+ Chapter 9: Testing a Claim Section 9.2 Tests About a Population Proportion The Practice of Statistics, 4th edition – For AP* STARNES, YATES, MOORE + Chapter 9 Testing a Claim 9.1 Significance Tests: The Basics 9.2 Tests about a Population Proportion 9.3 Tests about a Population Mean + Section 9.2 Tests About a Population Proportion Learning Objectives After this section, you should be able to… CHECK conditions for carrying out a test about a population proportion. CONDUCT a significance test about a population proportion. CONSTRUCT a confidence interval to draw a conclusion about for a two-sided test about a population proportion. Section 9.1 presented the reasoning of significance tests, including the idea of a P-value. In this section, we focus on the details of testing a claim about a population proportion. We’ll learn how to perform one-sided and two-sided tests about a population proportion. We’ll also see how confidence intervals and two-sided tests are related. Tests About a Population Proportion Confidence intervals and significance tests are based on the sampling distributions of statistics. That is, both use probability to say what would happen if we applied the inference method many times. + Introduction Out a Significance Test Does it provide convincing evidence against his claim? To find out, we must perform a significance test of H0: p = 0.80 Ha: p < 0.80 where p = the actual proportion of free throws the shooter makes in the long run. Check Conditions: In Chapter 8, we introduced three conditions that should be met before we construct a confidence interval for an unknown population proportion: Random, Normal, and Independent. These same three conditions must be verified before carrying out a significance test. Random We can view this set of 50 shots as a simple random sample from the population of all possible shots that the player takes. Normal Assuming H0 is true, p = 0.80. then np = (50)(0.80) = 40 and n (1 - p) = (50)(0.20) = 10 are both at least 10, so the normal condition is met. Independent In our simulation, the outcome of each shot does is determined by a random number generator, so individual observations are independent. Tests About a Population Proportion Recall our basketball player who claimed to be an 80% free-throw shooter. In an SRS of 50 free-throws, he made 32. His sample proportion of made shots, 32/50 = 0.64, is much lower than what he claimed. + Carrying Problem: Check the conditions for carrying out a significance test to determine if Jack should feel confident of victory in the mayoral election. Solution: The three required conditions are Random: A random sample of voters was selected for the poll. Normal: np0 = 100(0.5) = 50 ≥ 10 and n(1 – p0) = 100(1 – 0.5) = 50 ≥ 10 Independent: Assuming the poll was done confidentially, one response should not affect other responses. We must assume there are more than 10(100) = 1000 voters since we are sampling without replacement. + Tests About a Population Proportion Example – Can you be confident of victory? Alternate Out a Significance Test Calculations: Test statistic and P-value A significance test uses sample data to measure the strength of evidence against H0. Here are some principles that apply to most tests: • The test compares a statistic calculated from sample data with the value of the parameter stated by the null hypothesis. • Values of the statistic far from the null parameter value in the direction specified by the alternative hypothesis give evidence against H0. Definition: A test statistic measures how far a sample statistic diverges from what we would expect if the null hypothesis H0 were true, in standardized units. That is statistic - parameter test statistic = standard deviation of statistic Test About a Population Proportion If the null hypothesis H0 : p = 0.80 is true, then the player’s sample proportion of made free throws in an SRS of 50 shots would vary according to an approximately Normal sampling distribution with mean p(1 p) (0.8)(0.2) pˆ p 0.80 and standard deviation pˆ 0.0566 n 50 + Carrying Out a Hypothesis Test z 0.64 0.80 2.83 0.0566 The shaded area under the curve in (a) shows the P-value. (b) shows the corresponding area on the standard Normal curve, which displays the distribution of the z test statistic. Using Table A, we find that the P-value is P(z ≤ – 2.83) = 0.0023. So if H0 is true, and the player makes 80% of his free throws in the long run, there’s only about a 2-in-1000 chance that the player would make as few as 32 of 50 shots. Tests About a Population Proportion The test statistic says how far the sample result is from the null parameter value, and in what direction, on a standardized scale. You can use the test statistic to find the P-value of the test. In our free-throw shooter example, the sample proportion 0.64 is pretty far below the hypothesized value H0: p = 0.80. Standardizing, we get statistic - parameter test statistic = standard deviation of statistic + Carrying Example – How confident can you be? Problem: In an SRS of 100 voters, 56 favored Jack. (a) Calculate the test statistic. (b) Find and interpret the P-value. Solution: (a) Because the hypothesized value is = 0.5, the standardized test statistic is: 0.56 0.5 z 05(1 0.5) 100 1.20 (b) The shaded area under the curve below shows the P-value. Using the TI-84, the P-value = normalcdf(1.20,100) = 0.1151. If exactly 50% of all voters support Jack, there is about an 11.5% chance that 56% or more voters would support Jack in a random sample of size 100. Tests About a Population Proportion + Alternate One-Sample z Test for a Proportion State: What hypotheses do you want to test, and at what significance level? Define any parameters you use. Plan: Choose the appropriate inference method. Check conditions. Do: If the conditions are met, perform calculations. • Compute the test statistic. • Find the P-value. Conclude: Interpret the results of your test in the context of the problem. When the conditions are met—Random, Normal, and Independent, the sampling distribution ofpˆ is approximately Normal with mean p(1 p) pˆ p and standard deviation pˆ . n When performing a significance test, however, the null hypothesis specifies a value for p, which we will call p0. We assume that this value is correct when performing our calculations. Tests About a Population Proportion Significance Tests: A Four-Step Process + The One-Sample z Test for a Proportion One-Sample z Test for a Proportion Choose an SRS of size n from a large population that contains an unknown proportion p of successes. To test the hypothesis H0 : p = p0, compute the z statistic ˆ p p z Use this test p0 (1only p0 ) when the expected numbers n of successes and failures np0 and n(1 - p0) are Find the P-value by calculating the probability of getting a z statistic this large both at least 10 and the population or larger in the direction specified by the alternative hypothesis Ha: is at least 10 times as large as the sample. Tests About a Population Proportion The z statistic has approximately the standard Normal distribution when H0 is true. P-values therefore come from the standard Normal distribution. Here is a summary of the details for a one-sample z test for a proportion. + The One Potato, Two Potato State: We want to perform at test at the α = 0.10 significance level of H0: p = 0.08 Ha: p > 0.08 where p is the actual proportion of potatoes in this shipment with blemishes. Plan: If conditions are met, we should do a one-sample z test for the population proportion p. Random The supervisor took a random sample of 500 potatoes from the shipment. Normal Assuming H0: p = 0.08 is true, the expected numbers of blemished and unblemished potatoes are np0 = 500(0.08) = 40 and n(1 - p0) = 500(0.92) = 460, respectively. Because both of these values are at least 10, we should be safe doing Normal calculations. Independent Because we are sampling without replacement, we need to check the 10% condition. It seems reasonable to assume that there are at least 10(500) = 5000 potatoes in the shipment. Tests About a Population Proportion A potato-chip producer has just received a truckload of potatoes from its main supplier. If the producer determines that more than 8% of the potatoes in the shipment have blemishes, the truck will be sent away to get another load from the supplier. A supervisor selects a random sample of 500 potatoes from the truck. An inspection reveals that 47 of the potatoes have blemishes. Carry out a significance test at the α= 0.10 significance level. What should the producer conclude? + Example: One Potato, Two Potato Test statistic z pˆ 47/500 0.094. pˆ p0 0.094 0.08 1.15 p0 (1 p0 ) 0.08(0.92) n 500 P-value Using Table A or normalcdf(1.15,100), the desired P-value is P(z ≥ 1.15) = 1 – 0.8749 = 0.1251 Conclude: Since our P-value, 0.1251, is greater than the chosen significance level of α = 0.10, we fail to reject H0. There is not sufficient evidence to conclude that the shipment contains more than 8% blemished potatoes. The producer will use this truckload of potatoes to make potato chips. Tests About a Population Proportion Do: The sample proportion of blemished potatoes is + Example: Example: Better to be last? where p = the true proportion of American Idol fans who prefer the last performance they see. Plan: If conditions are met, we will perform a one-sample z test for p. Random: A random sample of American Idol fans was selected and the order in which the videos was viewed was randomized for each subject. Normal: np0 = (600) (1/12) = 50 ≥ 10, n(1 – p0)= (600)(1 – 1/12) = 550 ≥ 10. Independent: It is reasonable to assume that there are more than 10(600) = 6000 American Idol fans. Tests About a Population Proportion On shows like American Idol, contestants often wonder if there is an advantage to performing last. To investigate this, a random sample of 600 American Idol fans is selected and they are shown the audition tapes of 12 never-before-seen contestants. For each fan, the order of the 12 videos is randomly determined. Thus, if the order of performance doesn’t matter, we would