1. Testing Hypotheses about a Population Proportion Lecture 31 Sections 9.1 – 9.3 Wed, Mar 22, 2006

2. Discovering Characteristics of a Population Any question about a population must first be described in terms of a population parameter. Any question about a population must first be described in terms of a population parameter. Then the question about that parameter generally falls into one of two categories. Then the question about that parameter generally falls into one of two categories. Estimation Estimation What is the value of the parameter? What is the value of the parameter? Hypothesis testing Hypothesis testing Does the evidence support or refute a claim about the value of the parameter? Does the evidence support or refute a claim about the value of the parameter?

3. Example A standard assumption is that a newborn baby is as likely to be a boy as to be a girl. However, some people believe that boys are more likely. A standard assumption is that a newborn baby is as likely to be a boy as to be a girl. However, some people believe that boys are more likely. Suppose a random sample of 1000 live births shows that 520 are boys and 480 are girls. Suppose a random sample of 1000 live births shows that 520 are boys and 480 are girls. Use the data to estimate the proportion of male births. Use the data to estimate the proportion of male births. Does this evidence support the claim that a greater proportion of births are boys? Does this evidence support the claim that a greater proportion of births are boys?

4. Two Approaches for Hypothesis Testing Classical approach. Classical approach. Specify . Specify . Determine the critical value and the rejection region. Determine the critical value and the rejection region. See whether the statistic falls in the rejection region. See whether the statistic falls in the rejection region. Report the decision. Report the decision. p-Value approach. p-Value approach. Compute the p-value of the statistic. Compute the p-value of the statistic. Report the p-value. Report the p-value. If  is specified, then report the decision. If  is specified, then report the decision.

5. Classical Approach  H0H0

6.   H0H0

7.   0 z c H0H0

8.   0 z c Rejection Region Acceptance Region H0H0

9. Classical Approach   0 z c Rejection Region Acceptance Region H0H0

10. Classical Approach   0 z c Rejection Region Acceptance Region Reject z H0H0

11. Classical Approach   0 z c Rejection Region Acceptance Region H0H0

12. Classical Approach   0 z c Rejection Region Acceptance Region Accept z H0H0

13. p-Value Approach  H0H0

14.   H0H0

15.   0 z H0H0

16.   0 z Rejection Region Acceptance Region H0H0

17. p-Value Approach   0 z Rejection Region Acceptance Region H0H0

18. p-Value Approach   0 z Rejection Region Acceptance Region H0H0 z

19. p-Value Approach   0 z Rejection Region Acceptance Region Reject p-value <  H0H0 z

20. p-Value Approach   0 z Rejection Region Acceptance Region H0H0

21. p-Value Approach   0 z Rejection Region Acceptance Region H0H0 z

22. p-Value Approach   0 z Rejection Region Acceptance Region p-value >  H0H0 Accept z

23. The Steps of Testing a Hypothesis (p-Value Approach) The basic steps are The basic steps are 1. State the null and alternative hypotheses. 1. State the null and alternative hypotheses. 2. State the significance level. 2. State the significance level. 3. State the formula for the test statistic. 3. State the formula for the test statistic. 4. Compute the value of the test statistic. 4. Compute the value of the test statistic. 5. Compute the p-value. 5. Compute the p-value. 6. Make a decision. 6. Make a decision. 7. State the conclusion. 7. State the conclusion. See page 566. (The above steps are modified from what is in the book.) See page 566. (The above steps are modified from what is in the book.)

24. Step 1: State the Null and Alternative Hypotheses Let p = proportion of live births that are boys. Let p = proportion of live births that are boys. The null and alternative hypotheses are The null and alternative hypotheses are H 0 : p = 0.50. H 0 : p = 0.50. H 1 : p > 0.50. H 1 : p > 0.50.

25. State the Null and Alternative Hypotheses The null hypothesis should state a hypothetical value p 0 for the population proportion. The null hypothesis should state a hypothetical value p 0 for the population proportion. H 0 : p = p 0. H 0 : p = p 0. The alternative hypothesis must contradict the null hypothesis in one of three ways: The alternative hypothesis must contradict the null hypothesis in one of three ways: H 1 : p < p 0. (Direction of extreme is left.) H 1 : p < p 0. (Direction of extreme is left.) H 1 : p > p 0. (Direction of extreme is right.) H 1 : p > p 0. (Direction of extreme is right.) H 1 : p  p 0. (Direction of extreme is left and right.) H 1 : p  p 0. (Direction of extreme is left and right.)

26. Explaining the Data The observation is 520 males out of 1000 births, or 52%. That is, p ^ = 0.52. The observation is 520 males out of 1000 births, or 52%. That is, p ^ = 0.52. Since we observed 52%, not 50%, how do we explain the discrepancy? Since we observed 52%, not 50%, how do we explain the discrepancy? Chance, or Chance, or The true proportion is not 50%, but something larger, maybe 52%. The true proportion is not 50%, but something larger, maybe 52%.

