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(BQ) Part 2 book Introductory statistics hass contents: Confidence intervals for one population mean, inferences for two population means, inferences for population standard deviations, inferences for population proportions, ChiSquare procedures,...and other contents.
www.downloadslide.com CHAPTER Confidence Intervals for One Population Mean CHAPTER OBJECTIVES CHAPTER OUTLINE In this chapter, you begin your study of inferential statistics by examining methods for estimating the mean of a population As you might suspect, the statistic used to estimate ¯ Because of sampling error, you cannot the population mean, μ, is the sample mean, x expect x¯ to equal μ exactly Thus, providing information about the accuracy of the estimate is important, which leads to a discussion of confidence intervals, the main topic of this chapter In Section 8.1, we provide the intuitive foundation for confidence intervals Then, in Section 8.2, we present confidence intervals for one population mean when the population standard deviation, σ , is known Although, in practice, σ is usually unknown, we first consider, for pedagogical reasons, the case where σ is known In Section 8.3, we discuss confidence intervals for one population when the population standard deviation is unknown As a prerequisite to that topic, we introduce and describe one of the most important distributions in inferential statistics—Student’s t 8.1 Estimating a Population Mean 8.2 Confidence Intervals for One Population Mean When σ Is Known 8.3 Confidence Intervals for One Population Mean When σ Is Unknown CASE STUDY Bank Robberies: A Statistical Analysis In the article “Robbing Banks” (Significance, Vol 9, Issue 3, pp 17−21) B Reilly et al studied several aspects of bank robberies As these researchers state, “Robbing a bank is the staple crime of thrillers, movies and newspapers But bank robbery is not all it is cracked up to be.” The researchers concentrated on the factors that determine the amount of proceeds from bank robberies, and thus were able to work out both the economics of attempting one and of preventing one In particular, the researchers revealed that the return on an average bank robbery per person per raid is modest indeed—so modest that it is not worthwhile for the banks to spend too much money on such preventative measures as fast-rising screens at tellers’ windows The researchers obtained exclusive data from the British Bankers’ Association In one aspect of their study, they analyzed the data from a sample of 364 bank raids over a several-year period in the United Kingdom The following table repeats a portion of Table on page 31 of the article 353 www.downloadslide.com 354 CHAPTER Confidence Intervals for One Population Mean Variable Mean Std dev Amount stolen (pounds sterling) 20,330.5 Number of bank staff present 5.417 Number of customers present 2.000 Number of bank raiders 1.637 Travel time, in minutes from bank 4.557 to nearest police station After studying point estimates and confidence intervals in this chapter, you will be asked to use these 8.1 53,510.2 4.336 3.684 0.971 4.028 summary statistics to estimate the (population) means of the variables in the table Estimating a Population Mean A common problem in statistics is to obtain information about the mean, μ, of a population For example, we might want to know r the mean age of people in the civilian labor force, r the mean cost of a wedding, r the mean gas mileage of a new-model car, or r the mean starting salary of liberal-arts graduates If the population is small, we can ordinarily determine μ exactly by first taking a census and then computing μ from the population data If the population is large, however, as it often is in practice, taking a census is generally impractical, extremely expensive, or impossible Nonetheless, we can usually obtain sufficiently accurate information about μ by taking a sample from the population Point Estimate One way to obtain information about a population mean μ without taking a census is ¯ as illustrated in the next example to estimate it by a sample mean x, EXAMPLE 8.1 Point Estimate of a Population Mean Prices of New Mobile Homes The U.S Census Bureau publishes annual price figures for new mobile homes in Manufactured Housing Statistics The figures are obtained from sampling, not from a census A simple random sample of 36 new mobile homes yielded the prices, in thousands of dollars, shown in Table 8.1 Use the data to estimate the population mean price, μ, of all new mobile homes TABLE 8.1 Prices ($1000s) of 36 randomly selected new mobile homes 67.8 67.1 49.9 56.0 68.4 73.4 56.5 76.7 59.2 63.7 71.2 76.8 56.9 57.7 59.1 60.6 63.9 66.7 64.3 74.5 62.2 61.7 64.0 57.9 55.6 55.5 55.9 70.4 72.9 49.3 51.3 63.8 62.6 72.9 53.7 77.9 Solution We estimate the population mean price, μ, of all new mobile homes by ¯ of the 36 new mobile homes sampled From Table 8.1, the sample mean price, x, xi 2278 x¯ = = = 63.28 n 36 Interpretation Based on the sample data, we estimate the mean price, μ, of all new mobile homes to be approximately $63.28 thousand, that is, $63,280 Exercise 8.17 on page 359 An estimate of this kind is called a point estimate for μ because it consists of a single number, or point www.downloadslide.com 8.1 Estimating a Population Mean 355 As indicated in the following definition, the term point estimate applies to the use of a statistic to estimate any parameter, not just a population mean DEFINITION 8.1 ? What Does It Mean? Roughly speaking, a point estimate of a parameter is our best guess for the value of the unknown parameter based on sample data For instance, a sample mean is a point estimate of a population mean, and a sample standard deviation is a point estimate of a population standard deviation Point Estimate A point estimate of a parameter is the value of a statistic used to estimate the parameter In the previous example, the parameter is the mean price, μ, of all new mobile ¯ homes, which is unknown The point estimate of that parameter is the mean price, x, of the 36 mobile homes sampled, which is $63,280 In Section 7.2, we learned that the mean of the sample mean equals the population mean (μ x¯ = μ) In other words, on average, the sample mean equals the population mean For this reason, the sample mean is called an unbiased estimator of the population mean More generally, a statistic is called an unbiased estimator of a parameter if the mean of all its possible values equals the parameter; otherwise, the statistic is called a biased estimator of the parameter Ideally, we want our statistic to be unbiased and have small standard error In that case, chances are good that our point estimate (the value of the statistic) will be close to the parameter Confidence-Interval Estimate As you learned in Chapter 7, a sample mean is usually not equal to the population mean; generally, there is sampling error Therefore, we should accompany any point estimate of μ with information that indicates the accuracy of that estimate This information is called a confidence-interval estimate for μ, which we introduce in the next example EXAMPLE 8.2 Introducing Confidence Intervals Prices of New Mobile Homes Consider again the problem of estimating the (population) mean price, μ, of all new mobile homes by using the sample data in Table 8.1 Let’s assume that the population standard deviation of all such prices is $7.2 thousand, that is, $7200.† FIGURE 8.1 Normal score Normal probability plot of the price data in Table 8.1 ¯ that is, the sampling distribution of a Identify the distribution of the variable x, the sample mean for samples of size 36 b Use part (a) to show that approximately 95% of all samples of 36 new mobile homes have the property that the interval from x¯ − 2.4 to x¯ + 2.4 contains μ c Use part (b) and the sample data in Table 8.1 to find a 95% confidence interval for μ, that is, an interval of numbers that we can be 95% confident contains μ Solution a Figure 8.1 is a normal probability plot of the price data in Table 8.1 The plot shows we can reasonably presume that prices of new mobile homes are normally distributed Because n = 36, σ = 7.2, and prices of new mobile homes are normally distributed, Key Fact 7.2 on page 342 implies that r μx¯ = μ (which we don’t know), √ r σx¯ = σ/√n = 7.2/ 36 = 1.2, and r x¯ is normally distributed –1 –2 –3 50 55 60 65 70 75 80 Price ($1000s) In other words, for samples of size 36, the variable x¯ is normally distributed with mean μ and standard deviation 1.2 † We might know the population standard deviation from previous research or from a preliminary study of prices We examine the more usual case where σ is unknown in Section 8.3 www.downloadslide.com 356 CHAPTER Confidence Intervals for One Population Mean b Property of the empirical rule (Key Fact 6.6 on page 304) implies that, for a normally distributed variable, approximately 95% of all possible observations lie within two standard deviations to either side of the mean Applying this rule to the variable x¯ and referring to part (a), we see that approximately 95% of all samples of 36 new mobile homes have mean prices within 2.4 (i.e., · 1.2) of μ Equivalently, approximately 95% of all samples of 36 new mobile homes have the property that the interval from x¯ − 2.4 to x¯ + 2.4 contains μ c Because we are taking a simple random sample, each possible sample of size 36 is equally likely to be the one obtained From part (b), (approximately) 95% of all such samples have the property that the interval from x¯ − 2.4 to x¯ + 2.4 contains μ Hence, chances are 95% that the sample we obtain has that property Consequently, we can be 95% confident that the sample of 36 new mobile homes whose prices are shown in Table 8.1 has the property that the interval from x¯ − 2.4 to x¯ + 2.4 contains μ For that sample, x¯ = 63.28, so x¯ − 2.4 = 63.28 − 2.4 = 60.88 and x¯ + 2.4 = 63.28 + 2.4 = 65.68 Thus our 95% confidence interval is from 60.88 to 65.68 Interpretation We can be 95% confident that the mean price, μ, of all new mobile homes is somewhere between $60,880 and $65,680 We can be 95% confident that lies in here $60,880 Exercise 8.19 on page 359 $65,680 Note: Although this or any other 95% confidence interval may or may not contain μ, we can be 95% confident that it does because the method that we used to construct the confidence interval gives correct results 95% of the time With the previous example in mind, we now define confidence-interval estimate and related terms As indicated, the terms apply to estimating any parameter, not just a population mean ? DEFINITION 8.2 What Does It Mean? A confidence-interval estimate for a parameter provides a range of numbers along with a percentage confidence that the parameter lies in that range Confidence-Interval Estimate Confidence interval (CI): An interval of numbers obtained from a point estimate of a parameter Confidence level: The confidence we have that the parameter lies in the confidence interval (i.e., that the confidence interval contains the parameter) Confidence-interval estimate: The confidence level and confidence interval ¯ A confidence interval for a population mean depends on the sample mean, x, which in turn depends on the sample selected For example, suppose that the prices of the 36 new mobile homes sampled were as shown in Table 8.2 instead of as in Table 8.1 TABLE 8.2 Prices ($1000s) of another sample of 36 randomly selected new mobile homes 73.0 53.2 66.5 60.2 72.1 66.6 64.7 72.1 61.2 65.3 62.5 54.9 53.0 68.9 61.3 66.1 75.5 58.4 62.1 64.1 63.8 69.1 68.0 72.0 56.0 65.8 79.2 68.8 75.7 64.1 69.2 64.3 65.7 60.6 68.0 77.9 Then we would have x¯ = 65.83 so that x¯ − 2.4 = 65.83 − 2.4 = 63.43 and x¯ + 2.4 = 65.83 + 2.4 = 68.23 www.downloadslide.com 8.1 Estimating a Population Mean 357 In this case, the 95% confidence interval for μ would be from 63.43 to 68.23 We could be 95% confident that the mean price, μ, of all new mobile homes is somewhere between $63,430 and $68,230 Interpreting Confidence Intervals The next example stresses the importance of interpreting a confidence interval correctly It also illustrates that the population mean, μ, may or may not lie in the confidence interval obtained EXAMPLE 8.3 Interpreting Confidence Intervals Prices of New Mobile Homes Consider again the prices of new mobile homes As demonstrated in part (b) of Example 8.2, (approximately) 95% of all samples of 36 new mobile homes have the property that the interval from x¯ − 2.4 to x¯ + 2.4 contains μ In other words, if 36 new mobile homes are selected at random and their ¯ is computed, the interval from mean price, x, x¯ − 2.4 to x¯ + 2.4 (8.1) will be a 95% confidence interval for the mean price of all new mobile homes Illustrate that the mean price, μ, of all new mobile homes may or may not lie in the 95% confidence interval obtained Solution We used a computer to simulate 25 samples of 36 new mobile home prices each For the simulation, we assumed that μ = 65 (i.e., $65 thousand) and σ = 7.2 (i.e., $7.2 thousand) In reality, we don’t know μ; we are assuming a value for μ to illustrate a point For each of the 25 samples of 36 new mobile home prices, we did three things: ¯ used Equation (8.1) to obtain the 95% conficomputed the sample mean price, x; dence interval; and noted whether the population mean, μ = 65, actually lies in the confidence interval Figure 8.2 on the next page summarizes our results For each sample, we have drawn a graph on the right-hand side of Fig 8.2 The dot represents the sample ¯ in thousands of dollars, and the horizontal line represents the corresponding mean, x, 95% confidence interval Note that the population mean, μ, lies in the confidence interval only when the horizontal line crosses the dashed line Figure 8.2 reveals that μ lies in the 95% confidence interval in 24 of the 25 samples, that is, in 96% of the samples If, instead of 25 samples, we simulated 1000, we would probably find that the percentage of those 1000 samples for which μ lies in the 95% confidence interval would be even closer to 95% Hence we can be 95% confident that any computed 95% confidence interval will contain μ Margin of Error In Example 8.2(c), we found a 95% confidence interval for the mean price, μ, of all new mobile homes Looking back at the construction of that confidence interval on page 356, we see that the endpoints of the confidence interval are 60.88 and 65.68 (in thousands of dollars) These two numbers were obtained, respectively, by subtracting 2.4 from and adding 2.4 to the sample mean of 63.28 In other words, the endpoints of the confidence interval can be expressed as 63.28 ± 2.4 The number 2.4 is called the margin of error because it indicates how accurate ¯ is as an estimate for the value of the unknown parameter (in our guess (in this case, x) this case, μ) Here, we can be 95% confident that the mean price, μ, of all new mobile homes is within $2.4 thousand of the sample mean price of $63.28 thousand Using this terminology, we can express the (endpoints of the) confidence interval as follows: point estimate ± margin of error www.downloadslide.com 358 CHAPTER Confidence Intervals for One Population Mean FIGURE 8.2 Twenty-five confidence intervals for the mean price of all new mobile homes, each based on a sample of 36 new mobile homes 60 61 62 63 64 Sample – x 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 65.45 64.21 64.33 63.59 64.17 65.07 64.56 65.28 65.87 64.61 65.51 66.45 64.88 63.85 67.73 64.70 64.60 63.88 66.82 63.84 63.08 65.80 64.93 66.30 66.93 in Cl? 95% Cl 63.05 61.81 61.93 61.19 61.77 62.67 62.16 62.88 63.47 62.21 63.11 64.05 62.48 61.45 65.33 62.30 62.20 61.48 64.42 61.44 60.68 63.40 62.53 63.90 64.53 to to to to to to to to to to to to to to to to to to to to to to to to to 65 66 67 68 69 70 67.85 66.61 66.73 65.99 66.57 67.47 66.96 67.68 68.27 67.01 67.91 68.85 67.28 66.25 70.13 67.10 67.00 66.28 69.22 66.24 65.48 68.20 67.33 68.70 69.33 yes yes yes yes yes yes yes yes yes yes yes yes yes yes no yes yes yes yes yes yes yes yes yes yes This expression will be the form of most of the confidence intervals that we encounter in our study of statistics Observe that the margin of error is half the length of the confidence interval or, equivalently, the length of the confidence interval is twice the margin of error By the way, it is interesting to note that margin of error is analogous to tolerance in manufacturing and production processes Exercises 8.1 Understanding the Concepts and Skills 8.1 The value of a statistic used to estimate a parameter is called a of the parameter 8.2 What is a confidence-interval estimate of a parameter? Why is such an estimate superior to a point estimate? 8.3 When estimating an unknown parameter, what does the margin of error indicate? 8.4 Express the form of most of the confidence intervals that you will encounter in your study of statistics in terms of “point estimate” and “margin of error.” 8.5 Suppose that you take 1000 simple random samples from a population and that, for each sample, you obtain a 95% confidence interval for an unknown parameter Approximately how many of those confidence intervals will contain the value of the unknown parameter? 8.6 Suppose that you take 500 simple random samples from a population and that, for each sample, you obtain a 90% confidence interval for an unknown parameter Approximately how many of those confidence intervals will not contain the value of the unknown parameter? 8.7 A simple random sample is taken from a population and yields the following data for a variable of the population: 40 21 25 13 32 10 37 28 24 16 39 Find a point estimate for the population mean (i.e., the mean of the variable) 8.8 A simple random sample is taken from a population and yields the following data for a variable of the population: www.downloadslide.com 8.1 Estimating a Population Mean 20 16 12 21 359 b Is your point estimate in part (a) likely to equal μ exactly? Explain your answer Find a point estimate for the population mean (i.e., the mean of the variable) 8.9 Refer to Exercise 8.7 and find a point estimate for the population standard deviation (i.e., the standard deviation of the variable) 8.10 Refer to Exercise 8.8 and find a point estimate for the population standard deviation (i.e., the standard deviation of the variable) In each of Exercises 8.11–8.16, we provide a sample mean, sample size, and population standard deviation In each case, perform the following tasks a Find a 95% confidence interval for the population mean (Note: You may want to review Example 8.2, which begins on page 355.) b Identify and interpret the margin of error c Express the endpoints of the confidence interval in terms of the point estimate and the margin of error 8.11 x¯ = 20, n = 36, σ = 8.12 x¯ = 25, n = 36, σ = 8.13 x¯ = 31, n = 57, σ = 8.14 x¯ = 41, n = 57, σ = 8.15 x¯ = 50, n = 16, σ = 8.16 x¯ = 55, n = 16, σ = For Exercises 8.19–8.24, you may want to review Example 8.2, which begins on page 355 8.19 Wedding Costs A random sample of 20 recent weddings in a country yielded a mean wedding cost of $26,324.61 Assume that recent wedding costs in this country are normally distributed with a standard deviation of $8000 Complete parts (a) through (c) below a Determine a 95% confidence interval for the mean cost, μ, of all recent weddings in this country b Interpret your result in part (a) Choose the correct answer below c Does the mean cost of all recent weddings in this country lie in the confidence interval obtained in part (a)? Explain your answer 8.20 Cottonmouth Litter Size Refer to Exercise 8.18 Assume that σ = 2.4 a Obtain a 95% confidence interval for the mean number of young per litter of all female eastern cottonmouths b Interpret your result in part (a) c Does the mean number of young per litter of all female eastern cottonmouths lie in the confidence interval you obtained in part (a)? Explain your answer 8.21 A simple random sample of 20 new automobile models yielded the data shown to the right on fuel tank capacity, in gallons Applying the Concepts and Skills 15.1 21.6 15.9 20.8 8.17 Wedding Costs According to Bride’s Magazine, getting married these days can be expensive when the costs of the reception, engagement ring, bridal gown, pictures—just to name a few—are included A simple random sample of 20 recent U.S weddings yielded the following data on wedding costs, in dollars 19,496 27,806 30,098 32,269 23,789 21,203 13,360 40,406 18,312 29,288 33,178 35,050 14,554 34,081 42,646 21,083 18,460 27,896 24,053 19,510 a Use the data to obtain a point estimate for the population mean wedding cost, μ, of