1STA 291 - Lecture 22 1STA 291Lecture 22• Chapter 11 Testing Hypothesis – Concepts of Hypothesis TestingSTA 291 - Lecture 22 2• Bonus Homework, due in the lab April 20-22: Essay “How would you test the ‘hot hand’ theory in basketball games?” (~400-600 words / approximately one typed page)• Be as specific as you can: what data to collect? how many cases to collect? What hypothesis you are testing?STA 291 - Lecture 22 3Significance Tests• A significance test checks whether data agrees with a (null) hypothesis• A hypothesis is a statement about a characteristic of a population parameter or parameters• If the data is very unreasonable under the hypothesis, then we will rejectthe hypothesis• Usually, we try to find evidence against the hypothesis2STA 291 - Lecture 22 4Logical Procedure1. State a (null) hypothesis that you would like to find evidence against2. Get data and calculate a statistic (for example: sample proportion)3. The hypothesis (and CLT) determines the sampling distribution of our statistic4. If the calculated value in 2. is very unreasonable given 3 (i.e. almost impossible), then we conclude that the hypothesis was wrongSTA 291 - Lecture 22 5Example 1• Somebody makes the claim that “Nicotine Patch and Zyban has same effect on quitting smoke”• You don’t believe it. So you conduct the experiment and collect data: Patch: 244 subjects; 52 quit. Zyban: 244 subjects; 85 quit.• How (un)likely is this under the hypothesis of no difference?• The sampling distribution helps us quantify the (un)likeliness in terms of a probability (p-value)Example 2• Mr. Basketball was an 82% free throw shooter last season. This season so far in 59 free throws he only hit 40.• (null) Hypothesis: He is still an 82% shooter• alternative hypothesis: his percentage has changed. (not 82% anymore)STA 291 - Lecture 22 63Question:• How unlikely are we going to see 52/244 verses 85/244 if indeed Patch and Zyban are equally effective? (Probability = ?)• How unlikely for an 82% shooter to hit only 40 out of 59? ( Probability = ?)STA 291 - Lecture 22 7How small is too small?• A small probability imply very unlikely or impossible. (No clear cut, but Prob less than 0.01 is certainly small)• A larger probability imply this is likely and no surprise. (again, no clear boundary, but prob. > 0.1 is certainly not small)STA 291 - Lecture 22 8• For the Basketball data, we actually got Probability = 0.0045• For the Patch vs. Zyban data, we actually got Probability = 0.0013STA 291 - Lecture 22 94Usually we pick an alpha level• Suppose we pick alpha = 0.05, then Any probability below 0.05 is deemed “impossible” so this is evidence against the null hypothesis – we say that “we reject the null hypothesis”• Otherwise, we say “we cannot reject the null hypothesis” imply there is not enough• Evidence against the null hypothesisSTA 291 - Lecture 22 10• Notice “not enough evidence against null hypothesis” is different from • “validated the null hypothesis”, “accept null hypothesis”, • It could mean there is simply not enough data to reach any conclusion.STA 291 - Lecture 22 11• If the basketball data were 14 hits out of 20 shoots (14/20 = 0.7), the P-value would be 0.16247.• This probability is not small.• Usually we cut off ( that’s the alpha level) at 0.05 or 0.01 for P-valuesSTA 291 - Lecture 22 125STA 291 - Lecture 22 13Significance Test• A significance test is a way of statistically testing a hypothesis by comparing the data to values predicted by the hypothesis• Data that fall far from the predicted values provide evidence against the hypothesisSTA 291 - Lecture 22 14Elements of a Significance Test• Assumptions (about population dist.)• Hypotheses(about popu. Parameter. null and alternative)• Test Statistic (based on a SRS.)• P-value (a way of summarizing the strength of evidence.)• Conclusion (reject, or not reject, that is the question)STA 291 - Lecture 22 15Assumptions• What type of data do we have? – Qualitative or quantitative? – Different types of data require different test procedures– If we are comparing 2 population means, then how the SD differ?• What is the population distribution? – Is it normal? Or is it binomial?– Some tests require normal population distributions (t-test)6Assumptions-cont.• Which sampling method has been used?– We usually assume Simple Random Sampling• What is the sample size?– Some methods require a minimum sample size (like n >30)because of using CLTSTA 291 - Lecture 22 16STA 291 - Lecture 22 17Assumptions in the Example1• What type of data do we have? – Qualitative with two categories: Either “quit smoke” or “not quit smoke” • What is the population distribution? – It is Bernoulli type. It is definitely not normal since it can only take two values• Which sampling method has been used?– We assume simple random sampling• What is the sample size?– n=244STA 291 - Lecture 22 18Hypotheses• Hypotheses are statements about population parameter.• The null hypothesis (H0) is the hypothesis that we test (and try to find evidence against)• The name null hypothesis refers to the fact that it often (not always) is a hypothesis of “no effect” (no effect of a medical treatment, no difference in characteristics of populations, etc.)7• The alternative hypothesis (H1) is a hypothesis that contradicts the null hypothesis• When we reject the null hypothesis, we are in favor of the alternative hypothesis.• Often, the alternative hypothesis is the actual research hypothesis that we would like to “prove” by finding evidence against the null hypothesis (proof by contradiction) STA 291 - Lecture 22 19STA 291 - Lecture 22 20Hypotheses in the Example 1• Null hypothesis (H0):The percentage of quitting smoke with Patch and Zyban are the sameH0: Prop(patch) = Prop(zyban)• Alternative hypothesis (H1):The two proportions differ STA 291 - Lecture 22 21Hypotheses in the Example 2• Null hypothesis (H0):The percentage of free throw for Mr. Basketball is still 82% H0: Prop = 0.82• Alternative hypothesis (H1):The proportion differs from 0.828STA 291 - Lecture 22 22Test Statistic• The test statistic is a statistic that is calculated from the sample data • Formula will be given for test statistic, but you need to chose the right one.STA 291 - Lecture 22 23Test Statistic in the Example 2• Test statistic: