Chapter

1. Chapter 10 Hypothesis Tests Regarding a Parameter

2. Section 10.2 Hypothesis Tests for a Population Proportion

3. Objectives Explain the logic of hypothesis testing Test the hypotheses about a population proportion Test hypotheses about a population proportion using the binomial probability distribution.

4. Objective 1 Explain the Logic of Hypothesis Testing

5. A researcher obtains a random sample of 1000 people and finds that 534 are in favor of the banning cell phone use while driving, so = 534/1000. Does this suggest that more than 50% of people favor the policy? Or is it possible that the true proportion of registered voters who favor the policy is some proportion less than 0.5 and we just happened to survey a majority in favor of the policy? In other words, would it be unusual to obtain a sample proportion of 0.534 or higher from a population whose proportion is 0.5? What is convincing, or statistically significant, evidence?

6. When observed results are unlikely under the assumption that the null hypothesis is true, we say the result is statistically significant. When results are found to be statistically significant, we reject the null hypothesis.

7. To determine if a sample proportion of 0.534 is statistically significant, we build a probability model. Since np(1 – p) = 100(0.5)(1 – 0.5) = 250 ≥ 10 and the sample size (n = 1000) is sufficiently smaller than the population size, we can use the normal model to describe the variability in . The mean of the distribution of is and the standard deviation is

8. Sampling distribution of the sample proportion

9. We may consider the sample evidence to be statistically significant (or sufficient) if the sample proportion is too many standard deviations, say 2, above the assumed population proportion of 0.5. The Logic of the Classical Approach

10. Recall that our simple random sample yielded a sample proportion of 0.534, so standard deviations above the hypothesized proportion of 0.5. Therefore, using our criterion, we would reject the null hypothesis.

11. Why does it make sense to reject the null hypothesis if the sample proportion is more than 2 standard deviations away from the hypothesized proportion? The area under the standard normal curve to the right ofz= 2 is 0.0228.

12. If the null hypothesis were true (population proportion is 0.5), then 1 – 0.0228 = 0.9772 = 97.72% of all sample proportions will be less than .5 + 2(0.016) = 0.532 and only 2.28% of the sample proportions will be more than 0.532.

13. If the sample proportion is too many standard deviations from the proportion stated in the null hypothesis, we reject the null hypothesis. Hypothesis Testing Usingthe Classical Approach

14. Objective 2 Test hypotheses about a population proportion.

15. The best point estimate of p, the proportion of the population with a certain characteristic, is given by where x is the number of individuals in the sample with the specified characteristic and n is the sample size. Recall:

16. The sampling distribution of is approximately normal, with mean and standard deviation provided that the following requirements are satisfied: The sample is a simple random sample. np(1-p) ≥ 10. The sampled values are independent of each other. Recall:

17. To test hypotheses regarding the population proportion, we can use the steps that follow, provided that: The sample is obtained by simple random sampling. np0(1 – p0) ≥ 10. The sampled values are independent of each other. Testing Hypotheses Regarding a Population Proportion, p

18. Step 1:Determine the null and alternative hypotheses. The hypotheses can be structured in one of three ways:

19. Step 2:Select a level of significance, α, based on the seriousness of making a Type I error.

20. Classical Approach Step 3:Compute the test statistic Note: We use p0 in computing the standard error rather than . This is because, when we test a hypothesis, the null hypothesis is always assumed true.

21. (critical value) Classical Approach Use Table V to determine the critical value. Two-Tailed

22. (critical value) Classical Approach Left-Tailed

23. (critical value) Classical Approach Right-Tailed

24. Classical Approach Step 4:Compare the critical value with the test statistic:

25. Step 5:State the conclusion.

26. Parallel Example 1: Testing a Hypothesis about a Population Proportion: Large Sample Size In 1997, 46% of Americans said they did not trust the media “when it comes to reporting the news fully, accurately and fairly”. In a 2007 poll of 1010 adults nationwide, 525 stated they did not trust the media. At the α =0.05 level of significance, is there evidence to support the claim that the percentage of Americans that do not trust the media to report fully and accurately has increased since 1997? Source: Gallup Poll

27. Solution We want to know if p>0.46. First, we must verify the requirements to perform the hypothesis test: • This is a simple random sample. • np0(1– p0)=1010(0.46)(1 – 0.46)=250.8>10 • Since the sample size is less than 5% of the population size, the assumption of independence is met.

