Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Chapter Hypothesis Tests Regarding a Parameter 10

Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Section Hypothesis Tests for a Population Proportion 10.2

Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc Objectives 1.Explain the logic of hypothesis testing 2.Test the hypotheses about a population proportion 3.Test hypotheses about a population proportion using the binomial probability distribution.

Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc Objective Hypothesis Testing Using the P-Value Approach If the probability of getting a sample proportion as extreme or more extreme than the one obtained is small under the assumption the statement in the null hypothesis is true, reject the null hypothesis

Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc 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% (p > 0.5)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 or higher(p>0.534) from a population whose proportion is 0.5? What is convincing, or statistically significant, evidence?

Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc A second criterion we may use for testing hypotheses is to determine how likely it is to obtain a sample proportion of or higher from a population whose proportion is 0.5. If a sample proportion of or higher is unlikely (or unusual), we have evidence against the statement in the null hypothesis. Otherwise, we do not have sufficient evidence against the statement in the null hypothesis. The Logic of the P-Value Approach

Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc We can compute the probability of obtaining a sample proportion of or higher from a population whose proportion is 0.5 using the normal model.

Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc Recall ( test statistics ) So, we compute (right tailed test) The value is called the P-value, which means about 2 samples in 100 will give a sample proportion as high or higher than the one we obtained if the population proportion really is 0.5. Because these results are unusual, we take this as evidence against the statement in the null hypothesis.

Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc If the probability of getting a sample proportion as extreme or more extreme than the one obtained is small under the assumption the statement in the null hypothesis is true, reject the null hypothesis. Hypothesis Testing Using the P-value Approach

Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc Objective 2 Test hypotheses about a population proportion.

Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc 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.

Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc 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.

Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc 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. np 0 (1 – p 0 ) ≥ 10. The sampled values are independent of each other.

Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc Step 1: Determine the null and alternative hypotheses. The hypotheses can be structured in one of three ways:

Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc Step 2: Select a level of significance, α, based on the seriousness of making a Type I error.

Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc By Hand Step 3: Compute the test statistic. P-Value Approach

Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc P-Value Approach Two-Tailed Use Table V to estimate the P-value.

Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc P-Value Approach Left-Tailed

Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc P-Value Approach Right-Tailed

Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc Technology Step 3: Use a statistical spreadsheet or calculator with statistical capabilities to obtain the P-value. The directions for obtaining the P-value using the TI-83/84 Plus graphing calculator, MINITAB, Excel, and StatCrunch are in the Technology Step-by-Step in the text. P-Value Approach

Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc Step 4: If the P-value < α, reject the null hypothesis. P-Value Approach

Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc Step 5: State the conclusion.

Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc 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

Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc Solution We want to know if p > 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) = > 10 3.Since the sample size is less than 5% of the population size, the assumption of independence is met.

Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc Solution Step 1: H 0 : p = 0.46 versus H 1 : p > 0.46 Step 2: The level of significance is α = Step 3: The sample proportion is. The test statistic is then

Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc 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.

Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc 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.

Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc Problem Based on information form the National Cyber Security Alliance, 93% of computer owners believe that they have antivirus programs installed on their computers. In a random sample of 400 scanned computers, it is found that 380 of them actually have antivirus programs. Use the sample data from the scanned computers to test the claim that 93% of computers have antivirus programs.

Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc Solution- P-value approach Meets the requirements…. Step 1: H 0 : p = 0.93 versus H 1 : p 0.93 Step 2: The level of significance is α = Step 3: The Test Statistics

Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc Solution: P-Value Approach Step 4: Since this is a two-tailed test, the P- value is twice the area to the right of the test statistic z 0 =1.57. That is, P-value = 2*P(Z > 1.57) =2*( )=2*0.0582= Step 5: Since the P-value is greater than the level of significance, we fail to reject the null hypothesis.

Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc Step 6: There is not sufficient evidence at the α = 0.05 level of significance to reject the claim that 93% of the computers have antivirus programs

Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc Test hypotheses about a population proportion using the binomial probability distribution. Objective 3

Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc 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? We stated that an event was unusual if the probability of observing the event was less than This criterion is based on the P-value approach to testing hypotheses; the probability that we computed was the P-value. We use this same approach to test hypotheses regarding a population proportion for small samples.

Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc 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 α = 0.01 level of significance.

Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc 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.

Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc 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, α, we reject the null hypothesis.

Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc Solution Step 1: H 0 : p = versus H 1 : p < 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.

Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc 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

Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc 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%.