all recent U.S weddings (Note: The sum of the data is $526,538.) b Is your point estimate in part (a) likely to equal μ exactly? Explain your answer 8.18 Cottonmouth Litter Size In the article “The Eastern Cottonmouth (Agkistrodon piscivorus) at the Northern Edge of Its Range” (Journal of Herpetology, Vol 29, No 3, pp 391–398), C Blem and L Blem examined the reproductive characteristics of the eastern cottonmouth, a once widely distributed snake whose numbers have decreased recently due to encroachment by humans A simple random sample of 44 female cottonmouths yielded the following data on number of young per litter 10 8 12 14 8 12 12 7 11 11 12 10 10 11 3 a Use the data to obtain a point estimate for the mean number of young per litter, μ, of all female eastern cottonmouths (Note: xi = 334.) 16.5 20.7 21.8 16.2 22.9 16.2 16.6 21.6 16.7 18.8 22.9 17.6 18.9 20.2 22.7 21.8 a Find a point estimate for the mean fuel tank capacity for all new automobile models (Note: xi = 385.5 gallons.) b Determine a 95.44% confidence interval for the mean fuel tank capacity of all new automobile models Assume σ = 3.40 gallons c How would you decide whether fuel tank capacities for new automobile models are approximately normally distributed? d Must fuel tank capacities for new automobile models be exactly normally distributed for the confidence interval that you obtained in part (b) to be approximately correct? Explain your answer 8.22 Home Improvements The American Express Retail Index provides information on budget amounts for home improvements The following table displays the budgets, in dollars, of 45 randomly sampled home improvement jobs in the United States 3179 3915 2659 4503 2750 1032 4800 4660 2911 2069 1822 3843 3570 3605 3056 4093 5265 1598 2948 2550 2285 2467 2605 1421 631 1478 2353 3643 1910 4550 955 4200 2816 5145 5069 2773 514 3146 551 3125 3104 4557 2026 2124 1573 a Determine a point estimate for the population mean budget, μ, for such home improvement jobs Interpret your answer in words (Note: The sum of the data is $129,849.) b Obtain a 95% confidence interval for the population mean budget, μ, for such home improvement jobs and interpret your result in words Assume that the population standard deviation of budgets for home improvement jobs is $1350 www.downloadslide.com 360 CHAPTER Confidence Intervals for One Population Mean c How would you decide whether budgets for such home improvement jobs are approximately normally distributed? d Must the budgets for such home improvement jobs be exactly normally distributed for the confidence interval that you obtained in part (b) to be approximately correct? Explain your answer 8.23 Giant Tarantulas A tarantula has two body parts The anterior part of the body is covered above by a shell, or carapace In the paper “Reproductive Biology of Uruguayan Theraphosids” (The Journal of Arachnology, Vol 30, No 3, pp 571–587), F Costa and F Perez–Miles discussed a large species of tarantula whose common name is the Brazilian giant tawny red A simple random sample of 15 of these adult male tarantulas provided the following data on carapace length, in millimeters (mm) 15.7 19.2 16.4 18.3 19.8 16.8 19.7 18.1 18.9 17.6 18.0 18.5 19.0 20.9 19.5 a Obtain a normal probability plot of the data b Based on your result from part (a), is it reasonable to presume that carapace length of adult male Brazilian giant tawny red tarantulas is normally distributed? Explain your answer c Find and interpret a 95% confidence interval for the mean carapace length of all adult male Brazilian giant tawny red tarantulas The population standard deviation is 1.76 mm d In Exercise 6.97, we noted that the mean carapace length of all adult male Brazilian giant tawny red tarantulas is 18.14 mm Does your confidence interval in part (c) contain the population mean? Would it necessarily have to? Explain your answers 8.24 Serum Cholesterol Levels Information on serum total cholesterol level is published by the Centers for Disease Control and Prevention in National Health and Nutrition Examination Survey A simple random sample of 12 U.S females 20 years old or older provided the following data on serum total cholesterol level, in milligrams per deciliter (mg/dL) 8.2 260 169 289 173 190 191 214 178 110 129 241 185 a Obtain a normal probability plot of the data b Based on your result from part (a), is it reasonable to presume that serum total cholesterol level of U.S females 20 years old or older is normally distributed? Explain your answer c Find and interpret a 95% confidence interval for the mean serum total cholesterol level of U.S females 20 years old or older The population standard deviation is 44.7 mg/dL d In Exercise 6.98, we noted that the mean serum total cholesterol level of U.S females 20 years old or older is 206 mg/dL Does your confidence interval in part (c) contain the population mean? Would it necessarily have to? Explain your answers Extending the Concepts and Skills 8.25 New Mobile Homes A government bureau publishes annual price figures for new mobile homes A simple random sample of 36 new mobile homes yielded the following prices, in thousands of dollars Assume that