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Part II: Proportions

10-3 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.

10-4 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) ≥ 10. 3.The sampled values are independent of each other.

10-5 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 ) ≥ 10. 3.The sampled values are independent of each other.

10-6 Step 1: Determine the null and alternative hypotheses. The hypotheses can be structured in one of three ways:

10-7 Step 2: Select a level of significance, , based on the seriousness of making a Type I error.

10-8 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.

10-9 Step 4: Use Table V to estimate the P-value. P-Value Approach

10-10 P-Value Approach Two-Tailed

10-11 P-Value Approach Left-Tailed

10-12 P-Value Approach Right-Tailed

10-13 Step 5: If the P-value < , reject the null hypothesis. If the P-value ≥ α, fail to reject the null hypothesis P-Value Approach

10-14 Step 6: State the conclusion in the context of the problem.

10-15 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

10-16 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.

10-17 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

10-18 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.

10-19 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.