© 2010 Pearson Prentice Hall. All rights reserved Chapter Hypothesis Tests Regarding a Parameter 10

© 2010 Pearson Prentice Hall. All rights reserved Section Hypothesis Tests for a Population Proportion 10.4

© 2010 Pearson Prentice Hall. All rights reserved Objectives 1.Test the hypotheses about a population proportion 2.Test hypotheses about a population proportion using the binomial probability distribution.

© 2010 Pearson Prentice Hall. All rights reserved Objective 1 Test hypotheses about a population proportion.

© 2010 Pearson Prentice Hall. All rights reserved 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.

© 2010 Pearson Prentice Hall. All rights reserved Recall: The sampling distribution of is approximately normal, with mean and standard deviation provided that the following requirements are satisfied: 1.The sample is a simple random sample. 2. np(1-p) ≥ The sampled values are independent of each other.

© 2010 Pearson Prentice Hall. All rights reserved Testing Hypotheses Regarding a Population Proportion, p To test hypotheses regarding the population proportion, we can use the steps that follow, provided that: 1.The sample is obtained by simple random sampling. 2. np 0 (1-p 0 ) ≥ The sampled values are independent of each other.

© 2010 Pearson Prentice Hall. All rights reserved Step 1: Determine the null and alternative hypotheses. The hypotheses can be structured in one of three ways:

© 2010 Pearson Prentice Hall. All rights reserved Step 2: Select a level of significance, , based on the seriousness of making a Type I error.

© 2010 Pearson Prentice Hall. All rights reserved Step 3: Compute the test statistic Note: We use p 0 in computing the standard error rather than. This is because, when we test a hypothesis, the null hypothesis is always assumed true.

© 2010 Pearson Prentice Hall. All rights reserved Step 4: Use Table V to determine the critical value. Classical Approach

© 2010 Pearson Prentice Hall. All rights reserved Classical Approach Two-Tailed (critical value)

© 2010 Pearson Prentice Hall. All rights reserved Classical Approach Left-Tailed (critical value)

© 2010 Pearson Prentice Hall. All rights reserved Classical Approach Right-Tailed (critical value)

© 2010 Pearson Prentice Hall. All rights reserved Step 5: Compare the critical value with the test statistic: Classical Approach

© 2010 Pearson Prentice Hall. All rights reserved Step 4: Use Table V to estimate the P-value. P-Value Approach

© 2010 Pearson Prentice Hall. All rights reserved P-Value Approach Two-Tailed

© 2010 Pearson Prentice Hall. All rights reserved P-Value Approach Left-Tailed

© 2010 Pearson Prentice Hall. All rights reserved P-Value Approach Right-Tailed

© 2010 Pearson Prentice Hall. All rights reserved Step 5: If the P-value < , reject the null hypothesis. P-Value Approach

© 2010 Pearson Prentice Hall. All rights reserved Step 6: State the conclusion.

© 2010 Pearson Prentice Hall. All rights reserved 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

© 2010 Pearson Prentice Hall. All rights reserved Solution We want to know if p>0.46. First, we must verify the requirements to perform the hypothesis test: 1.This is a simple random sample. 2.np 0 (1-p 0 )=1010(0.46)(1-0.46)=250.8>10 3.Since the sample size is less than 5% of the population size, the assumption of independence is met.

© 2010 Pearson Prentice Hall. All rights reserved Solution Step 1: H 0 : p=0.46 versus H 1 : p>0.46 Step 2: The level of significance is  =0.05. Step 3: The sample proportion is. The test statistic is then

© 2010 Pearson Prentice Hall. All rights reserved 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 z 0.05 = Step 5: Since the test statistic, z 0 =3.83, is greater than the critical value 1.645, we reject the null hypothesis.

© 2010 Pearson Prentice Hall. All rights reserved 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 z 0 =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.

© 2010 Pearson Prentice Hall. All rights reserved 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.

© 2010 Pearson Prentice Hall. All rights reserved Objective 2 Test hypotheses about a population proportion using the binomial probability distribution.

© 2010 Pearson Prentice Hall. All rights reserved 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?

© 2010 Pearson Prentice Hall. All rights reserved 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.

© 2010 Pearson Prentice Hall. All rights reserved 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 np 0 (1-p 0 ) is greater than or equal to 10, where p 0 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.

© 2010 Pearson Prentice Hall. All rights reserved 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.

© 2010 Pearson Prentice Hall. All rights reserved Solution Step 1: H 0 : p=0.105 versus H 1 : p<0.105 Step 2: From the null hypothesis, we have p 0 = There were 34 mothers sampled, so np 0 (1-p 0 )=3.57<10. Thus, the sampling distribution of is not approximately normal.

© 2010 Pearson Prentice Hall. All rights reserved 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= We want to determine whether this result is unusual if the population mean is truly Thus, P-value = P(X ≤ 3 assuming p=0.105 ) = P(X = 0) + P(X =1) + P(X =2) + P(X = 3) = 0.51

© 2010 Pearson Prentice Hall. All rights reserved Solution Step 4: The P-value = 0.51 is greater than the level of significance so we do not reject H 0. 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%.