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## Chapter 10

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**Chapter 10**Asking and Answering Questions about a Population Proportion Created by Kathy Fritz**In the original study, 1385 women were sent a 19 question**survey. Of the 561 surveys returned, 229 women said they would like to choose the sex of a future child. Of these 229 women, 140 choose to have a girl. The article “Boy or Girl: Which Gender Baby Would You Pick?” (LiveScience, March 23, 2005, www.livescience.com) summarized a study that was published in Fertility and Sterility. The LiveScience article makes the following statement: “When given the opportunity to choose the sex of their baby, women are just as likely to choose pink socks as blue socks, a new study shows.” What is the population of interest? Are the 561 women who responded to the survey representative of the population? In this chapter, you will learn techniques for testing this claim. Does the high nonresponse rate pose a problem? Or, is observing a sample proportion as large as 0.611 very unlikely if the population proportion is 0.50?**Hypotheses and Possible Conclusions**Null Hypothesis Alternative Hypothesis**Hypotheses**Hypotheses are ALWAYS statements about the population characteristic – NEVER the sample statistic. In its simplest form, a hypothesis is a claim or statement about the value of a single population characteristic. The following are examples of hypotheses about population proportions:**What is a hypothesis test?**A hypothesis test uses sample data to choose between two competing hypotheses about a population characteristic. Suppose that a particular community college claims that the majority of students completing an associate’s degree transfer to a 4-year college. You would then want to determine if the sample data provide convincing evidence in support of the hypothesis p > 0.5. To test a claim: set up competing hypotheses p ≤ 0.5 and p > 0.5 Notice that the hypothesis of interest is one of the two competing hypotheses.**Hypothesis statements:**The null hypothesis, denoted by H0, is a claim about a population characteristic that is initially assumed to be true. The alternative hypothesis, denoted by Ha, is the competing claim. Two possible conclusions in a hypothesis test are: • Reject H0 • Fail to reject H0 If the sample data do not provide such evidence, H0 is not rejected. In carrying out a test of H0 versus Ha, the null hypothesis H0 is rejected in favor of the alternative hypothesis HaONLY if the sample data provide convincing evidence that H0 is false. Notice that the conclusions are made about the null hypothesis NOT about the alternative!**The Form of Hypotheses:**This one is considered a two-tailed test because you are interested in both directions. Null hypothesis H0: population characteristic = hypothesized value Alternative hypothesis Ha: population characteristic > hypothesized value Ha: population characteristic < hypothesized value Ha: population characteristic ≠ hypothesized value The null hypothesis always includes the equal case. This hypothesized value is a specific number determined by the context of the problem This sign is determined by the context of the problem. Notice that the alternative hypothesis uses the same population characteristic and the same hypothesized value as the null hypothesis. These are considered one-tailed tests because you are only interested in one direction. Let’s practice writing hypothesis statements.**This is what you assume is true before you begin.**Let’s consider a murder trial . . . What is the null hypothesis? What is the alternative hypothesis? H0: the defendant is innocent If there is not convincing evidence, then we would “fail to reject” the null hypothesis. We never end up determining the null hypothesis is true – only that there is not enough evidence to say it’s not true. Ha: the defendant is guilty To determine which hypothesis is correct, the jury will listen to the evidence. Only if there is “evidence beyond a reasonable doubt” would the null hypothesis be rejected in favor of the alternative hypothesis.**In a study, researchers were interested in determining if**sample data support the claim that more than one in four young adults live with their parents. Define the population characteristic: p = the proportion of young adults who live with their parents It is acceptable to write the null hypothesis as: H0: p ≤ 0.25 State the hypotheses : What is the hypothesized value? H0: p = 0.25 Ha: p > 0.25 What words indicate the direction of the alternative hypothesis?