Business Statistics: Understanding Hypothesis Testing & Significance Tests, Study notes of Statistics

Download Business Statistics: Understanding Hypothesis Testing & Significance Tests and more Study notes Statistics in PDF only on Docsity! Stat 226 – Introduction to Business Statistics I Spring 2009 Professor: Dr. Petrutza Caragea Section A Tuesdays and Thursdays 9:30-10:50 a.m. Chapter 6, Section 6.2 Test of Significance (Hypothesis testing) Stat 226 (Spring 2009) Introduction to Business Statistics I Section 6.2 1 / 27 Chapter 6.2 — Tests of Significance Tests of Significance Example: pick a jury of 12 people randomly out of a pool of 12 men and 12 women for a fair jury: 6 men and 6 women What about a selection of 5 men and 7 women? 4 men and 8 women? or even 1 man and 11 women? Where do we draw the line and no longer believe that the jury selection was truly random and fair? That is when do we start doubting that the chance of getting selected for each gender was truly 50/50? Stat 226 (Spring 2009) Introduction to Business Statistics I Section 6.2 2 / 27 Chapter 6.2 — Tests of Significance Some basic terminology Hypothesis A hypothesis is a claim or belief about a population parameter that we wish to test. In any test there are two competing hypotheses: the null hypothesis, denoted by H0, is a statement of what we assume to be true vs. the alternative hypothesis, denoted by Ha, which is a statement against H0 — this is what we want to show Stat 226 (Spring 2009) Introduction to Business Statistics I Section 6.2 3 / 27 Chapter 6.2 — Tests of Significance The philosophy behind a statistical hypothesis test is the same as in a jury trial. There are only two possibilities: “not guilty” corresponding to H0 vs. “guilty” corresponding to Ha Like in a jury trial the philosophy is: “innocent until proven guilty.” That is, we assume “not guilty” until we have enough evidence to determine “guilt”. Likewise we assume H0 is true until we have su!cient evidence in the data in favor of Ha. Stat 226 (Spring 2009) Introduction to Business Statistics I Section 6.2 4 / 27 Chapter 6.2 — Tests of Significance Both, null and alternative hypothesis are always stated in terms of the population parameter. Generally this will be µ for us. Example: A brewery claims that the average content (µ) of their cans of beer is 12 oz, but we suspect that the average content is less (getting ripped o") We want to test H0 : vs. Ha : Stat 226 (Spring 2009) Introduction to Business Statistics I Section 6.2 5 / 27 Chapter 6.2 — Tests of Significance Example: Developing a new diet to loose weight (we are interested in the average weight loss in lbs.) we want to see if the diet is e"ective. We want to test H0 : vs. Ha : Example: A machine “in control” should cut wood into 5 feet pieces. It is suspected that machine is “out of control”. We want to test H0 : vs. Ha : Stat 226 (Spring 2009) Introduction to Business Statistics I Section 6.2 6 / 27 Chapter 6.2 — Tests of Significance In summary we have three di"erent types of alternative hypotheses against the null hypothesis H0 : µ = µ0 1 2 3 Note, the “=” sign is always included in the null hypothesis, never in the alternative hypothesis. H0 and Ha always have to contradict each other. µ0 corresponds to the mean we assume under H0 Stat 226 (Spring 2009) Introduction to Business Statistics I Section 6.2 7 / 27 Chapter 6.2 — Tests of Significance Technically, we test H0 : µ ! µ0 vs. Ha : µ > µ0 (instead of H0 : µ = µ0) as well as H0 : µ " µ0 vs. Ha : µ < µ0 (instead of H0 : µ = µ0) For simplicity we will keep using H0 : µ = µ0. Example: Brewery claims the mean (average) content of a can of beer is 12 oz. We take a random sample of 36 beer cans and obtain the sample mean x̄ = 11.82 oz. If the standard deviation is known to be ! = 0.38 oz, do we have enough evidence that the brewery is making a false claim? earlier