1. Lecture 16 Section 8.1 Objectives: Testing Statistical Hypotheses − Stating hypotheses statements − Type I and II errors − Conducting a hypothesis test

2. Hypotheses and test procedures While confidence intervals are appropriate when our goal is to estimate a population parameter, Tests of hypotheses are used to assess the amount evidence the data has in favor or against some claim about the population. The hypothesis is a statement about the parameter in the population. A test of hypotheses is a method for using sample data to decide between the two competing hypotheses under consideration. We initially assume that one of the hypotheses, the null hypothesis, is correct; this is the “prior belief” claim. We then consider the evidence (sample data), and we reject the null hypothesis in favor of the competing claim, called the alternative hypothesis, only if there is convincing evidence against the null hypothesis.

3. Reasoning of Significance Tests We have seen that the properties of the sampling distribution of help us estimate a range of likely values for population mean . We can also rely on the properties of the sample distribution to test hypotheses. Example: You are in charge of quality control in your food company. You sample randomly four packs of cherry tomatoes, each labeled 1/2 lb. (227 g). The average weight from your four boxes is 222 g. Obviously, we cannot expect boxes filled with whole tomatoes to all weigh exactly half a pound. Thus,  Is the somewhat smaller weight simply due to chance variation?  Is it evidence that the calibrating machine that sorts cherry tomatoes into packs needs revision?

4. Stating hypotheses A test of statistical significance tests a specific hypothesis using sample data to decide on the validity of the hypothesis. In statistics, a hypothesis is an assumption or a theory about the characteristics of one or more variables in one or more populations. What you want to know: Does the calibrating machine that sorts cherry tomatoes into packs need revision? The same question reframed statistically: Is the population mean µ for the distribution of weights of cherry tomato packages equal to 227 g (i.e., half a pound)?

5. The null hypothesis is a very specific statement about a parameter of the population(s). It is labeled H 0. The alternative hypothesis is a more general statement about a parameter of the population(s) that is exclusive of the null hypothesis. It is labeled H a. Weight of cherry tomato packs: H 0 : µ = 227 g (µ is the average weight of the population of packs) H a : µ ≠ 227 g (µ is either larger or smaller)

6. One-sided and two-sided tests  A two-tail or two-sided test of the population mean has these null and alternative hypotheses: H 0 : µ = [a specific number] H a : µ  [a specific number]  A one-tail or one-sided test of a population mean has these null and alternative hypotheses: H 0 : µ = [a specific number] H a : µ < [a specific number] OR H 0 : µ = [a specific number] H a : µ > [a specific number] The FDA tests whether a generic drug has an absorption extent similar to the known absorption extent of the brand-name drug it is copying. Higher or lower absorption would both be problematic, thus we test: H 0 : µ generic = µ brand H a : µ generic  µ brand two-sided

7. Type I and II errors  A Type I error is made when we reject the null hypothesis and the null hypothesis is actually true (incorrectly reject a true H 0 ). The probability of making a Type I error is the significance level   A Type II error is made when we fail to reject the null hypothesis and the null hypothesis is false (incorrectly keep a false H 0 ). The probability of making a Type II error is labeled . The power of a test is 1 − .

8. Running a test of significance is a balancing act between the chance α of making a Type I error and the chance  of making a Type II error. Reducing α reduces the power of a test and thus increases . It might be tempting to emphasize greater power (the more the better).  However, with “too much power” trivial effects become highly significant.  A type II error is not definitive since a failure to reject the null hypothesis does not imply that the null hypothesis is wrong.

9. Remarks on Type I and II Errors  Ideally, we wish to have a test making both α and β as small as possible. However, as α decreases, β increases.  In a test of hypotheses, we control the significance level α to be employed in the test. Common significance levels are α = 0.10 or 0.05 or 0.01. A test with α=0.01 means that if H0 is actually true and the test procedure used repeatedly on samples, in the long run H0 would be incorrectly rejected only 1% of the time.  Then which value of significance level should be employed? If a type I error is much more serious than a type II error, a very small value of α is reasonable. When a type II error could have quite unpleasant consequences, it is better to use a larger α. But if there is no significant difference in the effects of these two errors, researchers often choose α=0.05.

10. Hypotheses and test procedures Steps for conducting a hypothesis test 1. Set up the null and alternative hypotheses 2. Calculate the test statistic 3. Calculate the p-value 4. Make a conclusion in terms of the problem Test statistic Test statistic is computed assuming that the null hypothesis is true. So, a test statistic measures compatibility between the null hypothesis and the data.

11. The P-value The packaging process has a known standard deviation  = 5 g. H 0 : µ = 227 g versus H a : µ ≠ 227 g The average weight from your four random boxes is 222 g. What is the probability of drawing a random sample such as yours if H 0 is true? Tests of statistical significance quantify the chance of obtaining a particular random sample result if the null hypothesis were true. This quantity is the P-value. This is a way of assessing the “believability” of the null hypothesis, given the evidence provided by a random sample.

12. Interpreting a P-value Could random variation alone account for the difference between the null hypothesis and observations from a random sample?  A small P-value implies that random variation due to the sampling process alone is not likely to account for the observed difference.  With a small p-value we reject H 0. The true property of the population is significantly different from what was stated in H 0. Thus, small P-values are strong evidence AGAINST H 0. But how small is small…?

13. Decision Rule What do you conclude from the test? We can compare the P-value with the desired significance level α. If P-value ≤ α then we reject H0 at significance level α. If P-value > α then we fail to reject H0 at significance level α.

14. Stating hypotheses A consumer advocate is interested in evaluating the claim that a new granola cereal contains “4 ounces of cashews in every bag.” The advocate recognizes that the amount of cashews will vary slightly from bag to bag, but she suspects that the mean amount of cashews per bag is less than 4 ounces. To check the claim, the advocate purchases a random sample of 40 bags of cereal and calculates a sample mean of 3.68 ounces of cashews. What alternative hypothesis does she want to test? a) b) c) d) e)

15. Statistical significance We reject the null hypothesis whenever a) P-value > . b) P-value  . c) P-value  . d) P-value  .

16. Statistical significance The significance level is denoted by a)  b)  c)  d) P-value

17. Calculating P-values To calculate the P-value for a significance test, we need to use information about the a) Sample distribution b) Population distribution c) Sampling distribution of

18. Conclusions Suppose the P-value for a hypothesis test is 0.304. Using  = 0.05, what is the appropriate conclusion? a) Reject the null hypothesis. b) Reject the alternative hypothesis. c) Do not reject the null hypothesis. d) Do not reject the alternative hypothesis.

19. Conclusions Suppose the P-value for a hypothesis test is 0.0304. Using  = 0.05, what is the appropriate conclusion? a) Reject the null hypothesis. b) Reject the alternative hypothesis. c) Do not reject the null hypothesis. d) Do not reject the alternative hypothesis.

20. Stating hypotheses If we test H 0 :  = 40 vs. H a :  < 40, this test is a) One-sided (left tail). b) One-sided (right tail). c) Two-sided.

21. Stating hypotheses If we test H 0 :  = 40 vs. H a :   40, this test is a) One-sided (left tail). b) One-sided (right tail). c) Two-sided.