the population standard deviation of all such prices is $10.2 thousand, that is, $10,200 Use the data to obtain a 99.7% confidence interval for the mean price of all new mobile homes Prices of New Mobile Homes Prices ($1000s) of 36 Randomly Selected New Mobile Homes 68.7 67.7 50.1 55.2 68.7 73.5 56.7 76.3 59.7 64.5 71.1 78.2 58.0 56.9 59.7 61.1 64.8 65.8 63.1 74.3 61.2 61.0 64.1 57.3 56.4 55.8 56.3 71.3 72.1 49.3 51.1 64.9 61.8 74.4 52.7 76.8 8.26 New Mobile Homes Refer to Examples 8.1 and 8.2 Use the data in Table 8.1 on page 354 to obtain a 68% confidence interval for the mean price of all new mobile homes (Hint: Proceed as in Example 8.2, but use Property of the empirical rule on page 304 instead of Property 2.) Confidence Intervals for One Population Mean When σ Is Known In Section 8.1, we showed how to find a 95% confidence interval for a population mean, that is, a confidence interval at a confidence level of 95% In this section, we generalize the arguments used there to obtain a confidence interval for a population mean at any prescribed confidence level To begin, we introduce some general notation used with confidence intervals Frequently, we want to write the confidence level in the form − α, where α is a number between and 1; that is, if the confidence level is expressed as a decimal, α is the number that must be subtracted from to get the confidence level To find α, we simply subtract the confidence level from If the confidence level is 95%, then α = − 0.95 = 0.05; if the confidence level is 90%, then α = − 0.90 = 0.10; and so on Next, recall from Section 6.2 that the symbol z α denotes the z-score that has area α to its right under the standard normal curve So, for example, z 0.05 denotes the z-score that has area 0.05 to its right, and z α/2 denotes the z-score that has area α/2 to its right Note that, for any normally distributed variable, 100 (1 − α)% of all possible observations lie within z α/2 standard deviations to either side of the mean You should draw a graph to verify that result www.downloadslide.com 8.2 Confidence Intervals for One Population Mean When σ Is Known 361 Obtaining Confidence Intervals for a Population Mean When σ Is Known We now develop a step-by-step procedure to obtain a confidence interval for a population mean when the population standard deviation is known In doing so, we assume that the variable under consideration is normally distributed Because of the central limit theorem, however, the procedure will also work to obtain an approximately correct confidence interval when the sample size is large, regardless of the distribution of the variable The basis of our confidence-interval procedure is stated in Key Fact 7.2: If x is a normally distributed variable with mean μ and standard deviation σ , then, for samples of size n, the√variable x¯ is also normally distributed and has mean μ and standard deviation σ/ n As in Section 8.1, we can use Property of the empirical rule√to conclude that approximately 95% of all samples of size n have means within · σ/ n of μ, as depicted in Fig 8.3(a) FIGURE 8.3 (a) Approximately 95% of all samples have means within standard deviations of μ; (b) 100(1 − α )% of all samples have means within zα/2 standard deviations of μ 0.025 0.95 −2• +2• √n −2 0.025 √n – x z 1− /2 −z /2 +z • −z √n /2 (a) /2 z /2 • √n /2 – x z (b) More generally (and more precisely), √ we can say that 100(1 − α)% of all samples of size n have means within z α/2 · σ/ n of μ, as depicted in Fig 8.3(b) Equivalently, we can say that 100(1 − α)% of all samples of size n have the property that the interval from σ σ x¯ − z α/2 · √ to x¯ + z α/2 · √ n n contains μ Consequently, we have Procedure 8.1, called the one-mean z-interval procedure, or, when no confusion can arise, simply the z-interval procedure.† PROCEDURE 8.1 One-Mean z-Interval Procedure Purpose To find a confidence interval for a population mean, μ Assumptions Simple random sample Normal population or large sample σ known Step For a confidence level of − α, use Table II to find zα/2 Step The confidence interval for μ is from σ σ to x¯ + zα/2 · √ , x¯ − zα/2 · √ n n where z α/2 is found in Step 1, n is the sample size, and x¯ is computed from the sample data Step Interpret the confidence interval Note: The confidence interval is exact for normal populations and is approximately correct for large samples from nonnormal populations † The one-mean z-interval procedure is also known as the one-sample z-interval procedure and the one-variable z-interval procedure We prefer “one-mean” because it makes clear the parameter being estimated www.downloadslide.com 362 CHAPTER Confidence Intervals for One Population Mean Note: By saying that the confidence interval is exact, we mean that the true confidence level equals − α; by saying that the confidence interval is approximately correct, we mean that the true confidence level only approximately equals − α Before applying Procedure 8.1, we need to make several comments about it and the assumptions for its use r We use the term normal population as an abbreviation for “the variable under consideration is normally distributed.” r The z-interval procedure works reasonably well even when the variable is not normally distributed and the sample size is small or moderate, provided the variable is not too far from being normally distributed Thus we say that the z-interval procedure is robust to moderate violations of the normality assumption.