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STAB22-LEC19-(20,21).docx

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Department
Statistics
Course
STAB22H3
Professor
Ken Butler
Semester
Fall

Description
LEC19 CHAPTER 20: TESTING HYPOTHESES ABOUT PROPORTIONS (COVERS 20,21) QUESTION: "A newsletter reported tha 90% of adults drink milk. However, a survey from a certain region found that 652 of 750 (86.93%) randomly selected adults drink milk. Is this evidence that the 90% figure is NOT accurate for this region?" ADDRESSING THIS QUESTION - diff. of 86.93% from survey vs. 90% from newsletter may be due to chance - ex. could have got signif. # of non-milk drinkers which caused % to go down ONE APPROACH: CONFIDENCE INTERVAL - find the standard error: = = 0.012308127 ~ 0.0123 - now, calculate confidence interval: a) 95% 95% CI: (0.8693) ± (1.96)(0.0123) = (0.845192, 0.893408) - we are 95% confident that the true prop. of milk drinkers is between about 84.5% to 89% b) 99% (0.8693) ± (1.645)(0.0123) = (0.838, 0.901) - we are 99% confident that the true prop. of milk drinkers is between about 84% and 90% - based on confidence interval results, it is most likely that % of milk drinkers is below 90% - do we believe this 90% figure or not? CI is one way to determine if its plausible What is another way to test the plausibility of this 90% figure? - hypothesis test [234] BETTER: HYPOTHESIS TESTING; THINK ABOUT LOGIC [OF HYPOTHESIS TESTING] FIRST BY ANALOGY This: is analogous to: Court of law analogy - H0 (null hypothesis): accused is innocent - HA (alternative hypothesis): accused is guilty - b/c we do not know for sure whether innocent/guilty, have to go thro. process in court to decide the individual's status. - analogous to having to go thro. statistical calculations to see if null model holds or not - if person is found not guilty, this does not mean they definitely did NOT do it, but they may have done it after all but it could not have been proven - analgous to failing to reject the H0, when in fact it was incorrect (2. TYPE II error: H0 is false, but is failed to be rejected) => diff. b/ween they didn't do it vs. u could not prove they did it - if person innocent and not guilty, then right decision made - analagous to making the correct decision of failing to reject H0 - always chance of making error - one error is to say that person is actually innocent, but found guilty (serious error; - analogous to this type of error: 1. TYPE I: H0 is true, but is mistakingly rejected) - in this legal sys, person is presumed innocent until proven guilty - analagous to assuming H0 is true, until it is proven that we should reject it. [240] HYPOTHESIS TESTING - H0 (Null hypothesis) - analagous to "presumption of innocence" - HA (Alternative hypothesis) - states that H0 is false - but to reject H0 in favour of HA, we req. sufficient evidence (data) - in hypo. testing, you ask: "what do I need evidence for"? What evidence is needed for is HA (EX) MILK EXAMPLE Recall: "A newsletter reported tha 90% of adults drink milk. However, a survey from a certain region found that 652 of 750 (86.93%) randomly selected adults drink milk. Is this evidence that the 90% figure is NOT accurate for this region?" ADDRESSING THE QUESTION WITH HYPOTHESIS TESTING - H0, HA - statements made, don't know if true/not - make them and see if true or not - H0: p = 0.90 - statement that ppl who drink is 90% - going to believe that newsletter about milk until we know better - HA: p ≠ 0.90 - do not say what p is, but say it is NOT 0.90 - saying H0 is wrong => trying to prove that 90% is not correct for this region => hypo. is either true or false alternative hypothesis =- what are we trying to prove - what are we trying to gather evidence in favour of) - prop. of ppl drinking milk is sth OTHER than 0.90 - have to give evidence to backup what is meant by evidence? - data [241] HOW TO ASSESS WHETHER WE BELIEVE H0? 