Chapter 20 Testing Hypotheses for Proportions

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1 Chapter 20 Testing Hypotheses for Proportions A hypothesis proposes a model for the world. Then we look at the data. If the data are consistent with that model, we have no reason to disbelieve the hypothesis. But if the data are inconsistent with the model, what then? Notice the difference between the two possibilities. If the facts are consistent with the model, they lend support to the hypothesis. Does this prove the hypothesis is true? No. Lending support is not the same as proving something. Even if the hypothesis is true, we can never prove it. When the hypothesis is false, we might be able to recognize it. When the data are glaringly inconsistent with the hypothesis, it becomes clear that the hypothesis can't be right. 1

2 P value: The probability that the event we've just witnessed could have happened by chance. We use the model proposed by the hypothesis to calculate the probability. It quantifies how surprised we are. Null Hypothesis, H 0 : specifies a population parameter of interest and proposes a value for the parameter. The null hypothesis is the ordinary state of affairs, so it's the alternative to the null hypothesis that we consider unusual and for which we must marshal evidence. Form: H 0 : parameter = hypothesized value Alternative Hypothesis, H A : proposes what we should conclude if we find the null hypothesis to be unlikely. The alternative hypothesis contains the values of the parameter we accept if we reject the null. 2

3 Interpreting P values Null Hypothesis is also the claim that we seek evidence against (think criminal trial). Alternative Hypothesis the claim about the population we seek evidence for Can be one sided parameter is < or > H 0 value Can be two sided parameter is different from ( ) H 0 (value could be larger or smaller.) Failure to find evidence against H 0 means only that the data are consistent with H 0, not that we have clear evidence that H 0 is true. P value the probability that measures the strength of the evidence against H 0 The probability of getting a sample result at least as extreme as the one we did if H 0 were true. The smaller the P value, the stronger the evidence against H 0 provided by the data. They say that the observed result is unlikely to occur when H 0 is true. Large P values fail to give convincing evidence against H 0 because they say that the observed result is likely to occur by chance when H 0 is true. 3

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5 2. Write the null and alternative hypothesis you would use to test each of the following situations. a. In the 1950s only about 40% of high school graduates went on to college. Has the percentage changed? b. 20% of cars of a certain model have needed costly transmission work after being driven between 50,000 and 100,000 miles. The manufacturer hopes that redesign of a transmission component has solved this problem. c. We field test a new flavor soft drink, planning to market it only if we are sure that over 60% of the people like the flavor. 5

6 4. Dice. The seller of a loaded die claims that it will favor the outcome 6. We don't believe that claim, and roll the die 200 times to test an appropriate hypothesis. Our P value turns out to be Which conclusion is appropriate? Explain. a. There's a 3% chance that the die is fair. b. There's a 97% chance that the die is fair. c. There's a 3% chance that a loaded die could randomly produce the results we observed, so it's reasonable to conclude that the die is fair. d. There's a 3% chance that a fair die could randomly produce the results we observed, so it's reasonable to conclude that the die is loaded. 6

7 5. Relief. A company's old antacid formula provided relief for 70% of the people who used it. The company tests a new formula to see if it is better, and gets a P value of Is it reasonable to conclude that the new formula and the old one are equally effective? Explain. 7

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9 HW p. 476; 1,3,6,8,10 ANS 9

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