expect approximately 1/12 of the fans to prefer the last contestant they view. In this study, 59 of the 600 fans preferred the last contestant they viewed. Does this data provide convincing evidence that there is an advantage to going last? State: We want to perform at test at the α = 0.05 significance level of H0: p = 1/12 Ha: p > 1/12 + Alternate Example: Better to be last? Test statistic z pˆ p0 p0 (1 p0 ) n 0.098 0.083 1.33 0.083(1 0.083) 600 P-value Using Table A or normalcdf(1.33,100), the desired P-value is P(z > 1.33) = 0.0918 Conclude: Since the P-value is greater than α (0.0918 > 0.05), we fail to reject the null hypothesis. There is not convincing evidence to conclude that there is an advantage to performing last in American Idol. Tests About a Population Proportion Do: The sample proportion of fans who preferred the last contestant is pˆ = 59/600 = 0.098. + Alternate Tests State: We want to perform at test at the α = 0.05 significance level of H0: p = 0.50 Ha: p ≠ 0.50 where p is the actual proportion of students in Taeyeon’s school who would say they have never smoked cigarettes. Plan: If conditions are met, we should do a one-sample z test for the population proportion p. Random Taeyeon surveyed an SRS of 150 students from his school. Normal Assuming H0: p = 0.50 is true, the expected numbers of smokers and nonsmokers in the sample are np0 = 150(0.50) = 75 and n(1 - p0) = 150(0.50) = 75. Because both of these values are at least 10, we should be safe doing Normal calculations. Independent We are sampling without replacement, we need to check the 10% condition. It seems reasonable to assume that there are at least 10(150) = 1500 students a large high school. Tests About a Population Proportion According to the Centers for Disease Control and Prevention (CDC) Web site, 50% of high school students have never smoked a cigarette. Taeyeon wonders whether this national result holds true in his large, urban high school. For his AP Statistics class project, Taeyeon surveys an SRS of 150 students from his school. He gets responses from all 150 students, and 90 say that they have never smoked a cigarette. What should Taeyeon conclude? Give appropriate evidence to support your answer. + Two-Sided Tests + Two-Sided pˆ 60/150 0.60. Test statistic z pˆ p0 0.60 0.50 2.45 p0 (1 p0 ) 0.50(0.50) n 150 P-value To compute this P-value, we find the area in one tail and double it. Using Table A or normalcdf(2.45, 100) yields P(z ≥ 2.45) = 0.0071 (the right-tail area). So the desired P-value is 2(0.0071) = 0.0142. Conclude: Since our P-value, 0.0142, is less than the chosen significance level of α = 0.05, we have sufficient evidence to reject H0 and conclude that the proportion of students at Taeyeon’s school who say they have never smoked differs from the national result of 0.50. Tests About a Population Proportion Do: The sample proportion is Example – Benford’s law and fraud State: We want to perform at test at the α = 0.05 significance level of H0: p = 0.301 Ha: p ≠ 0.301 where p = the true proportion of expenses that begin with the digit 1. Plan: If conditions are met, we will perform a one-sample z test for p. Random: A random sample of expenses was selected. Normal: np0 = (300) (0.301) = 90.3 ≥ 10, n(1 – p0) = (300)(1 – 0.301) = 209.7 ≥ 10. Independent: It is reasonable to assume that there are more than 10(300) = 3000 expenses in this company’s financial records. Tests About a Population Proportion When the accounting firm AJL and Associates audits a company’s financial records for fraud, they often use a test based on Benford’s law. Benford’s law states that the distribution of first digits in many real-life sources of data is not uniform. In fact, when there is no fraud, about 30.1% of the numbers in financial records begin with the digit 1. However, if the proportion of first digits that are 1 is significantly different from 0.301 in a random sample of records, AJL and Associates does a much more thorough investigation of the company. Suppose that a random sample of 300 expenses from a company’s financial records results in only 68 expenses that begin with the digit 1. Should AJL and Associates do a more thorough investigation of this company? + Alternate Example – Benford’s law and fraud pˆ 68 / 300 0.227. Test statistic z pˆ p0 p0 (1 p0 ) n 0.227 0.301 2.79 0.301(1 0.301) 300 P-value P-value: 2P(z < –2.79) = 2normalcdf(–100,–2.79) = 2(0.0026) = 0.0052 Conclude: Since the P-value is less than α (0.0052 < 0.05), we reject the null hypothesis. There is convincing evidence that the proportion of expenses that have first digit of 1 is not 0.301. Therefore, AJL and Associates should do a more thorough investigation of this company. Tests About a Population Proportion Do: The sample proportion of expenses that began with the digit 1 is + Alternate Confidence Intervals Give More Information Taeyeon found that 90 of an SRS of 150 students said that they had never smoked a cigarette. Before we construct a confidence interval for the population proportion p, we should check