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381 Hypothesis Testing (Introduction-I) QSCI 381 – Lecture 25 (Larson and Farber, Sect 7.1) Illustrative Example-I 381 It is claimed that 20% of a certain species of rockfish spawns each year. We sample 780 fish and find 125 show evidence for spawning. Is the sample statistic ( pˆ 125 / 780 0.160 ) “sufficiently different” from the claim that we can reject the claim? Illustrative Example-II 381 We construct the sampling distribution for samples of size 780 if the population proportion was really 0.2. data 15 10 0 5 Density 20 25 0.10 Is the difference “big enough”? 0.15 0.20 0.25 Claim 0.30 Sampling Distributions 381 A is a (continuous) probability distribution for a sample statistic (given values for the population parameters). One can think of the sampling distribution for the sample mean as a histogram for the value the sample mean could take given the collection of (very) many samples of size n. Hypothesis Tests-I 381 A is a process that uses sample statistics to test a claim about the value of a population parameter. Colloquially, what we are going to do is to see whether the data we have “could have happened” if the claim was true. Hypothesis Tests-II 381 A claim about a population parameter is called a . To test a statistical hypothesis, we state a pair of hypotheses – one that represents the claim and the other its complement. We use statistical methods to determine whether or not we can reject the claim. Hypothesis Tests-III 381 A H0 is a statistical hypothesis that contains a statement of equality, e.g. , , or =. The is the complement of the null hypothesis. It is a statement that must be true if H0 is false. H0 is read as “H subzero” or “H naught”, Ha is read as “H sub-a”. Note that it is sometimes necessary to define the claim as the alternative hypothesis. Null and Alternative Hypotheses 381 First determine the claim and hence the null hypothesis. H0 : k; Ha : k H0 : k; Ha : k H0 : k; Ha : k The first two hypotheses have one-sided alternatives while the third hypothesis has a two-sided alternative. Null and Alternative Hypotheses 381 What are the null and alternative hypotheses for: The density of salmon is 100 fish / ha. The escapement is 20%. More than 60% of the population is mature. The number of recaptures is 7. The number of recaptures is not 7. Null and Alternative Hypotheses 381 Notes: The null hypothesis and the alternative hypothesis should be determined before the data are collected. Always use a two-sided alternative unless there are good theoretical reasons for using a one-sided alternative. This is particularly true if the hypotheses are constructed after the data are collected. Testing Hypotheses 381 To test a hypothesis: There are two outcomes from this: We assume the null hypothesis is true. We determine the sampling distribution for the data. We compare the data with the sampling distribution for the data if the null hypothesis was true. We reject the null hypothesis. We fail to reject the null hypothesis. Note: we do not accept the null hypothesis – why not? Type I and Type II Errors-I 381 A occurs if the null hypothesis is rejected when it is actually true. A occurs if the null hypothesis is not rejected when it is actually false. Errors occur because the sample is not the same as the population. 381 Type I and Type II Errors-II Actual Truth of H0 Decision H0 is true Do not reject H0 Correct decision Reject H0 Type I error (Prob=) H0 is false Type II error (Prob = ) Correct decision (Power = 1-) Type I and Type II Errors-III 381 The consequences of Type I and Type II errors can be quite different and very important. Consider the claims: By implementing this management measure, the rate of recovery will be at least 1% per annum. We sampled 4 clams and claim that the proportion of the population with a given disease is 3% or less. We claim that male and female gnu grow at the same rate. We measure 10,000 gnu and compare the mean lengths of males and females. How would you balance Type I and Type II errors in these cases. Significance 381 In a hypothesis test, the is the maximum allowable probability of making a type I error (denoted ). The probability of making a type II error is denoted by . There are three commonly used levels of significance (=0.1, 0.05 and 0.01) – they are all arbitrary.