Class Note for STAT 528 at OSU 53
Class Note for STAT 528 at OSU 53
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Date Created: 02/06/15
Stat 528 Autumn 2008 Tests of signi cance Reading Sections 62 63 0 An example of a test 0 The components of a test The null and alternative hypotheses Errors and probabilities of error in tests of signi cance De ning a test statistic Calculating the p value and reaching a conclusion 0 Carrying out the Z test in MlNlTAB o lssues in testing Failing to reject The perils of a 005 signi cance level Statistical versus practical signi cance Validity of tests and multiple testing Indirect proofs o A prime number is a positive integer that is divisible only by itself and 1 eg 2 3 5 7 11 o Prove that there is no largest prime number 0 The simplest proof dates back to ancient times in Greece lt is an indirect proof 0 Suppose Work and consequences Contradic tion Conclusions 0 A hypothesis test mirrors the indirect proof 0 Compare the indirect proof to our hypothesis test The two arguments match exactly until the Contradiction step lndirect proof There is a mathematical contradiction Hypothesis test lt is implausible to explain the data as anything but a contradiction Hypotheses o A hypothesis is a claim or statement about the value of a population characteristic parameter or characteristics Examples 1 Let u mean ball bearing weight on a production line Hypothesis 1 100 grams 2 Let p proportion of OSU researchers who use statistics in their research Hypothesis p 05 o Hypotheses should not depend on the sample data These are not hypotheses 1 E 95 2p0 The null and alternative hypotheses o The null hypothesis H0 is the Claim about one or more populations or population Characteristics that is initially as sumed to be true We Will assume the null hypothesis is of the form H0 population parameter some value 0 The alternative or alternate hypothesis denoted by Ha is the Claim to which Wish we compare H0 some people use H1 instead of Ha We Will assume that Ha has one of three forms 1 Ha population parameter y some value 2 Ha population parameter lt some value 3 Ha population parameter gt some value These alternative hypotheses are one or two sided Hypothesis examples ln each of the following situations a signi cance test for a popu lation mean u is called for State the null hypothesis H0 and the alternative hypothesis Ha in each case 0 Experiments on learning in animals sometimes measure how long it takes a mouse to nd its way through a maze The mean time is 20 seconds for one particular maze A researcher thinks that playing rap music will cause the mice to complete the maze faster She measures how long each of 12 mice takes with a rap music as a stimulus 0 ln a botanical study there is interest in measuring the av erage nitrogen percentage of plants of the species Leucaena luecocephala grown in a laboratory It is known that the average nitrogen percentage for the species found in nature is 3 The researcher believes that the average percentage may be higher for lab grown species Pollution example 0 Thirty samples were taken from a stream and the pollution level in parts per million ppm was recorded for each sam ple The average pollution level for the data was 3 101 ppm Suppose that the population standard deviation is 27 ppm The investigator who collected the samples is inter ested in determining whether or not there is evidence that the population mean pollution level is greater than 95 ppm A hypothesis test asks and answers the question ls this belief about the world plausible in light of the data The test procedure c We see how well the sample data supports the null hy pothesis c As a conclusion we either reject H0 or fail to reject H0 0 Although you may hear people say it we never accept Hal This will become clear when we discuss con dence intervals Errors in tests of signi cance 0 Consider the truth about the world The truth H0 is true H 0 is false We reject H 0 Type I error We fail to reject H0 0 Type I error we reject H0 when H0 is actually true 0 Type II error Type ll error we fail to reject H0 when H0 is actually false 0 When we base our inference on data from a study we cannot completely eliminate either type of error The probabilities of making each error o The signi cance level or size of the test is or P making a type l error Pltreject H0 given that H0 is true eg For an or 005 test if H0 is true and the test was repeatedly run on different random samples from the same population in the long run H0 would be rejected 5 of the time o The other probability of error is 6 Pmallte a type ll error Pfail to reject H0 given that H0 is false A compromise in errors 0 In practice or increases as 6 decreases and vice versa 0 Common strategy 1 Select a test statistic that allows you to distinguish be tween the null and alternative hypotheses 2 Choose the signi cance level oz for your test eg people often use or 05 or or 01 3 Find a test procedure that leads to a small 6 given this choice of oz 0 The tests in this class were constructed in this way 10 De ning a test statistic o A test statistic is a function of the data which is calculated and used as a judge between H0 and Ha c We calculate the test statistic for the sample data the ob served value and ask the question based on the sampling distribution of the test statistic under the assumption that H0 holds how likely is it to obtain data that clashes with H0 at least as much as the observed value does 11 Pollution example cont c We know that the sample