**A study included data from a survey of 1752 people ages 13**to 39. One of the survey questions asked participants how satisfied they were with their current financial situation. Suppose you want to determine if the survey data provide convincing evidence that fewer than 10% of adults 19 to 39 are very satisfied with their current financial situation. Define the population characteristic: p = the proportion of adults ages 19 to 39 who are very satisfied with their current financial situation State the hypotheses : What words indicate the direction of the alternative hypothesis? H0: p = 0.10 Ha: p < 0.10**The manufacturer of M&Ms claims that 40% of plain M&Ms are**brown. A sample of M&Ms will be used to determine if the proportion of brown M&Ms is different from what the manufacturer claims. Define the population characteristic: p = the proportion of plain M&Ms that are brown State the hypotheses : What words indicate the direction of the alternative hypothesis? H0: p = 0.40 Ha: p ≠ 0.40**For each pair of hypotheses, indicate which are not**legitimate and explain why Must use a population characteristic! Ha does NOT include equality statement! Must use same number as in H0! H0 MUST include the equality statement!**Potential Errors in Hypothesis Testing**Type I Errors Type II Errors Significance Level**When you perform a hypothesis test you make a decision:**reject H0or fail to reject H0 Each could possibly be a wrong decision; therefore, there are two types of errors.**Type I error**A Type I error is the error of rejecting H0when H0 is true. The probability of a Type I error is denoted by a. In a hypothesis test, the probability of a Type I error, a,is also called the significance level. This is the lower-case Greek letter “alpha”.**Type II error**A Type II error is the error of failing to reject H0 when H0 is false. The probability of a Type II error is denoted by b. This is the lower-case Greek letter “beta”.**The U.S. Bureau of Transportation Statistics reports that**for 2008, 65.3% of all domestic passenger flights arrived on time (meaning within 15 minutes of its scheduled arrival time). Suppose that an airline with a poor on-time record decides to offer its employees a bonus if the airline’s proportion of on-time flights exceeds the overall industry rate of 0.653 in an upcoming month. Let p = the actual proportion of the airline’s flights that are on time during the month of interest. The hypotheses are: State a Type I error in context. State a Type II error in context. A Type I error is concluding that the airline on-time rate exceeds the overall industry rating when in fact the airline does not have a better on-time record. A Type II is not concluding that the airline’s on-time proportion is better than the industry proportion when the airline really did have a better on-time record. H0: p = 0.653 Ha: p > 0.653**Boston Scientific developed a new heart stent used to treat**arteries blocked by heart disease. The new stent, called the Liberte, is made of thinner metal than heart stents currently in use, making it easier for doctors to direct the stent to a blockage. In order to obtain approval to sell the new Liberte stent, the Food and Drug Administration (FDA) required Boston Scientific to provide evidence that the proportion of patients receiving the Liberte stent who experienced a re-blocked artery was less than 0.1. Letp= the proportion of patients receiving the Liberte stent who experience a re-blocked artery H0: p = 0.1 Ha: p < 0.1 A consequence of making a Type I error would be that the new stent is approved for sale. More patients will experience re-blocked arteries. A Type I error would be to conclude that the re-block proportion for the new stent is less than 0.1 when it really is 0.1 (or greater). State a Type I error in the context of this problem.**Boston Scientific developed a new heart stent used to treat**arteries blocked by heart disease. The new stent, called the Liberte, is made of thinner metal than heart stents currently in use, making it easier for doctors to direct the stent to a blockage. In order to obtain approval to sell the new Liberte stent, the Food and Drug Administration (FDA) required Boston Scientific to provide evidence that the proportion of patients receiving the Liberte stent who experienced a re-blocked artery was less than 0.1. Letp= the proportion of patients receiving the Liberte stent who experience a re-blocked artery H0: p = 0.1 Ha: p < 0.1 A consequence of making a Type II error would be that the new stent is not approved for sale. Patients and doctors will not benefit from the new design. A Type II error would be that you are not convinced that the re-block proportion for the new stent is less than 0.1 when it really is less than 0.1. State a Type II error in the context of this problem.**The relationship between a and b**The ideal test procedure would result in both a = 0 (probability of a Type I error) and b = 0 (probability of a Type II error). This is impossible to achieve since we must base our decision on sample data. Standard test procedures allow us to select a, the significance level of the test, but we have no direct control over b. Selecting a significance level a = 0.05 results in a test procedure that, used over and over with different samples, rejects a trueH0 about 5 times in 100. So why not always choose a small a (like a = 0.05 or a = 0.01)?