative hypotheses that we will encounter in this chapter: population parameter < claimed value population parameter > claimed value population parameter ≠ claimed value

10. Copyright © 2014 Pearson Education. All rights reserved. 9.1-10 Slide 9.1- 10 It is useful to give names to the three different types of alternative hypotheses. The first form (“less than”) leads to what is called a left- tailed hypothesis test, because it requires testing whether the population parameter lies to the left (lower values) of the claimed value. Similarly, the second form (“greater than”) leads to a right- tailed hypothesis test, because it requires testing whether the population parameter lies to the right (higher values) of the claimed value. The third form (“not equal to”) leads to a two-tailed hypothesis test, because it requires testing whether the population parameter lies significantly far to either side of the claimed value.

11. Copyright © 2014 Pearson Education. All rights reserved. 9.1-11 Slide 9.1- 11 Null and Alternative Hypotheses The null hypothesis, or H 0, is the starting assumption for a hypothesis test. For the types of hypothesis tests in this chapter, the null hypothesis always claims a specific value for a population parameter and therefore takes the form of an equality: H 0 (null hypothesis): population parameter = claimed value The alternative hypothesis, or H a, is a claim that the population parameter has a value that differs from the value claimed in the null hypothesis. It may take one of the following forms: (left-tailed) H a : population parameter < claimed value (right-tailed) H a : population parameter > claimed value (two-tailed) H a : population parameter ≠ claimed value

12. Copyright © 2014 Pearson Education. All rights reserved. 9.1-12 Slide 9.1- 12 In each case, identify the population parameter about which a claim is made, state the null and alternative hypotheses for a hypothesis test, and indicate whether the hypothesis test will be left-tailed, right-tailed, or two-tailed. EXAMPLE 1 Identifying Hypotheses a. The manufacturer of a new model of hybrid car advertises that the mean fuel consumption is equal to 62 miles per gallon on the highway. A consumer group claims that the mean is less than 62 miles per gallon.

13. Copyright © 2014 Pearson Education. All rights reserved. 9.1-13 Slide 9.1- 13 Possible Outcomes of a Hypothesis Test Two Possible Outcomes of a Hypothesis Test 1.Reject the null hypothesis, H 0, in which case we have evidence in support of the alternative hypothesis. 2.Not reject the null hypothesis, H 0, in which case we do not have enough evidence to support the alternative hypothesis. There are two possible outcomes to a hypothesis test:

14. Copyright © 2014 Pearson Education. All rights reserved. 9.1-14 Slide 9.1- 14 Notice that “accepting the null hypothesis” is not a possible outcome, because the null hypothesis is always the starting assumption. The hypothesis test may not give us reason to reject this starting assumption, but it cannot by itself give us reason to conclude that the starting assumption is true. The fact that there are only two possible outcomes makes it extremely important that the null and alternative hypotheses be chosen in an unbiased way. In particular, both hypotheses must always be formulated before a sample is drawn from the population for testing. Otherwise, data from the sample might inappropriately bias the selection of the hypotheses to test.

15. Copyright © 2014 Pearson Education. All rights reserved. 9.1-15 Slide 9.1- 15 TIME OUT TO THINK The idea that a hypothesis test cannot lead us to accept the null hypothesis is an example of the old dictum that “absence of evidence is not evidence of absence.” As an illustration of this idea, explain why it would be easy in principle to prove that some legendary animal (such as Bigfoot or the Loch Ness Monster) exists, but it is nearly impossible to prove that it does not exist.

16. Copyright © 2014 Pearson Education. All rights reserved. 9.1-16 Slide 9.1- 16 For each of the three cases from Example 1 (slides 12-17), describe the possible outcomes of a hypothesis test and how we would interpret these outcomes. EXAMPLE 2 Hypothesis Test Outcomes

17. Copyright © 2014 Pearson Education. All rights reserved. 9.1-17 Slide 9.1- 17 Drawing a Conclusion from a Hypothesis Test Statistical Significance If the probability of a particular result is 0.05 or less, we say that the result is statistically significant at the 0.05 level; if the probability is 0.01 or less, the result is statistically significant at the 0.01 level. The 0.01 level therefore means stronger significance than the 0.05 level. We will look at two ways to make the decision about rejecting or not rejecting the null hypothesis.

18. Copyright © 2014 Pearson Education. All rights reserved. 9.1-18 Slide 9.1- 18 Hypothesis Test Decisions Based on Levels of Statistical Significance If the chance of a sample result at least as extreme as the observed result is less than 1 in 100 (or 0.01), then the test is statistically significant at the 0.01 level and offers strong evidence for rejecting the null hypothesis. If the chance of a sample result at least as extreme as the observed result is less than 1 in 20 (or 0.05), then the test is statistically significant at the 0.05 level and offers moderate evidence for rejecting the null hypothesis. We decide the outcome of a hypothesis test by comparing the actual sample result (mean or proportion) to the result expected if the null hypothesis is true. We must choose a significance level for the decision.

19. Copyright © 2014 Pearson Education. All rights reserved. 9.1-19 Slide 9.1- 19 Hypothesis Test Decisions Based on Levels of Statistical Significance (cont.) If the chance of a sample result at least as extreme as the observed result is greater than the chosen level of significance (0.05 or 0.01), then we do not reject the null hypothesis.