that both the number of successes and failures are at least 10. The number of successes and the number of failures in the sample are 90 and 60, respectively, so we can proceed with calculations. Our 95% confidence interval is: pˆ z * pˆ (1 pˆ ) 0.60(0.40) 0.60 1.96 0.60 0.078 (0.522,0.678) n 150 We are 95% confident that the interval from 0.522 to 0.678 captures the true proportion of students at Taeyeon’s high school who would say that they have never smoked a cigarette. Tests About a Population Proportion The result of a significance test is basically a decision to reject H0 or fail to reject H0. When we reject H0, we’re left wondering what the actual proportion p might be. A confidence interval might shed some light on this issue. + Why Example – Benford’s law and fraud Solution: (a) State: We want to estimate p = the true proportion of expenses that begin with the digit 1 at the 95% confidence level. Plan: We will use a one-sample z interval for p if the following conditions are satisfied. Random: A random sample of expenses was selected. Normal: np 68 10 and n(1 p) 232 10 Independent: It is reasonable to assume that there are more than 10(300) = 3000 expenses in this company’s financial records. Plan: p(1 p) 0.227(1 0.227) p z* 0.227 1.96 0.227 0.047 (0.180,0.274) n 300 Conclude: We are 95% confident that the interval from 0.176 to 0.270 captures the true proportion of expenses at this company that begin with the digit 1. (b) Since 0.301 is not in the interval from (a), 0.301 is not a plausible value for the true proportion of expenses that begin with the digit 1. Thus, this company should be investigated for fraud. Tests About a Population Proportion (a) Find and interpret a confidence interval for the true proportion of expenses that begin with the digit 1 for the company in the previous alternate example. (b) Use your interval from (a) to decide whether this company should be investigated for fraud. + Alternate Problem: Intervals and Two-Sided Tests level α (say, α = 0.05) and a 100(1 – α)% confidence interval (a 95% confidence interval if α = 0.05) give similar information about the population parameter. the sample proportion falls in the IfHowever, if the sample proportion “fail H0” region, like the fallstoinreject the “reject H0” region, the green value in confidence the figure, the resulting 95% interval resulting confidence interval would not95% include p0. In that case, would include p0. In that both both the significance testcase, and the the significance testwould and the confidence interval provide confidence interval would evidence that p0 is not thebe unable to rule out pvalue. 0 as a plausible parameter parameter value. Tests About a Population Proportion There is a link between confidence intervals and two-sided tests. The 95% confidence interval gives an approximate range of p0’s that would not be rejected by a two-sided test at the α = 0.05 significance level. The link isn’t perfect because the standard error used for the confidence interval is based on the sample proportion, while the denominator of the test statistic is based on the value p0 from the null hypothesis. A two-sided test at significance + Confidence + Section 9.2 Tests About a Population Proportion Summary In this section, we learned that… As with confidence intervals, you should verify that the three conditions— Random, Normal, and Independent—are met before you carry out a significance test. Significance tests for H0 : p = p0 are based on the test statistic z pˆ p0 p0 (1 p0 ) n with P-values calculated from the standard Normal distribution. The one-sample z test for a proportion is approximately correct when (1) the data were produced by random sampling or random assignment; (2) the population is at least 10 times as large as the sample; and (3) the sample is large enough to satisfy np0 ≥ 10 and n(1 - p0) ≥ 10 (that is, the expected numbers of successes and failures are both at least 10). + Section 9.2 Tests About a Population Proportion Summary In this section, we learned that… Follow the four-step process when you carry out a significance test: STATE: What hypotheses do you want to test, and at what significance level? Define any parameters you use. PLAN: Choose the appropriate inference method. Check conditions. DO: If the conditions are met, perform calculations. • Compute the test statistic. • Find the P-value. CONCLUDE: Interpret the results of your test in the context of the problem. Confidence intervals provide additional information that significance tests do not—namely, a range of plausible values for the true population parameter p. A two-sided test of H0 : p = p0 at significance level α gives roughly the same conclusion as a 100(1 – α)% confidence interval. + Looking Ahead… In the next Section… We’ll learn how to test a claim about a population mean. We’ll learn about Carrying out a significance test The one-sample t test for a mean Two-sided tests and confidence intervals Paired data and one-sample t procedures

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