**.5**The relationship between a and b If the null hypothesis is false and the alternative hypothesis is true, then the true proportion is believed to be greater than 0.5 – so the curve should really be shifted to the right. Let’s consider the following hypotheses: H0: p = 0.5 Ha: p > 0.5 Let a = 0.05 This tail would represent b, the probability of failing to reject a false H0. This is the part of the curve that represents a or the Type I error.**The relationship between a and b**If the null hypothesis is false and the alternative hypothesis is true, then the true proportion is believed to be greater than 0.5 – so the curve should really be shifted to the right. Notice that as a gets smaller, b gets larger! Let’s consider the following hypotheses: H0: p = 0.5 Ha: p > 0.5 Let a = 0.01 This tail would represent b, the probability of failing to reject a false H0.**How does one decide what a level to use?**After assessing the consequences of type I and type II errors, identify the largesta that is tolerable for the problem. Then employ a test procedure that uses this maximum acceptable value – rather than anything smaller – as the level of significance. Remember, using a smaller a increases b.**Heart Stents Revisited . . .**Letp= the proportion of patients receiving the Liberte stent who experience a re-blocked artery H0: p = 0.1 versus Ha: p < 0.1 A consequence of making a Type I error would be that the new stent is approved for sale. More patients will experience re-blocked arteries. A Type I error has a more serious consequence. Because this represents an unnecessary risk to patients (given that other stents with lower re-block proportions are available), a small value for a, such as 0.01, would be selected. A consequence of making a Type II error would be that the new stent is not approved for sale. Patients and doctors will not benefit from the new design. Which type of error has the more serious consequence?**The Logic of Hypothesis Testing**An Informal Example**In June 2006, an Associated Press survey was conducted to**investigate how people use the nutritional information provided on food packages. Interviews were conducted with 1003 randomly selected adult Americans, and each participant was asked a series of questions, including the following two: Based on these data, is it reasonable to conclude that a majority of adult Americans frequently check nutritional labels when purchasing packaged foods? Question 1: When purchasing packaged food, how often do you check the nutritional labeling on the package? Question 2: How often do you purchase food that is bad for you, even after you’ve checked the nutrition labels? It was reported that 582 responded “frequently” to the question about checking labels and 441 responded “very often” or “somewhat often” to the question about purchasing bad foods even after checking the labels.**Nutritional Labels Continued . . .**H0: p = 0.5 Ha: p > 0.5 p = true proportion of adult Americans who frequently check nutritional labels For this sample: We use p > 0.5 to test for a majority of adult Americans who frequently check nutritional labels.**Thus, we have convincing evidence to suggest that the null**hypothesis is not true. We would reject H0.**A Procedure for Carrying Out a Hypothesis Test**Test Statistic P-value**Test Statistic**A test statistic is computed using sample data. The value of the test statistic is used to determine the P-value associated with the test.**P-values**TheP-value (also sometimes called the observed significance level) is a measure of inconsistency between the null hypothesis and the observed sample. It is the probability, assuming that H0 is true, of obtaining a test statistic value at least as inconsistent with H0 as what actually resulted. You reject the null hypothesis when the P-value is small.**Using P-values to make a decision:**A decision in a hypothesis test is based on comparing the P-value to the chosen significance level a. H0 is rejected if the P-value ≤ a. H0 is not rejected if the P-value > a. For example, suppose that P-value = 0.0352 and a = 0.05. Then, because 0.0352 ≤ 0.05 H0 would be rejected. What decision would be made if a = 0.01?**Recall the 5 Steps for Performing a Hypothesis Test**• Identify the appropriate test and test statistic. • Select a significance level for the test. Verify that any conditions for the selected test are met. • Find the values of any sample statistics needed to calculate the value of the test statistic. • Calculate the value of the test statistic. • Determine the P-value for the test. • Compare the P-value to the selected significance level and make a decision to either reject H0 or fail to reject H0. • Provide a conclusion in words that is in context and addresses the question of interest.