20. Copyright © 2014 Pearson Education. All rights reserved. 9.1-20 Slide 9.1- 20 Consider the Gender Choice example in which the sample size is n = 100 and the sample proportion is p = 0.64. Using techniques that we will discuss later in this chapter, it is possible to calculate the probability of randomly choosing such a sample (or a more extreme sample with p > 0.64) under the assumption that the null hypothesis (p = 0.50) is true; the result is that the probability is 0.0026. Based on this result, should you reject or not reject the null hypothesis? EXAMPLE 3 Statistical Significance in Hypothesis Testing pˆ pˆ pˆ

21. Copyright © 2014 Pearson Education. All rights reserved. 9.1-21 Slide 9.1- 21 Drawing a Conclusion from a Hypothesis Test Statistical Significance P-Values P-value is short for probability value; notice the capital P, used to avoid confusion with the lowercase p that stands for a population proportion. We will discuss the actual calculation of P-values in Sections 9.2 and 9.3; here, we focus only on their interpretation.

22. Copyright © 2014 Pearson Education. All rights reserved. 9.1-22 Slide 9.1- 22 Hypothesis Test Decisions Based on P-Values A small P-value (such as less than or equal to 0.05) indicates that the sample result is unlikely, and therefore provides reason to reject the null hypothesis. A large P-value (such as greater than 0.05) indicates that the sample result could easily occur by chance, so we cannot reject the null hypothesis The P-value (probability value) for a hypothesis test of a claim about a population parameter is the probability of selecting a sample at least as extreme as the observed sample, assuming that the null hypothesis is true:

23. Copyright © 2014 Pearson Education. All rights reserved. 9.1-23 Slide 9.1- 23 You suspect that a coin may have a bias toward landing tails more often than heads, and decide to test this suspicion by tossing the coin 100 times. The result is that you get 40 heads (and 60 tails). A calculation (not shown here) indicates that the probability of getting 40 or fewer heads in 100 tosses with a fair coin is 0.0228. Find the P-value and level of statistical significance for your result. Should you conclude that the coin is biased against heads? EXAMPLE 4 Fair Coin?

24. Copyright © 2014 Pearson Education. All rights reserved. 9.1-24 Slide 9.1- 24 Putting It All Together The Hypothesis Test Process Step 1. Formulate the null and alternative hypotheses, each of which must make a claim about a population parameter, such as a population mean (μ) or a population proportion (p); be sure this is done before drawing a sample or collecting data. Based on the form of the alternative hypothesis, decide whether you will need a left-, right-, or two- tailed hypothesis test. Step 2. Draw a sample from the population and measure the sample statistics, including the sample size (n) and the relevant sample statistic, such as the sample mean (x) or sample proportion (p). ˆ p ¯ x

25. Copyright © 2014 Pearson Education. All rights reserved. 9.1-25 Slide 9.1- 25 The Hypothesis Test Process (cont.) Step 3. Determine the likelihood of observing a sample statistic (mean or proportion) at least as extreme as the one you found under the assumption that the null hypothesis is true. The precise probability of such an observation is the P-value (probability value) for your sample result. Step 4. Decide whether to reject or not reject the null hypothesis, based on your chosen level of significance (usually 0.05 or 0.01, but other significance levels are sometimes used).

26. Copyright © 2014 Pearson Education. All rights reserved. 9.1-26 Slide 9.1- 26 A lowercase p represents a population proportion—that is, the true proportion among a complete population. A lowercase p (“p-hat”) represents a sample proportion—that is, the proportion found in a sample drawn from a population. An uppercase P stands for probability in a P-value. Again, be sure to avoid confusion among the three different uses of the letter p: pˆ

27. Copyright © 2014 Pearson Education. All rights reserved. 9.1-27 Slide 9.1- 27 In the United States, the average car is driven about 12,000 miles each year. The owner of a large rental car company suspects that for his fleet, the mean distance is greater than 12,000 miles each year. He selects a random sample of n = 225 cars from his fleet and finds that the mean annual mileage for this sample is miles. A calculation shows that if you assume the fleet mean is the national mean of 12,000 miles, then the probability of selecting a 225-car sample with a mean annual mileage of at least 12,375 miles is 0.01. Based on these data, describe the process of conducting a hypothesis test and drawing a conclusion. EXAMPLE 5 Mean Rental Car Mileage x = 12,375 ¯ x

28. Copyright © 2014 Pearson Education. All rights reserved. 9.1-28 Slide 9.1- 28 A Legal Analogy of Hypothesis Testing In American courts of law, the fundamental principle is that a defendant is presumed innocent until proven guilty. Because the starting assumption is innocence, this represents the null hypothesis: The job of the prosecutor is to present evidence so compelling that the jury is persuaded to reject the null hypothesis and find the defendant guilty. If the prosecutor does not make a sufficiently compelling case, then the jury will not reject H 0 and the defendant will be found “not guilty.” H 0 : The defendant is innocent. H a : The defendant is guilty.

29. Copyright © 2014 Pearson Education. All rights reserved. 9.1-29 Slide 9.1- 29 TIME OUT TO THINK Consider two situations. In one, you are a juror in a case in which the defendant could be fined a maximum of $2,000. In the other, you are a juror in a case in which the defendant could receive the death penalty. Compare the significance levels you would use in the two situations. In each situation, what are the consequences of wrongly rejecting the null hypothesis?.

30. Copyright © 2014 Pearson Education. All rights reserved. 9.1-30 Slide 9.1- 30 The End