6/3/11 Lecture 17 1 STOR 155 Introductory Statistics Lecture 17: Hypothesis Testing Section 6.2 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL6/3/11 Lecture 17 2 • 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 Procedures for statistical inference6/3/11 Lecture 17 3 Confidence Interval • Point estimate with margin of error • Confidence interval for a population mean – Assumption: the population variance is known – Confidence level C determines z* nzxnzx**,6/3/11 Lecture 17 4 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’’6/3/11 Lecture 17 5 “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)6/3/11 Lecture 17 6 • 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 Concepts of Hypothesis Testing6/3/11 Lecture 17 7 “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-sided6/3/11 Lecture 17 8 • A sprinkler system maker claims that the true average system-activation 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 ? Sprinkler6/3/11 Lecture 17 9 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)6/3/11 Lecture 17 10 • A sprinkler system maker claims that the true average system-activation 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 Sprinkler (continued) 16.25.008.19/5.113008.131z6/3/11 Lecture 17 11 • 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 P-value6/3/11 Lecture 17 12 • 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. Implication of P-value (rule of thumb)6/3/11 Lecture 17 13 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 .6/3/11 Lecture 17 14 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 .6/3/11 Lecture 17 15 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.6/3/11 Lecture 17 166/3/11 Lecture 17 17 Sprinkler (continued) • Test the hypotheses: H0: m = 130 vs. Ha: m 130 at significance level = 0.01 ? • Test statistic • 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 ? .16.25./08.1)9/5.1/()13008.131()//()(0 nxzm6/3/11 Lecture 17 18 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
View Full Document
Unlocking...