**Computing P-values**The calculation of the P-value depends on the form of the inequality in the alternative hypothesis. Ha: p > hypothesize value z curve P-value = area in upper tail Calculated z**Computing P-values**The calculation of the P-value depends on the form of the inequality in the alternative hypothesis. Ha: p < hypothesize value z curve P-value = area in lower tail Calculated z**Computing P-values**The calculation of the P-value depends on the form of the inequality in the alternative hypothesis. Ha: p ≠ hypothesize value P-value = sum of area in two tails z curve Calculated -z and z**A Large-Sample Test for a Population Proportion**Appropriate when the following conditions are met: • The sample is a random sample from the population of interest or the sample is selected in a way that would result in a representative sample. • The sample size n is large. This condition is met when both np> 10 and n (1 - p) >10. When these conditions are met, the following test statistic can be used: Where p0 is the hypothesized value from the null hypothesis**A Large-Sample Test for a Population Proportion Continued .**. . Null hypothesis: H0: p = p0 Ha: p < p0 Area under the z curve to the left of the calculated value of the test statistic Ha: p ≠ p0 2·(area to the right of z) if z is positive Or 2·(area to the left of z) if z is negative**In a study, 2205 adolescents ages 12 to 19 took a**cardiovascular treadmill test. The researchers conducting the study believed that the sample was representative of adolescents nationwide. Of the 2205 adolescents tested, 750 had a poor level of cardiovascular fitness. Does this sample provide convincing evidence that more than thirty percent of adolescents have a poor level of cardiovascular fitness? Hypothesis: Let p = proportion of American adolescents who have a poor level of cardiovascular fitness H0: p = 0.30 Ha: p> 0.30**Cardiovascular Fitness Continued . . .**H0: p = 0.30 versus Ha: p > 0.30 Method: Because the answers to the four key questions are 1) hypothesis testing, 2) sample data, 3) one categorical variable, and 4) one sample, consider a large sample hypothesis test for a population proportion. To select a significance level, you must consider the consequences of potential Type I and Type II errors. What are the Type I and Type II errors? Which has the most serious consequence? Significance level: a = 0.05 Because neither type of error is much worse than the other, you might choose a value of 0.05.**Cardiovascular Fitness Continued . . .**H0: p = 0.30 versus Ha: p > 0.30 Check: The researchers believed the sample to be representative of adolescents nationwide. The sample size is large enough because np0 = 2205(0.3) = 661.5 ≥ 10 and n (1 - p0) = 2205(0.7) = 1543.5 ≥ 10**Cardiovascular Fitness Continued . . .**H0: p = 0.30 versus Ha: p > 0.30 z = 4.00 P-value ≈ 0 Communicate Results: Decision: 0 < 0.05, Reject H0 Conclusion: The sample provides convincing evidence that more than 30% of adolescents have a poor fitness level. Notice that the conclusion answers the question that was posed in the problem.**A Few Final Things to Consider**• What about Small Samples? In np ≥ 10 and n (1 – p) ≥ 10, the standard normal distribution is a reasonable approximation to the distribution of the z test statistic when the null hypothesis is true. If the sample size is not large enough to satisfy the large sample conditions, the distribution of the test statistic may be quite different from the standard normal distribution. Thus, you can’t use the standard normal distribution to calculate P-values.**A Few Final Things to Consider**2. Choosing a Potential Method Take a look back at Table 7.1 (on page 420 and also on the inside back cover of the text). As you get into the habit of answering the four key questions for each new situation that you encounter, it will become easier to use this table to select an appropriate method in a given situation.**Because organically grown produce costs more than produce**that is not organically grown, the manager thinks this expansion will be profitable only if the proportion of store customers who would pay more for organic produce is greater than 0.30. Suppose that the manager of a grocery store is thinking about expanding the store’s selection of organically grown produce. What is a Type I error for this context? What is a Type II error for this context? The manager plans to ask each person in a random sample of customers if he or she would be willing to pay more for organic produce. He will then use the resulting data to test H0: p = 0.30 versus Ha: p > 0.30 using a significance level of a = 0.05.**How likely is it that the null hypothesis is rejected?**Grocery Store Problem Continued . . . H0: p = 0.30 versus Ha: p > 0.30**How likely is it that the null hypothesis is rejected?**Grocery Store Problem Continued . . . H0: p = 0.30 versus Ha: p > 0.30 Let’s investigate this further.