The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STOR 155 Introductory Statistics Lecture 17 Hypothesis Testing Section 6 2 6 4 10 Lecture 17 1 Procedures for statistical inference Point estimation to estimate a parameter quick and easy Confidence interval to estimate a parameter with measures for reliability and accuracy attached Hypothesis testing hypothesis a statement about the parameters to assess whether the data provide enough evidence for some claim about the population 6 4 10 Lecture 17 2 Confidence Interval Point estimate with margin of error Confidence interval for a population mean x z x z n n Assumption the population variance is known Confidence level C determines z 6 4 10 Lecture 17 3 Hypothesis Testing Sometimes you may not be interested in estimating the parameter value Rather you have some claim belief about the parameter and you want to see whether the data support the claim or not i e to choose between two decisions support versus contradict 6 4 10 Lecture 17 4 Biased one Euro Coin A group of statistics students spun the Belgian one Euro coin 250 times and heads came up 140 times p P H in each spin Claim the coin is biased more specifically p is greater than 0 5 6 4 10 Lecture 17 5 Concepts of Hypothesis Testing Two hypotheses H0 the null hypothesis the statement of no effect or no difference the statement we try to find evidence against Ha the alternative hypothesis the statement we hope or suspect to be true Usually Ha is based on a hint from samples 6 4 10 Lecture 17 6 Biased one Euro Coin continued Sample 140 heads among 250 spins of a Belgian one Euro coin a hint p P H in each spin H0 p 0 5 vs Ha p 0 5 one sided H0 p 0 5 vs Ha p 0 5 two sided 6 4 10 Lecture 17 7 Sprinkler A sprinkler system maker claims that the true average systemactivation temperature is 130o A sample of 9 systems is tested and yields an average activation temperature of 131 08o If the distribution of activation temperatures is normal with 1 5o does the data contradict the claim The population consists of all sprinkler systems made We want to show that the mean activation temp is different from 130 Ha m 130 The null hypothesis must specify a single value of the parameter m H0 m 130 How do we proceed 6 4 10 Lecture 17 8 Specify a test statistic A test is based on a statistic often a point estimate of the parameter that appears in the hypotheses Values of the estimate far from the parameter value in H0 give evidence against H0 Ha determines which direction will be counted as far from the parameter value Using standardization the test statistic usually has the form z estimate hypothesized value standard deviation of the estimate 6 4 10 Lecture 17 9 Sprinkler continued A sprinkler system maker claims that the true average systemactivation temperature is 130o A sample of 9 systems is tested and yields an average activation temperature of 131 08o If the distribution of activation temperatures is normal with 1 5o does the data contradict the claim The test statistic is 131 08 130 1 08 z 2 16 0 5 1 5 9 6 4 10 Lecture 17 10 P value Definition the probability of observing a test statistic value as extreme or more extreme than the actually observed value given that H0 is true extreme means far from what we would expect from H0 P value indicates how strong the statistical evidence is against the null hypothesis the smaller a P value the stronger evidence against H0 6 4 10 Lecture 17 11 Implication of P value rule of thumb If the P value is less than 1 there is overwhelming evidence against the null hypothesis If the P value is between 1 and 5 there is strong evidence against the null hypothesis If the P value is between 5 and 10 there is weak evidence against the null hypothesis If the P value exceeds 10 there is no evidence against the null hypothesis 6 4 10 Lecture 17 12 Significance Level and Statistical Significance We need to make a conclusion after carrying out a hypothesis test What do we conclude We compare the P value with a fixed value that we regard as decisive This amounts to deciding in advance how much evidence against H0 we require in order to reject H0 The decisive value is called the significance level of the test It is denoted by and the corresponding test is called a level test Statistical Significance If the P value we say that the data are statistically significant at level 6 4 10 Lecture 17 13 and P value P value and significance level Reject H0 if the P value Do not reject H0 if the P value When is the evidence against H0 stronger large P value or small P value The smaller the P value the stronger the evidence against H0 When is it easier to reject H0 large or small We need a lot more evidence to reject H0 for small than for large 6 4 10 Lecture 17 14 4 steps in hypothesis testing Define the hypotheses and identify the significance level given in the problem Find the value of the test statistic Calculate the P value based on the data State your conclusion Reject the null hypothesis if the P value if the P value the data do not provide sufficient evidence to reject the null hypothesis 6 4 10 Lecture 17 15 6 4 10 Lecture 17 16 Sprinkler continued Test the hypotheses H0 m 130 vs Ha m 130 at significance level 0 01 Test statistic z x m0 n 131 08 130 1 5 9 1 08 5 2 16 P value 2 P Z 2 16 0 0308 0 01 H0 is not rejected at significance level 0 01 Question What is the 99 confidence interval for the activation temperature m 6 4 10 Lecture 17 17 CI 2 Sided Tests A level 2 sided test accepts or rejects H0 m m0 exactly when the value m0 falls inside or outside a level 1 confidence interval for m Confidence interval can be used to test hypotheses Calculate the 1 level confidence interval If m0 falls within the interval do not reject the null hypothesis Otherwise reject the null hypothesis 6 4 10 Lecture 17 18 SAT In a discussion of SAT scores someone comments Because only a minority of students take the test the scores overestimate the ability of typical seniors The mean SAT M score is about 475 but I think if all seniors took the test the mean would be 450 You gave the test to an SRS of 500 seniors from California These students had an average score of 461 Assume the SAT M score follows a normal distribution with a standard deviation of 100 Is there sufficient evidence against the claim that the mean for all California seniors is 450 under a significance level of 0 05 Give …
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