we set up the following hypotheses: Stat 226 (Spring 2009) Introduction to Business Statistics I Section 6.2 8 / 27 Chapter 6.2 — Tests of Significance If we chose " = 0.01, we would commit the error only 1% of the times, but it would be harder to reject the null hypothesis (x̄ will have to be more extreme before we can reject H0) Stat 226 (Spring 2009) Introduction to Business Statistics I Section 6.2 17 / 27 Chapter 6.2 — Tests of Significance The choice of " is somewhat subjective — How much of an error probability are we willing to accept? This is equivalent to how strong your evidence against H0 has to be before you are willing to reject H0. Decision Rule if p-value ! ", reject H0 in favor of Ha We say: We have statistically significant evidence against H0 and have reason to believe in Ha if p-value > ", fail to reject H0 We say: We do not have su!cient evidence against H0 and no reason to believe in Ha Stat 226 (Spring 2009) Introduction to Business Statistics I Section 6.2 18 / 27 Chapter 6.2 — Tests of Significance Example: " = 0.05 (level of significance) We say any p-value ! 0.05 is statistically significant at the 0.05 level. Example: Suppose p-value=0.03 if " = 0.05: if " = 0.01: Stat 226 (Spring 2009) Introduction to Business Statistics I Section 6.2 19 / 27 Chapter 6.2 — Tests of Significance A technical & philosophical note: the decision is always in terms of the null hypothesis H0; we either are able to“reject H0” or we “fail to reject H0” we never prove neither H0 nor Ha, we just collect evidence against H0. If we fail to find strong evidence against H0, we will “stick to H0”. This does not imply that H0 is necessarily true, maybe we just did not have a su!ciently large sample size On the other hand, rejecting H0 in favor of Ha does not guarantee that Ha is true despite very strong evidence. For any given hypothesis test, there is two kind of errors we can commit, but we also a very high chance of making a correct decision: Stat 226 (Spring 2009) Introduction to Business Statistics I Section 6.2 20 / 27 Chapter 6.2 — Tests of Significance Type I and Type II error in Hypotheses Tests Stat 226 (Spring 2009) Introduction to Business Statistics I Section 6.2 21 / 27 Chapter 6.2 — Tests of Significance Handouts (“z-procedure” & Examples) Stat 226 (Spring 2009) Introduction to Business Statistics I Section 6.2 22 / 27 Chapter 6.2 — Tests of Significance Connection between confidence intervals and two-sided hypotheses tests (p.394/395) Recall the example of water bottling company Water bottles are supposed to contain 710 ml on average, ! = 6 ml and a sample of 90 bottles yielded an average of 708 ml. Example: Is the bottling process still on target? We constructed a 98% CI for µ and obtained (706.53 ; 709.47) We concluded intuitively, that this is a good indicator that the process is not on target any longer. — Was this intuitive decision justified? Stat 226 (Spring 2009) Introduction to Business Statistics I Section 6.2 23 / 27 Chapter 6.2 — Tests of Significance Let’s see what decision we will obtain by conducting the corresponding hypothesis test: Stat 226 (Spring 2009) Introduction to Business Statistics I Section 6.2 24 / 27 Chapter 6.2 — Tests of Significance Stat 226 (Spring 2009) Introduction to Business Statistics I Section 6.2 25 / 27 Chapter 6.2 — Tests of Significance What is the connection? A two-sided hypothesis test using a significance level " and a (1$ ") % 100% confidence interval are equivalent. That is, a two-sided hypothesis test rejects the null hypothesis H0 exactly when the value µ0 falls outside the corresponding (1$ ") % 100% confidence interval Stat 226 (Spring 2009) Introduction to Business Statistics I Section 6.2 26 / 27 Chapter 6.2 — Tests of Significance Practical versus Statistical Significance Handout Stat 226 (Spring 2009) Introduction to Business Statistics I Section 6.2 27 / 27