† r Watch for outliers because their presence calls into question the normality assumption Moreover, even for large samples, outliers can sometimes unduly affect a z-interval because the sample mean is not resistant to outliers Key Fact 8.1 lists some general guidelines for use of the z-interval procedure KEY FACT 8.1 When to Use the One-Mean z-Interval Procedure‡ r For small samples—say, of size less than 15—the z-interval procedure should be used only when the variable under consideration is normally distributed or very close to being so r For samples of moderate size—say, between 15 and 30—the z-interval procedure can be used unless the data contain outliers or the variable under consideration is far from being normally distributed r For large samples—say, of size 30 or more—the z-interval procedure can be used essentially without restriction However, if outliers are present and their removal is not justified, you should compare the confidence intervals obtained with and without the outliers to see what effect the outliers have If the effect is substantial, use a different procedure or take another sample, if possible r If outliers are present but their removal is justified and results in a data set for which the z-interval procedure is appropriate (as previously stated), the procedure can be used Key Fact 8.1 makes it clear that you should conduct preliminary data analyses before applying the z-interval procedure More generally, the following fundamental principle of data analysis is relevant to all inferential procedures KEY FACT 8.2 ? What Does It Mean? Always look at the sample data (by constructing a histogram, normal probability plot, boxplot, etc.) prior to performing a statisticalinference procedure to help check whether the procedure is appropriate A Fundamental Principle of Data Analysis Before performing a statistical-inference procedure, examine the sample data If any of the conditions required for using the procedure appear to be violated, not apply the procedure Instead use a different, more appropriate procedure, if one exists Even for small samples, where graphical displays must be interpreted carefully, it is far better to examine the data than not to Remember, though, to proceed cautiously † A statistical procedure that works reasonably well even when one of its assumptions is violated (or moderately violated) is called a robust procedure relative to that assumption ‡ Statisticians also consider skewness Roughly speaking, the more skewed the distribution of the variable under consideration, the larger is the sample size required for the validity of the z-interval procedure See, for instance, the paper “How Large Does n Have to Be for Z and t Intervals?” by D Boos and J Hughes-Oliver (The American Statistician, Vol 54, No 2, pp 121–128) www.downloadslide.com TABLE IV Values of t ␣ t␣ NOTE: See the version of Table IV in Appendix A for additional values of t df t 0.10 t 0.05 t 0.025 t 0.01 t 0.005 df 3.078 1.886 1.638 1.533 6.314 2.920 2.353 2.132 12.706 4.303 3.182 2.776 31.821 6.965 4.541 3.747 63.657 9.925 5.841 4.604 1.476 1.440 1.415 1.397 1.383 2.015 1.943 1.895 1.860 1.833 2.571 2.447 2.365 2.306 2.262 3.365 3.143 2.998 2.896 2.821 4.032 3.707 3.499 3.355 3.250 10 11 12 13 14 1.372 1.363 1.356 1.350 1.345 1.812 1.796 1.782 1.771 1.761 2.228 2.201 2.179 2.160 2.145 2.764 2.718 2.681 2.650 2.624 3.169 3.106 3.055 3.012 2.977 10 11 12 13 14 15 16 17 18 19 1.341 1.337 1.333 1.330 1.328 1.753 1.746 1.740 1.734 1.729 2.131 2.120 2.110 2.101 2.093 2.602 2.583 2.567 2.552 2.539 2.947 2.921 2.898 2.878 2.861 15 16 17 18 19 20 21 22 23 24 1.325 1.323 1.321 1.319 1.318 1.725 1.721 1.717 1.714 1.711 2.086 2.080 2.074 2.069 2.064 2.528 2.518 2.508 2.500 2.492 2.845 2.831 2.819 2.807 2.797 20 21 22 23 24 25 26 27 28 29 1.316 1.315 1.314 1.313 1.311 1.708 1.706 1.703 1.701 1.699 2.060 2.056 2.052 2.048 2.045 2.485 2.479 2.473 2.467 2.462 2.787 2.779 2.771 2.763 2.756 25 26 27 28 29 30 35 40 50 60 1.310 1.306 1.303 1.299 1.296 1.697 1.690 1.684 1.676 1.671 2.042 2.030 2.021 2.009 2.000 2.457 2.438 2.423 2.403 2.390 2.750 2.724 2.704 2.678 2.660 30 35 40 50 60 70 80 90 100 1000 2000 1.294 1.292 1.291 1.290 1.282 1.282 1.667 1.664 1.662 1.660 1.646 1.646 1.994 1.990 1.987 1.984 1.962 1.961 2.381 2.374 2.369 2.364 2.330 2.328 2.648 2.639 2.632 2.626 2.581 2.578 70 80 90 100 1000 2000 1.282 1.645 1.960 2.326 2.576 z 0.10 z 0.05 z 0.025 z 0.01 z 0.005 www.downloadslide.com TABLE II Second decimal place in z Areas under the standard normal curve 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00 z 3.9 3.8 3.7 3.6 3.5 † z 0.0001 0.0001 0.0001 0.0002 0.0001 0.0001 0.0001 0.0002 0.0001 0.0001 0.0001 0.0002 0.0001 0.0001 0.0001 0.0002 0.0001 0.0001 0.0001 0.0002 0.0001 0.0001 0.0001 0.0002 0.0001 0.0001 0.0001 0.0002 0.0001 0.0001 0.0001 0.0002 0.0001 0.0001 0.0002 0.0002 0.0000 0.0001 