27. Step 2: State the Significance Level The significance level  should be given in the problem. The significance level  should be given in the problem. If it isn’t, then use  = 0.05. If it isn’t, then use  = 0.05. In this example, we will use  = 0.05. In this example, we will use  = 0.05.

28. The Sampling Distribution of p ^ To decide whether the sample evidence is significant, we will compare the p-value to . To decide whether the sample evidence is significant, we will compare the p-value to . If we were using the classical approach, we would use  to find the critical value(s). If we were using the classical approach, we would use  to find the critical value(s).  is the probability that the value of the test statistic is at least as extreme as the critical value(s), if the null hypothesis is true.  is the probability that the value of the test statistic is at least as extreme as the critical value(s), if the null hypothesis is true.

29. The Sampling Distribution of p ^ Therefore, when we compute the p-value, we do it under the assumption that H 0 is true, i.e., that p = p 0. Therefore, when we compute the p-value, we do it under the assumption that H 0 is true, i.e., that p = p 0.

30. The Sampling Distribution of p ^ We know that the sampling distribution of p ^ is normal with mean p and standard deviation We know that the sampling distribution of p ^ is normal with mean p and standard deviation Thus, we assume that p ^ has mean p 0 and standard deviation: Thus, we assume that p ^ has mean p 0 and standard deviation:

31. Step 3: The Test Statistic Test statistic – The z-score of p ^, under the assumption that H 0 is true. Test statistic – The z-score of p ^, under the assumption that H 0 is true. Thus, Thus, Write this

32. The Test Statistic In our example, we compute In our example, we compute Therefore, the test statistic is Therefore, the test statistic is Now, to find the value of the test statistic, all we need to do is to collect the sample data and substitute the value of p ^. Now, to find the value of the test statistic, all we need to do is to collect the sample data and substitute the value of p ^.

33. Step 4: Compute the Test Statistic In the sample, p ^ = 0.52. In the sample, p ^ = 0.52. Thus, Thus,

34. Step 5: Compute the p-value To compute the p-value, we must first check whether it is a one-tailed or a two-tailed test. To compute the p-value, we must first check whether it is a one-tailed or a two-tailed test. We will compute the probability that Z would be at least as extreme as the value of our test statistic. We will compute the probability that Z would be at least as extreme as the value of our test statistic. If the test is two-tailed, then we must take into account both tails of the distribution to get the p- value. (Double the value in one tail.) If the test is two-tailed, then we must take into account both tails of the distribution to get the p- value. (Double the value in one tail.)

35. Compute the p-value In this example, the test is one-tailed, with the direction of extreme to the right. In this example, the test is one-tailed, with the direction of extreme to the right. So we compute So we compute p-value = P(Z > 1.265) = 0.1029.

36. Compute the p-value An alternative is to evaluate An alternative is to evaluate normalcdf(0.52, E99, 0.50, 0.01581) on the TI-83. It should give the same answer (except for round-off). It should give the same answer (except for round-off).

37. Step 6: Make a Decision Since the p-value is greater than , our decision is: Do not reject the null hypothesis. Since the p-value is greater than , our decision is: Do not reject the null hypothesis. The decision is stated in statistical jargon. The decision is stated in statistical jargon.

38. Step 7: State the Conclusion State the conclusion in a sentence: State the conclusion in a sentence: The data do not support the claim, that more than 50% of live births are male. The data do not support the claim, that more than 50% of live births are male. The conclusion must relate the decision to the context of the problem. It should not use statistical jargon. The conclusion must relate the decision to the context of the problem. It should not use statistical jargon.

39. Summary 1.H 0 : p = 0.50 H 1 : p > 0.50 2.  = 0.05. 3.Test statistic: 4.z = (0.52 – 0.50)/0.0158 = 1.26. 5.p-value = P(Z > 1.26) = 0.1038. 6.Do not reject H 0. 7.The data do not indicate that the proportion of boys is greater than the proportion of girls among newborns.

40. Summary 1.H 0 : p = 0.50 H 1 : p > 0.50 2.  = 0.05. 3.Test statistic: 4.z = (0.52 – 0.50)/0.0158 = 1.26. 5.p-value = P(Z > 1.26) = 0.1038. 6.Do not reject H 0. 7.The data do not indicate that the proportion of boys is greater than the proportion of girls among newborns. Before collecting data

41. Summary 1.H 0 : p = 0.50 H 1 : p > 0.50 2.  = 0.05. 3.Test statistic: 4.z = (0.52 – 0.50)/0.0158 = 1.26. 5.p-value = P(Z > 1.26) = 0.1038. 6.Do not reject H 0. 7.The data do not indicate that the proportion of boys is greater than the proportion of girls among newborns. Before collecting data After collecting data

42. Testing Hypotheses on the TI-83 The TI-83 has special functions designed for hypothesis testing. The TI-83 has special functions designed for hypothesis testing. Press STAT. Press STAT. Select the TESTS menu. Select the TESTS menu. Select 1-PropZTest… Select 1-PropZTest… Press ENTER. Press ENTER. A window with several items appears. A window with several items appears.

