Stat 31, Section 1, Last Time

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## Stat 31, Section 1, Last Time

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**Stat 31, Section 1, Last Time**• Statistical Inference • Confidence Intervals: • Range of Values to reflect uncertainty • Bracket true value in 95% of repetitions • Choice of sample size • Choose n to get desired error • Hypothesis Testing • Yes – No questions, under uncertainty**Reading In Textbook**Approximate Reading for Today’s Material: Pages 400-416, 425-428 Approximate Reading for Next Class: Pages 431-439, 450-471**Hypothesis Tests**E.g. A fast food chain currently brings in profits of $20,000 per store, per day. A new menu is proposed. Would it be more profitable? Test: Have 10 stores (randomly selected!) try the new menu, let = average of their daily profits.**Hypothesis Testing**Note: Can never make a definite conclusion, Instead measure strength of evidence. Reason: have to deal with uncertainty But: Can quantify uncertainty**Hypothesis Testing**Approach I: (note: different from text) Choose among 3 Hypotheses: H+: Strong evidence new menu is better H0: Evidence in inconclusive H-: Strong evidence new menu is worse**Caution!!!**• Not following text right now • This part of course can be slippery • I am “breaking this down to basics” • Easier to understand (If you pay careful attention) • Will “tie things together” later • And return to textbook approach later**Fast Food Business Example**Base decision on best guess: Will quantify strength of the evidence using probability distribution of E.g. Choose H+ Choose H0 Choose H-**Fast Food Business Example**How to draw line? (There are many ways, here is traditional approach) Insist that H+ (or H-) show strong evidence I.e. They get burden of proof (Note: one way of solving gray area problem)**Fast Food Business Example**Suppose observe: , based on Note , but is this conclusive? or could this be due to natural sampling variation? (i.e. do we risk losing money from new menu?)**Fast Food Business Example**Assess evidence for H+ by: H+ p-value = Area**Fast Food Business Example**Computation in EXCEL: Class Example 22, Part 1: http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg24.xls P-value = 0.094 i.e. About 10% Is this “small”? (where do we draw the line?)**Fast Food Business Example**View 1: Even under H0, just by chance, see values like , about 10% of the time, • i.e. 1 in 10, • so not “terribly convincing”??? • Could be a “fluke”? But where is the boundary line?**P-value cutoffs**View 2: Traditional (and even “legal”) cutoff, called here the yes-no cutoff: Say evidence is strong, when P-value < 0.05 • Just a commonly agreed upon value, but very widely used: • Drug testing • Publication of scientific papers**P-value cutoffs**• Say “results are statistically significant” when this happens, i.e. P-value < 0.05 • Can change cutoff value 0.05, to some other level, often called Greek “alpha” E.g. your airplane safe to fly, want E.g. often called strongly significant**P-value cutoffs**View 3: Personal idea about cutoff, called gray level (vs. yes-no above) P-value < 0.01: “quite strong evidence” 0.01 < P-value < 0.1: “weaker evidence but stronger for smaller P-val.” 0.1 < P-value: “very weak evidence, at best”**Gray Level Cutoffs**View 3: gray level (vs. yes-no above) Note: only about interpretation of P-value E.g.: When P-value is given: HW: 6.40 & (d) give gray level interp. (no, no, relatively weak evidence) 6.41 & (d) give gray level interp. (yes, not, moderately strong evidence)**Caution!!!**• Gray level viewpoint not in text • Will see it is more sensible • Hence I teach this • Suggest you use this later in life • Will be on HW & exams**Fast Food Business Example**P-value of 0.094 for H+, Is “quite weak evidence for H+”, i.e. “only a mild suggestion” This happens sometimes: not enough information in data for firm conclusion**Fast Food Business Example**Flip side: could also look at “strength of evidence for H-”. Expect: very weak, since saw Quantification: H- P-value = $20,000 $21,000**Fast Food Business Example**EXCEL Computation: Class Example 24, part 1 http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg24.xls H- P-value = 0.906 >> ½, so no evidence at all for H- (makes sense)**Fast Food Business Example**A practical issue: Since , May want to gather more data… Could prove new menu clearly better (since more data means more information, which could overcome uncertainty)**Fast Food Business Example**Suppose this was done, i.e. n = 10 is replaced by n = 40, and got the same: Expect: 4 times the data ½ of the SD Impact on P-value? Class Example 24, Part 2 http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg24.xls**Fast Food Business Example**How did it get so small, with only ½ the SD? mean = $20,000, observed $21,000 P-value = 0.094 P-value = 0.004**Hypothesis Testing**HW: C20 For each of the problems: • A box label claims that on average boxes contain 40 oz. A random sample of 12 boxes shows on average 39 oz., with s = 2.2. Should we dispute the claim?