Hypothesis Test for Population Proportion: Large and Small Sample Sizes

1. Chapter 10 Hypothesis Tests Regarding a Parameter

2. Section 10.4 Hypothesis Tests for a Population Proportion

3. Objectives • Test the hypotheses about a population proportion • Test hypotheses about a population proportion using the binomial probability distribution.

4. Objective 1 • Test hypotheses about a population proportion.

5. Recall: • 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.

6. Recall: • 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.

7. Testing Hypotheses Regarding a Population Proportion, p 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.

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

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

10. 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.

11. Classical Approach Step 4:Use Table V to determine the critical value.

12. (critical value) Classical Approach Two-Tailed

13. (critical value) Classical Approach Left-Tailed

14. (critical value) Classical Approach Right-Tailed

15. Classical Approach Step 5:Compare the critical value with the test statistic:

16. P-Value Approach Step 4:Use Table V to estimate the P-value.

17. P-Value Approach Two-Tailed

18. P-Value Approach Left-Tailed

19. P-Value Approach Right-Tailed

20. P-Value Approach Step 5:If the P-value < , reject the null hypothesis.

21. Step 6:State the conclusion.

22. 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 =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

23. 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.

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

25. Solution: Classical Approach Step 4: Since this is a right-tailed test, we determine the critical value at the =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.

26. Solution: P-Value Approach Step 4: Since this is a right-tailed test, the P- value is the area under the standard normal distribution to the right of the test statistic z0=3.83. That is, P-value = P(Z > 3.83)≈0. Step 5: Since the P-value is less than the level of significance, we reject the null hypothesis.

27. Solution Step 6: There is sufficient evidence at the =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.

28. Objective 2 • Test hypotheses about a population proportion using the binomial probability distribution.

29. For the sampling distribution of to be approximately normal, we require np(1-p) be at least 10. What if this requirement is not met?

30. Parallel Example 4: Hypothesis Test for a Population Proportion: Small Sample Size In 2006, 10.5% of all live births in the United States were to mothers under 20 years of age. A sociologist claims that births to mothers under 20 years of age is decreasing. She conducts a simple random sample of 34 births and finds that 3 of them were to mothers under 20 years of age. Test the sociologist’s claim at the  = 0.01 level of significance.

31. Parallel Example 4: Hypothesis Test for a Population Proportion: Small Sample Size 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.

32. Parallel Example 4: Hypothesis Test for a Population Proportion: Small Sample Size Approach: Step 3: Compute the P-value. For right-tailed tests, the P-value is the probability of obtaining x or more successes. For left-tailed tests, the P- value is the probability of obtaining x or fewer successes. The P-value is always computed with the proportion given in the null hypothesis. Step 4: If the P-value is less than the level of significance, , we reject the null hypothesis.

33. Solution Step 1: H0: p=0.105 versus H1: p<0.105 Step 2: From the null hypothesis, we have p0=0.105. There were 34 mothers sampled, so np0(1-p0)=3.57<10. Thus, the sampling distribution of is not approximately normal.

34. Solution Step 3: Let X represent the number of live births in the United States to mothers under 20 years of age. We have x=3 successes in n=34 trials so = 3/34= 0.088. We want to determine whether this result is unusual if the population mean is truly 0.105. Thus, P-value = P(X ≤ 3 assuming p=0.105 ) = P(X = 0) + P(X =1) + P(X =2) + P(X = 3) = 0.51

35. Solution Step 4: The P-value = 0.51 is greater than the level of significance so we do not reject H0. There is insufficient evidence to conclude that the percentage of live births in the United States to mothers under the age of 20 has decreased below the 2006 level of 10.5%.