Chapter 6.3

1. Chapter 6.3 Hypothesis Tests

2. Idea • Suppose data has been collected to show that a particular effect exists, and the data exhibits a pattern that suggests the effect dose exist. Could the pattern have been produced by variation, without the effect being present at all? • Yes -> then data are conventionally regarded as, at best, weak statistical evidence in favor of the existence of the effect • No -> explanations for the pattern exhibited by data without the presence of the effect is not available, in another word, the data is conventionally regarded as strong statistical evidence in favor of the existence of the effect.

3. An example • Researchers have postulate that because of differences in diet, Japanese children have a lower mean blood cholesterol level than U.S. children. It is known that the mean level for U.S. children is 170. We take 40 Japanese children randomly and their blood cholesterol are: • 157.212 149.967 135.686 161.626 148.623 119.150 182.806 …………………………….. 170.880 153.276 Descriptive Statistics Variable N Mean Median TrMean StDev SE Mean C1 40 155.96 153.75 155.90 16.48 2.61 Variable Minimum Maximum Q1 Q3 C1 119.15 192.72 144.98 168.68

4. FORMAL STRUCTURE • Hypothesis Tests are based on an “presumed innocent until proven guilty” form of argument. • Specifically, we make an assumption and then attempt to show that assumption leads to an absurdity or contradiction, hence the assumption is wrong.

5. FORMAL STRUCTURE • The null hypothesis, denoted H0 is a statement or claim about a population characteristic that is initially assumed to be true. • The alternate hypothesis, denoted by Ha is the competing claim.It is a statement about the same population characteristic that is used in the null hypothesis.

6. FORMAL STRUCTURE • Rejection of the null hypothesis will imply the acceptance of this alternative hypothesis. • Assume H0 is true and attempt to show this leads to an absurdity, hence H0 is false and Ha is true. • Reject H0 Equivalent to saying that Ha is correct or true 2. Fail to reject H0 Equivalent to saying that we have failed to show a statistically significant deviation from the claim of the null hypothesis

7. AN ANALOGY • The Statistical Hypothesis Testing process can be compared very closely with a judicial trial. • Assume a defendant is innocent (H0) • Present evidence to show guilt • Try to prove guilt beyond a reasonable doubt(Ha) • Two Hypotheses are then created. • H0: Innocent • Ha: Not Innocent (Guilt)

8. Case study: By chance alone, the subjects should be able to identify 50% of the placements correctly, and Emily's hypothesis was that the proportion of correct answers would be 0.50. H0:  =0.5 Ha: 0.5 (Two-sided alternative) Examples of Hypotheses

9. The students entering into the math program used to have a mean SAT quantitative score of 525. Are the current students poorer (as measured by the SAT quantitative score)? H0:  =525 (Really:   525) Ha:  < 525 (One-sided alternative) Examples of Hypotheses

10. Do the “16 ounce” cans of peaches canned and sold by DelMonte meet the claim on the label (on the average)? Notice, the real concern would be selling the consumer less than 16 ounces of peaches. H0:  =16 (Really:   16) Ha: < 16 Examples of Hypotheses

11. Is the proportion of defective parts produced by a manufacturing process more than 5%? H0: p =0.05 (Really, p 0.05) Ha: p > 0.05 Examples of Hypotheses

12. One-sided Two-sided Hypothesis Form • The form of the null hypothesis is H0: population characteristic = hypothesized value • Where the hypothesized value is a specific number determined by the problem context. • The alternative (or alternate) hypothesis will have one of the following three forms: Ha: population characteristic > hypothesized value Ha: population characteristic < hypothesized value Ha: population characteristic  hypothesized value

13. Comments on Hypothesis Form • The null hypothesis must contain the equal sign. This is absolutely necessary because the test requires the null hypothesis to be assumed to be true and the value attached to the equal sign is then the value assumed to be true. • The alternate hypothesis should be what you are really attempting to show to be true. This is not always possible.

14. Caution • When you set up a hypothesis test, the result is either • Strong support for the alternate hypothesis (if the null hypothesis is rejected) • There is not sufficient evidence to refute the claim of the null hypothesis (you are stuck with it, but there is only a lack of strong evidence against the null hypothesis.Never accept null hypothesis!

15. Error Analogy • Consider a medical test where the hypotheses are equivalent to H0: the patient has a specific disease Ha: the patient doesn’t have the disease • Then, Type I error is equivalent to a false negative (I.e., Saying the patient does not have the disease when in fact, he does.) Type II error is equivalent to a false positive (I.e., Saying the patient has the disease when, in fact, he does not.)

16. Error level of significance

17. Comment of Process • Look at the consequences of type I and type II errors and then identify the largest a that is tolerable for the problem. • Employ a test procedure that uses this maximum acceptable value of a (rather than anything smaller) as the level of significance (because using a smaller a increases b).

18. P-value A test statistic is the function of sample data on which a conclusion to reject or fail to reject H0 is based. • The P-value is the probability of a result as extreme as the observed test statistic, assuming that H0 is true. Remark:The P-value (also called the observed significance level) is a measure of inconsistency between the hypothesized value for a population characteristic and the observed sample.

19. Decision Criteria • A decision as to whether H0 should be rejected results from comparing the P-value to the chosen : • H0 should be rejected if P-value a. • H0 should not be rejected if P-value > a.

20. Steps in a Hypothesis-Testing Analysis • Describe (determine) the population characteristic about which hypotheses are to be tested. • State the null hypothesis H0. • State the alternate hypothesis Ha. • Select the significance level a for the test. • Check to make sure that any assumptions required for the test are reasonable. • Display the test statistic to be used, with substitution of the hypothesized value identified in step 2. And compute the test statistic. • Determine the P-value associated with the observed value of the test statistic based on Ha • State the conclusion in the context of the problem, including the level of significance.