mean is an unbiased estimator of the true population mean u 0 Thus we will base our test on the sample mean X For the sample size we have n 30 it seems reasonable to assume that Y is approximately 0 Standardizing we have Y M 2 am is approximately NO1 0 Z is called the test statistic The test is called the ztest 12 The observed test statistic 0 Question is the observed test statistic for our data com patible with H0 0 For the example E 1010 27 and n 30 c When H0 is true the population mean u 95 0 Does this value of 2 support the null hypothesis or not We answer this question by calculating a P Value 13 Under the null hypothesis 0 Remember we are testing H0 u 95 versus Ha u gt 95 Y 0 Under H0 Z H has a NO1 distribution a o How does the observed test statistic 2 compare with this distribution 04 03 02 01 00 14 The p value o The p value is the probability that the test statistic takes a value as extreme or more extreme than the observed test statistic The probability calculation is based on the sampling distribution of the test statistic assuming H0 is true The smaller the p value the more evidence in the data against H0 0 For a test of signi cance at the level oz lf p Value 3 oz we reject H0 lf p Value gt oz we fail to reject H0 0 The form of p value calculation will depend on Ha not H0 0 Let 2 be the observed test statistic and let Z be a standard normal RV the distribution of our test statistic under H0 We Will illustrate the cases When 2 is negative or positive 15 Calculating the p value one sided Ha o For Ha 21 lt 0 the p value is PltZ g quotC quotC 0 0 391 391 0 0 N N 0 0 0 0 Q Q 0 0 3 2Z 1 0 1 2 3 o For Ha 21 gt 0 the p value is PltZ Z quotC quotC 0 0 391 391 0 0 N N 0 0 0 0 Q Q 0 0 3 2Z 1 0 1 2 3 16 Calculating the p value two sided Ha o For Ha 21 y 0 the p value is 2PltZ Z quotC o 03 02 01 00 17 Testing example The runners You measure the weights of 24 male runners You do not actually choose a SR8 but are willing to assume that these runners are a random sample from the population of male runners in your town or city The mean of the sample is given by 6179 kg Suppose that the standard deviation of the population is known to be a 45 kg ls there evidence that the population mean weight is not equal to 64 0 Write down the hypotheses for your test Make sure you de ne all the quantities in your hypothesis 0 Calculate the value of the observed 2 test statistic 0 Calculate the p value for your test 0 ls the result signi cant at the 5 level ie oz 005 At the 1 level 0 Interpret your result in words 18 The steps in MINITAB 1 Load the dataset into MlNlTAB 2 Examine the data with graphs and summary statistics 3 Select Stat gt Basic Statistics gt 1Sample Z 0 Variable Cl 0 Standard deviation 45 o Tick Perform hypothesis test 0 Test mean 63 0 Under Options select an Alternative of not equal to One Sample Z runner weights Test of mu 64 vs not 64 The assumed standard deviation 45 Variable N Mean StDev SE Mean 95 CI Z P runner weights 24 61792 4808 0919 59991 63592 240 0016 19 Thinking about hypothesis tests Which of the following does a test of signi cance answer 0 ls the sample or experiment properly collected or designed o ls the observed effect important 0 ls the observed effect due to chance 0 Could the observed effect be due to chance 0 ls the hypothesized parameter value consistent with the data 20 Failing to reject 0 Suppose we test H0 u 0 versus Ha u y MO o If we fail to reject H0 then either H0 is true OR there is not enough evidence in the data to reject H0 0 The second case is related to the power of a test 7 the probability of detecting an effect of the size you hope to nd yet to come 0 Remember a lack of signi cance is still a positive result Journals tend to disagree 21 The perils of a 005 signi cance level Suppose that SAT math scores vary normally with a 100 One hundred students go through a rigorous training program to raise their scores by improving their mathematics skills We carry out a test of H0 21 475 versus Ha 21 gt 475 0 Suppose the average student score is E 4914 Then 2 164 lt 1645 ie not signi cant at the 005 level 0 Suppose the average student score is E 4915 Then 2 165 gt 1645 ie signi cant at the 005 level Beware attempts to treat or 005 as a sacred number 22 Statistical versus practical signi cance 0 ln a hypothesis test when the p value is smaller than the chosen signi cance level oz we say the result is statistically signi cant 0 The observed deviation that was expected under H 0 cannot be attributed to sampling variation alone 0 But is the actual difference signi cant in practice Example ln comparing two drugs for alleviating cold symp toms investigators nd that the rst drug works signi cantly quicker than the second drug by 20 minutes on the average Suppose drug A costs twice as much as drug B Which drug is better in practice 0 Practical signi cance will be speci c to each problem 23 Validity of hypothesis tests o The following causes can destroy the validity of a hypothesis test Bad survey or experimental design eg lack of control badly worded survey questions Faulty data collection eg we do not have a random sample Poor approximations for sampling distributions eg for a Z test based on E the data contain outliers the population distribution is heavily skewed a is unknown 0 We need to adjust our testing scheme if we carry out mul tiple tests at once Even when H 0 is true in each case it is possible to produce a signi cant result by chance alone 24
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