**Hypothesis Testing**• We know from long experience that Farmer A’s pigs average 570 lbs. A sample of 16 pigs from Farmer B averages 590 lbs, with an SD of 110. Is it safe to say B’s pigs are heavier on average? • Same as (b) except “lighter on average”. • Same as (b) except that B’s average is 630 lbs.**Hypothesis Testing**Do: • Define the population mean of interest. • Formulate H+, H0, and H-, in terms of mu. • Give the P-values for both H+ and H-. (a. 0.942, 0.058, b. 0.234, 0.766, c. 0.234, 0.766, d. 0.015, 0.985) • Give a yes-no answer to the questions. (a. H- don’t dispute b. H- not safe c. H- not safe d. H- safe)**Hypothesis Testing**• Give a gray level answer to the questions. (a. H- moderate evidence against b. H- no strong evidence c. H- seems to go other way d. H- strong evidence, almost very strong)**And now for somethingcompletely different….**An amazing movie clip: http://abfhm.free.fr/basket.htm Thanks to Trent Williamson**Hypothesis Testing**Hypo Testing Approach II: 1-sided testing (more conventional & is version in text) Idea: only one of H+ and H- is usually relevant, so combine other with H0**Attention!!!**• Now return to textbook presentation • H-, H0, and H+ ideas are building blocks • Will combine these • In two different ways • To get more conventional hypothesis • As developed in text**Hypothesis Testing**Approach II: New Hypotheses Null Hypothesis: H0 = “H0 or ” Alternate Hypothesis: HA = opposite of Note: common notation for HA is H1 Gets “burden of proof”, I might accidentally put this i.e. needs strong evidence to prove this**Hypothesis Testing**Weird terminology: Firm conclusion is called “rejecting the null hypothesis” Basics of Test: P-value = Note: same as H0 in H+, H0, H- case, so really just same as above**Fast Food Business Example**Recall: New menu more profitable??? Hypo testing setup: P-val = Same as before. See: Class Example 24, part 3: http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg24.xls**Hypothesis Testing**HW: 6.55, 6.61 Interpret with bothyes-no and gray level AlternateTerminology: “Significant at the 5% level” = = P-value < 0.05 “Test Statistic z” = N(0,1) cutoff**Hypothesis Testing**Hypo Testing Approach III: 2-sided tests Main idea: when either of H+ or H- is conclusive, then combine them E.g. Is population mean equal to a given value, or different? Note either bigger or smaller is strong evidence**Hypothesis Testing**Hypo Testing Approach III: “Alternative Hypothesis” is: HA = “H+ or H-” General form: Specified Value**Hypothesis Testing, III**Note: “ ” always goes in HA, since cannot have “strong evidence of =”. i. e. cannot be sure about difference between and + 0.000001 while can have convincing evidence for “ ” (recall HA gets “burden of proof”)**Hypothesis Testing, III**Basis of test: (now see why this distribution form is used) observed value of “more conclusive” is the two tailed area**Fast Food Business Example**Two Sided Viewpoint: $1,000 $1,000 P-value = $20,000 $21,000 mutually exclusive “or” rule**Fast Food Business Example**P-value = =NORMDIST… See Class Example 24, part 4 http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg24.xls = 0.188 So no strong evidence, Either yes-no or gray-level**Fast Food Business Example**Shortcut: by symmetry 2 tailed Area = 2 x Area See Class Example 24, part 4 http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg24.xls**Hypothesis Testing, III**HW: 6.62 - interpret both yes-no & gray-level (-2.20, 0.0278, rather strong evidence)**Hypothesis Testing, III**A “paradox” of 2-sided testing: Can get strange conclusions (why is gray level sensible?) Fast food example: suppose gathered more data, so n = 20, and other results are the same**Hypothesis Testing, III**One-sided test of: P-value = … = 0.031 Part 5 ofhttp://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg24.xls Two-sided test of: P-value = … = 0.062**Hypothesis Testing, III**Yes-no interpretation: Have strong evidence But no evidence !?! (shouldn’t bigger imply different?)**Hypothesis Testing, III**Notes: • Shows that yes-no testing is different from usual logic (so be careful with it!) • Reason: 2-sided admits more uncertainty into process (so near boundary could make a difference, as happened here) • Gray level view avoids this: (1-sided has stronger evidence, as expected)**Hypothesis Testing, III**Lesson: 1-sided vs. 2-sided issues need careful: • Implementation (choice does affect answer) • Interpretation (idea of being tested depends on this choice) Better from gray level viewpoint**Hypothesis Testing, III**CAUTION: Read problem carefully to distinguish between: One-sided Hypotheses - like: Two-sided Hypotheses - like:**Hypothesis Testing**Hints: • Use 1-sided when see words like: • Smaller • Greater • In excess of • Use 2-sided when see words like: • Equal • Different • Always write down H0 and HA • Since then easy to label “more conclusive” • And get partial credit….**Hypothesis Testing**E.g. Text book problem 6.34: In each of the following situations, a significance test for a population mean, is called for. State the null hypothesis, H0 and the alternative hypothesis, HA in each case….