Chapter 10: Testing Hypotheses About Proportions ESP experiment: Cards with 5 different symbols. “Sender” draws randomly a card and concentrates on it. The “receiver” tries to determine which card it is. Suppose this experiment is repeated 100 times with one pair of subjects. How many would you expect the receiver to get right if she were just guessing? Would it surprise you if she got 25 right out of 100? 30 right? 35 right? 50 right? What calculations could you do to help you make these decisions? We can view this situation as a test of two hypotheses about a parameter: the long-run proportion p of correct responses by a person in this experiment (we could also think of this parameter as the probability that she would respond correctly on any single trial). If the person is guessing, then p = .2. If the person has some ESP ability, then p > .2. We are interested in whether the sample data provide evidence that the person is doing better than guessing; that is, is there any evidence to support the hypothesis that p > .2? In order to assess the strength of the evidence, we ask ourselves: how likely is the sample outcome if the person were guessing? In formal terms, we have • the null hypothesis which is that p = .2. • the alternative hypothesis which is that p > .2. We abbreviate these H0 : p = .2 HA: p > .2 The alternative hypothesis is what we’re trying to prove. The null hypothesis is what one would assume is true unless proven otherwise: the status quo. To assess the evidence against H0 and in favor of HA, we compute the probability, if H0 were true, of observing a sample result as extreme or more extreme as the one we got. This probability is called the P-value for the test.2Example: What is the P-value for the test of the hypotheses above for someone who gets 25 right out of 100? 30 right? 35 right? 50 right? The smaller the P-value, the stronger the evidence against H0 and in favor of HA. In this case, the smaller the P-value, the stronger the evidence that someone really does better than guessing. How small a P-value would convince you that HA is true, that is, would cause you to reject H0 in favor of HA? That’s an individual decision that should be based on how strongly you believe H0 a priori, and what the costs of making a wrong decision are. • We can never prove the null hypothesis wrong because, no matter how small the P-value, it’s always possible that the sample results occurred just by chance. • A large P-value does not “prove” the null hypothesis; it only means that the data are consistent with the null hypothesis. But the data may be consistent with many other models also that we are not testing. • The P-value is a probability; its value must always be between 0 and 1.3In the ESP example, the calculation of the P-value depended on some assumptions which need to be checked: • Independence assumption: Under the null hypothesis that the person is guessing, the outcomes of the trials are independent of each other • Random sampling condition: these 100 trials can be reasonably assumed to be representative of all possible trials • 10% condition: the “population” is infinite so this assumption is satisfied • Success/failure condition: This condition is to ensure that the normal model foris justified. Since the P-value is computed under the null hypothesis, nppˆ0 = 100(.2) = 20 and nq0 = 100(.8) = 80 are both greater than 10. Note on the interpretation of a P-value: the P-value is the probability, if the null hypothesis were true, of getting a sample result as extreme or more extreme in the direction of the alternative hypothesis. It is not the probability that the null hypothesis is true. The null hypothesis is either true or it isn’t. The P-value is P(sample result | H0), not P(H0 | sample result). The hypothesis test we used above has a name: the one-proportion z-test. The elements of a hypothesis test are: • A parameter of interest; in this chapter, it’s a population proportion p. • The hypotheses. The null hypothesis has the form 00: ppH= where is some specific number between 0 and 1. The alternative hypothesis has one of the following three forms: , or , or 0p0: ppHA>0: ppHA<0: ppHA≠. Which form you choose depends on what you’d like to show. • A random sample of size n from the population of interest from which you can compute the sample proportion. pˆ• The model: in the case of a proportion, we use the normal model for the sampling distribution of . We have to check that the assumptions for using this approximation are satisfied. pˆ• The mechanics of the test. For the one-proportion z-test, we first compute the standard deviation of the sampling distribution of when Hpˆ0 is true: SD( ) = pˆnqp00 • The test statistic is)ˆ(ˆ0pSDppz−=. • The P-value is the probability of observing a test statistic as extreme or more extreme in the direction of the alternative if H0 were true. The meaning of this depends on what the alternative hypothesis is: 0: ppHA> , P-value = P(Z ≥ z) 0: ppHA< , P-value = P(Z ≤ z) 0: ppHA≠ . P-value = P(Z ≤ -|z| or Z ≥ |z|) = 2P(Z ≥ |z|) • A conclusion in the context of the problem. Note: the hypotheses are always formulated before the data are collected.4Example: Use-availability Neu et al. (1974) (cited in Manly et al. 1993, Resource Selection by Animals, Chapman and Hall) examined the use of habitat by moose on a site surrounding the Little Sioux Burn in Minnesota during the winter of 1971-72. They determined the proportion of the area in four habitat categories (in burn; in burn, edge; out of burn, edge; out of burn, further) using a planimeter on a map; these proportions are considered “known”. They then classified 117 observations of moose or groups of moose or moose tracks using the same categories (groups of moose or moose tracks were considered one observation). Consider one category: the interior of the burn. It was determined that the proportion of the area in this habitat was .340. Of the117 moose locations, 25 were in this habitat. Do these data provide evidence that moose use of this habitat type is different from its availability? What is the parameter of interest? What are the null and alternative hypotheses? What is the estimate of the parameter of interest? What is the test statistic? Compute
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