43. Testing Hypotheses on the TI-83 Enter the value of p 0. Press ENTER and the down arrow. Enter the value of p 0. Press ENTER and the down arrow. Enter the numerator x of p ^. Press ENTER and the down arrow. Enter the numerator x of p ^. Press ENTER and the down arrow. Enter the sample size n. Press ENTER and the down arrow. Enter the sample size n. Press ENTER and the down arrow. Select the type of alternative hypothesis. Press the down arrow. Select the type of alternative hypothesis. Press the down arrow. Select Calculate. Press ENTER. Select Calculate. Press ENTER. (You may select Draw to see a picture.) (You may select Draw to see a picture.)

44. Testing Hypotheses on the TI-83 The display shows The display shows The title “1-PropZTest” The title “1-PropZTest” The alternative hypothesis. The alternative hypothesis. The value of the test statistic Z. The value of the test statistic Z. The p-value. The p-value. The value of p ^. The value of p ^. The sample size. The sample size. We are interested in the p-value. We are interested in the p-value.

45. The Classical Approach The seven steps The seven steps 1. State the null and alternative hypotheses. 1. State the null and alternative hypotheses. 2. State the significance level. 2. State the significance level. 3. Write the formula for the test statistic. 3. Write the formula for the test statistic. 4. State the decision rule. 4. State the decision rule. 5. Compute the value of the test statistic. 5. Compute the value of the test statistic. 6. State the decision. 6. State the decision. 7. State the conclusion. 7. State the conclusion. (Do not compute the p-value.) (Do not compute the p-value.)

46. The Classical Approach The seven steps The seven steps 1. State the null and alternative hypotheses. 1. State the null and alternative hypotheses. 2. State the significance level. 2. State the significance level. 3. Write the formula for the test statistic. 3. Write the formula for the test statistic. 4. State the decision rule. 4. State the decision rule. 5. Compute the value of the test statistic. 5. Compute the value of the test statistic. 6. State the decision. 6. State the decision. 7. State the conclusion. 7. State the conclusion. (Do not compute the p-value.) (Do not compute the p-value.) Before collecting data

47. The Classical Approach The seven steps The seven steps 1. State the null and alternative hypotheses. 1. State the null and alternative hypotheses. 2. State the significance level. 2. State the significance level. 3. Write the formula for the test statistic. 3. Write the formula for the test statistic. 4. State the decision rule. 4. State the decision rule. 5. Compute the value of the test statistic. 5. Compute the value of the test statistic. 6. State the decision. 6. State the decision. 7. State the conclusion. 7. State the conclusion. (Do not compute the p-value.) (Do not compute the p-value.) Before collecting data After collecting data

48. Example of the Classical Approach Test the hypothesis that there are more male births than female births. Test the hypothesis that there are more male births than female births. Let p = the proportion of live births that are male. Let p = the proportion of live births that are male. Step 1: State the hypotheses. Step 1: State the hypotheses. H 0 : p = 0.50 H 0 : p = 0.50 H 1 : p > 0.50 H 1 : p > 0.50

49. Example of the Classical Approach Step 2: State the significance level. Step 2: State the significance level. Let  = 0.05. Let  = 0.05. Step 3: Define the test statistic. Step 3: Define the test statistic.

50. Example of the Classical Approach Step 4: State the decision rule. Step 4: State the decision rule. Find the critical value. Find the critical value. On the standard scale, the value z 0 = 1.645 cuts off an upper tail of area 0.05. On the standard scale, the value z 0 = 1.645 cuts off an upper tail of area 0.05. This is a normal percentile problem. This is a normal percentile problem. Use invNorm(0.95) on the TI-83 or use the table. Use invNorm(0.95) on the TI-83 or use the table. Therefore, we will reject H 0 if z > 1.645. Therefore, we will reject H 0 if z > 1.645. The decision rule

51. Example of the Classical Approach Step 5: Compute the value of the test statistic. Step 5: Compute the value of the test statistic.

52. Example of the Classical Approach Step 6: State the decision. Step 6: State the decision. Since z < 1.645, our decision is to accept H 0. Since z < 1.645, our decision is to accept H 0. Step 7: State the conclusion. Step 7: State the conclusion. Our conclusion is that the proportion of male births is the same as the proportion of female births. Our conclusion is that the proportion of male births is the same as the proportion of female births.