28. Solution Step 1:H0: p = 0.46 versus H1: p > 0.46 Step 2: The level of significance is α= 0.05. Step 3: The sample proportion is . The test statistic is then

29. Solution: Classical Approach Step 4: Since this is a right-tailed test, we determine the critical value at the α= 0.05 level of significance to be z0.05 = 1.645. Step 5: Since the test statistic, z0 = 3.83, is greater than the critical value 1.645, we reject the null hypothesis.

30. Solution Step 6: There is sufficient evidence at the α =0.05 level of significance to conclude that the percentage of Americans that do not trust the media to report fully and accurately has increased since 1997.

31. Objective 3 Test hypotheses about a population proportion using the binomial probability distribution.

32. Parallel Example 4: Hypothesis Test for a Population Proportion A recent survey found that 68.6% of the population own their homes. In a random sample of 150 heads of households, 92 responded that they owned their homes. At the α= 0.01level of significance, does that suggest a difference from the national proportion(different than 68.6%)?

33. Parallel Example 4: Hypothesis Test for a Population Proportion: Approach: Step 1: Determine the null and alternative hypotheses Step 2: Check whether np0(1–p0) is greater than or equal to 10, where p0 is the proportion stated in the null hypothesis. If it is, then the sampling distribution of is approximately normal and we can use the steps for a large sample size. Otherwise we use the following Steps 3 and 4.

34. Solution Check the requirements: From the null hypothesis, we have p0=0.686. n=150, so np0(1– p0)=32.31>10. And the sample size is less than 5% of the population size.Thus, the sampling distribution of is approximately normal. Step 1: H0: p=0.686 versus H1: p#0.686 Step 2: The level of significance is α= 0.01.

35. Solution Step 3: The sample proportion is . The test statistic is then … Step 4: Since this is a two-tailed test, we determine the critical value at the α= 0.01 level of significance to be z0.005 = 2.58 and -z0.005 =- 2.58. Step 5: Since the test statistic, z0 = -1.93 does not fall within the critical region, we do not reject the null hypothesis.

36. Step 6: There is insufficient evidence at the α =0.01 level of significance to conclude that the proportion of … is different than 0.686

37. Section 10.3 Hypothesis Tests for a Population Mean

38. Objectives Test hypotheses about a mean Understand the difference between statistical significance and practical significance.

39. Objective 1 Test Hypotheses about a Mean

40. To test hypotheses regarding the population mean assuming the population standard deviation is unknown, we use the t-distribution rather than the Z-distribution. When we replace σ with s, follows Student’s t-distribution with n –1 degrees of freedom.

41. The t-distribution is different for different degrees of freedom. Properties of the t-Distribution

42. The t-distribution is different for different degrees of freedom. The t-distribution is centered at 0 and is symmetric about 0. Properties of the t-Distribution

43. The t-distribution is different for different degrees of freedom. The t-distribution is centered at 0 and is symmetric about 0. The area under the curve is 1. Because of the symmetry, the area under the curve to the right of 0 equals the area under the curve to the left of 0 equals 1/2. Properties of the t-Distribution

44. 4. As t increases (or decreases) without bound, the graph approaches, but never equals, 0. Properties of the t-Distribution

45. As t increases (or decreases) without bound, the graph approaches, but never equals, 0. The area in the tails of thet-distribution is a little greater than the area in the tails of the standard normal distribution because using s as an estimate of σ introduces more variability to the t-statistic. Properties of the t-Distribution

46. 6. As the sample size n increases, the density curve of t gets closer to the standard normal density curve. This result occurs because as the sample size increases, the values of s get closer to the values of σby the Law of Large Numbers. Properties of the t-Distribution

47. To test hypotheses regarding the population mean, we use the following steps, provided that: The sample is obtained using simple random sampling. The sample has no outliers, and the population from which the sample is drawn is normally distributed or the sample size is large (n ≥ 30). The sampled values are independent of each other. Testing Hypotheses Regarding aPopulation Mean

48. Step 1:Determine the null and alternative hypotheses. Again, the hypotheses can be structured in one of three ways:

49. Step 2:Select a level of significance, α, based on the seriousness of making a Type I error.

50. Classical Approach Step 3:Compute the test statistic which follows the Student’s t-distribution with n – 1 degrees of freedom.