1. ASSESS EVIDENCE - EVIDENCE = DATA - specifically, the SAMPLE PROPORTION p P-value = probab. of being as far/further from H0 than the value observed (Milk example) - H0: p = 0.90 - p = 652/750 = 0.8693 - null model is given by Normal model distribution - ex. mean = 0.90 (p), SD =? - SD = \sqrt{(0.90)(0.10)/750} = 0.0110 - the 'p' we are using the hypothesized value of the H0 - p = 0.90 - this is NOT the true prop. of the popn - now, use SD and mean to get test statistic (z) z = (0.8693 - 0.9) / (0.0110) = -2.79 This corresponds to P(z ≤ -2.79) = 0.0026 below - 0.0026*2 = P-value = 0.0052, b/c we could have observed p to be above 90% as well - so this P-value is telling us that the probability of observing a proportion of about 87% being milk drinkers, given that null model is true, is about 0.52% Implication - what is done is that we use info from sample (n = 750, p = 0.90, p = 0.87) to see if we believe H0, or not P-value is evidence against H0 - b/c it is so small for this ex, it is saying that it is highly unlikely to have observed the val we got for p given that H0 is true - calculation showing that if 0.9 is correct, then observing 0.8693, even though its only 3% pts below, is pretty unlikely - so P is v.small => strong evidence against null, so h0 is false beyond reasonable doubt => it is very unlikely that, if the null model is true, we would get a proportion of about 87% being milk drinkers. We do not believe that this was a fluke, after all, we gathered a lot of evidence to come to this percentage. So, instead of believing that, we reject H0 in favour of HA. [242] "BEYOND A REASONABLE DOUBT" (Milk example) - P-value = 0.0052 - is this "beyond a reasonable doubt"? - P-value that we got says that value of p we got from data is v.unlikely if H0: p = 0.90 is true - So either: a) we observed sth v.unlikely (it was a fluke) b) H0: p = 0.90 is not true at all - When P-value is signifiantly small (like in this case), we prefer to believe b) => we reject H0 in favour of HA: p ≠ 0.90 Suppose we got P-value = 0.7789 - saying that there is about a 78% likelihood of getting data we got, given that null model is true - this P-value is not small at all - says that if H0 were true, then what is observed is entirely consistent w/ that - it could have been observed given H0 is true, so no reason to doubt it NOTE: - when have large P val, have NOT proved that H0 is true, b/c that is what we started w/ having assumed - instead we say we could not prove that it wasn't false - opp. of reject is acept - not accepting H0, b/c don't believe it is true - rather we could not prove that it was wrong though - If - P is small, then reject H0 in favour of HA - P is large, then fail to reject H0 - this does NOT prove that H0 true, rather all it declares is that we cannot reject H0. [243] ONE-SIDED AND TWO-SIDED TESTS [!] On final, if we are asked to do problems with these tests, we will only be doing two-sided tests. However, the professor still wants us to understand why one-sided test might be useful. (Milk example) - HA: p ≠ 0.90 - this is a two-sided test b/c val's ofar above or far below 0.90 can make us reject H0 - suppose we got p = 0.92 - we compute z-score for this, using p = 0.90, SD = 0.0110, and we get 1.82 - this z-score corresponds to prop. of 0.9656 less - but, we need the prop of the two tails of normal model: - for purpose of this course, were focusing on 2-SIDED TESTS One-sided tests (Milk example) - we might be looking for evidence that p is SMALLER than 0.90 - ie. - H0: p = 0.90 - HA: p < 0.90 HOW TO GET P-value? 1. check if p we retrieve is within the interval of the HA - ex. p = 0.92 is NOT less than 0.90, so immediately we fail to reject H0 - ex. p = 0.8693 IS less than 0.90, so we proceed 2. - calculate z-score for p (ex. using p = 0.8693) - P-value is probab. less - for p = 0.8693, z = -2.79, and we get P = 0.0026 - DO NOT MULTIPLY BY 2 NOTE ABOUT ONE-SIDED TEST (can skip 1-sided test portion) - just recog. when one-sided might be useful, but WILL NOT BE ASKED TO ACTUALLY DO IT [245] ONE-SIDED TEST (EX2) - H0: p = 0.90 - HA: p > 0.90 HOW TO GET P-value? 1. check if p we retrieve is within the interval of the HA - ex. p = 0.8693 is NOT greater than 0.90, so immediately we fail to reject H0 - ex. p = 0.92 IS greater than 0.90, so we proceed 2. - calculate z-score for p (ex. using p = 0.92) - P-value is probab. greater - for p = 0.92, z = 1.82, P = 1 - 0.9656 = 0.0344 (remember: we want the tail ends of the normal distribution) - DO NOT MULTIPLY BY 2 [246] WHEN DOING A HYPOTHESIS TEST 1. ALWAYS STATE P-value - allows reader to draw their own conclusion about truthfulness or falsity of H0 2. FOLLOW UP REJECTION OF H0 WITH CONFIDENCE INTERVAL - allows reader to get idea of size of parameter (effect size) => get an idea of whether result is important vs. merely statistically sig
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