0.0001 0.0002 0.0002 0.0002 0.0003 0.0005 0.0007 0.0010 0.0003 0.0004 0.0005 0.0007 0.0010 0.0003 0.0004 0.0005 0.0008 0.0011 0.0003 0.0004 0.0006 0.0008 0.0011 0.0003 0.0004 0.0006 0.0008 0.0011 0.0003 0.0004 0.0006 0.0008 0.0012 0.0003 0.0004 0.0006 0.0009 0.0012 0.0003 0.0005 0.0006 0.0009 0.0013 0.0003 0.0005 0.0007 0.0009 0.0013 0.0003 0.0005 0.0007 0.0010 0.0013 3.4 3.3 3.2 3.1 3.0 0.0014 0.0019 0.0026 0.0036 0.0048 0.0014 0.0020 0.0027 0.0037 0.0049 0.0015 0.0021 0.0028 0.0038 0.0051 0.0015 0.0021 0.0029 0.0039 0.0052 0.0016 0.0022 0.0030 0.0040 0.0054 0.0016 0.0023 0.0031 0.0041 0.0055 0.0017 0.0023 0.0032 0.0043 0.0057 0.0018 0.0024 0.0033 0.0044 0.0059 0.0018 0.0025 0.0034 0.0045 0.0060 0.0019 0.0026 0.0035 0.0047 0.0062 2.9 2.8 2.7 2.6 2.5 0.0064 0.0084 0.0110 0.0143 0.0183 0.0066 0.0087 0.0113 0.0146 0.0188 0.0068 0.0089 0.0116 0.0150 0.0192 0.0069 0.0091 0.0119 0.0154 0.0197 0.0071 0.0094 0.0122 0.0158 0.0202 0.0073 0.0096 0.0125 0.0162 0.0207 0.0075 0.0099 0.0129 0.0166 0.0212 0.0078 0.0102 0.0132 0.0170 0.0217 0.0080 0.0104 0.0136 0.0174 0.0222 0.0082 0.0107 0.0139 0.0179 0.0228 2.4 2.3 2.2 2.1 2.0 0.0233 0.0294 0.0367 0.0455 0.0559 0.0239 0.0301 0.0375 0.0465 0.0571 0.0244 0.0307 0.0384 0.0475 0.0582 0.0250 0.0314 0.0392 0.0485 0.0594 0.0256 0.0322 0.0401 0.0495 0.0606 0.0262 0.0329 0.0409 0.0505 0.0618 0.0268 0.0336 0.0418 0.0516 0.0630 0.0274 0.0344 0.0427 0.0526 0.0643 0.0281 0.0351 0.0436 0.0537 0.0655 0.0287 0.0359 0.0446 0.0548 0.0668 1.9 1.8 1.7 1.6 1.5 0.0681 0.0823 0.0985 0.1170 0.1379 0.0694 0.0838 0.1003 0.1190 0.1401 0.0708 0.0853 0.1020 0.1210 0.1423 0.0721 0.0869 0.1038 0.1230 0.1446 0.0735 0.0885 0.1056 0.1251 0.1469 0.0749 0.0901 0.1075 0.1271 0.1492 0.0764 0.0918 0.1093 0.1292 0.1515 0.0778 0.0934 0.1112 0.1314 0.1539 0.0793 0.0951 0.1131 0.1335 0.1562 0.0808 0.0968 0.1151 0.1357 0.1587 1.4 1.3 1.2 1.1 1.0 0.1611 0.1867 0.2148 0.2451 0.2776 0.1635 0.1894 0.2177 0.2483 0.2810 0.1660 0.1922 0.2206 0.2514 0.2843 0.1685 0.1949 0.2236 0.2546 0.2877 0.1711 0.1977 0.2266 0.2578 0.2912 0.1736 0.2005 0.2296 0.2611 0.2946 0.1762 0.2033 0.2327 0.2643 0.2981 0.1788 0.2061 0.2358 0.2676 0.3015 0.1814 0.2090 0.2389 0.2709 0.3050 0.1841 0.2119 0.2420 0.2743 0.3085 0.9 0.8 0.7 0.6 0.5 0.3121 0.3483 0.3859 0.4247 0.4641 0.3156 0.3520 0.3897 0.4286 0.4681 0.3192 0.3557 0.3936 0.4325 0.4721 0.3228 0.3594 0.3974 0.4364 0.4761 0.3264 0.3632 0.4013 0.4404 0.4801 0.3300 0.3669 0.4052 0.4443 0.4840 0.3336 0.3707 0.4090 0.4483 0.4880 0.3372 0.3745 0.4129 0.4522 0.4920 0.3409 0.3783 0.4168 0.4562 0.4960 0.3446 0.3821 0.4207 0.4602 0.5000 0.4 0.3 0.2 0.1 0.0 † For z 3.90, the areas are 0.0000 to four decimal places www.downloadslide.com TABLE II (cont.) Second decimal place in z Areas under the standard normal curve z † z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 0.1 0.2 0.3 0.4 0.5000 0.5398 0.5793 0.6179 0.6554 0.5040 0.5438 0.5832 0.6217 0.6591 0.5080 0.5478 0.5871 0.6255 0.6628 0.5120 0.5517 0.5910 0.6293 0.6664 0.5160 0.5557 0.5948 0.6331 0.6700 0.5199 0.5596 0.5987 0.6368 0.6736 0.5239 0.5636 0.6026 0.6406 0.6772 0.5279 0.5675 0.6064 0.6443 0.6808 0.5319 0.5714 0.6103 0.6480 0.6844 0.5359 0.5753 0.6141 0.6517 0.6879 0.5 0.6 0.7 0.8 0.9 0.6915 0.7257 0.7580 0.7881 0.8159 0.6950 0.7291 0.7611 0.7910 0.8186 0.6985 0.7324 0.7642 0.7939 0.8212 0.7019 0.7357 0.7673 0.7967 0.8238 0.7054 0.7389 0.7704 0.7995 0.8264 0.7088 0.7422 0.7734 0.8023 0.8289 0.7123 0.7454 0.7764 0.8051 0.8315 0.7157 0.7486 0.7794 0.8078 0.8340 0.7190 0.7517 0.7823 0.8106 0.8365 0.7224 0.7549 0.7852 0.8133 0.8389 1.0 1.1 1.2 1.3 1.4 0.8413 0.8643 0.8849 0.9032 0.9192 0.8438 0.8665 0.8869 0.9049 0.9207 0.8461 0.8686 0.8888 0.9066 0.9222 0.8485 0.8708 0.8907 0.9082 0.9236 0.8508 0.8729 0.8925 0.9099 0.9251 0.8531 0.8749 0.8944 0.9115 0.9265 0.8554 0.8770 0.8962 0.9131 0.9279 0.8577 0.8790 0.8980 0.9147 0.9292 0.8599 0.8810 0.8997 0.9162 0.9306 0.8621 0.8830 0.9015 0.9177 0.9319 1.5 1.6 1.7 1.8 1.9 0.9332 0.9452 0.9554 0.9641 0.9713 0.9345 0.9463 0.9564 0.9649 0.9719 0.9357 0.9474 0.9573 0.9656 0.9726 0.9370 0.9484 0.9582 0.9664 0.9732 0.9382 0.9495 0.9591 0.9671 0.9738 0.9394 0.9505 0.9599 0.9678 0.9744 0.9406 0.9515 0.9608 0.9686 0.9750 0.9418 0.9525 0.9616 0.9693 0.9756 0.9429 0.9535 0.9625 0.9699 0.9761 0.9441 0.9545 0.9633 0.9706 0.9767 2.0 2.1 2.2 2.3 2.4 0.9772 0.9821 0.9861 0.9893 0.9918 0.9778 0.9826 0.9864 0.9896 0.9920 0.9783 0.9830 0.9868 0.9898 0.9922 0.9788 0.9834 0.9871 0.9901 0.9925 0.9793 0.9838 0.9875 0.9904 0.9927 0.9798 0.9842 0.9878 0.9906 0.9929 0.9803 0.9846 0.9881 0.9909 0.9931 0.9808 0.9850 0.9884 0.9911 0.9932 0.9812 0.9854 0.9887 0.9913 0.9934 0.9817 0.9857 0.9890 0.9916 0.9936 2.5 2.6 2.7 2.8 2.9 0.9938 0.9953 0.9965 0.9974 0.9981 0.9940 0.9955 0.9966 0.9975 0.9982 0.9941 0.9956 0.9967 0.9976 0.9982 0.9943 0.9957 0.9968 0.9977 0.9983 0.9945 0.9959 0.9969 0.9977 0.9984 0.9946 0.9960 0.9970 0.9978 0.9984 0.9948 0.9961 0.9971 0.9979 0.9985 0.9949 0.9962 0.9972 0.9979 0.9985 0.9951 0.9963 0.9973 0.9980 0.9986 0.9952 0.9964 0.9974 0.9981 0.9986 3.0 3.1 3.2 3.3 3.4 0.9987 0.9990 0.9993 0.9995 0.9997 0.9987 0.9991 0.9993 0.9995 0.9997 0.9987 0.9991 0.9994 0.9995 0.9997 0.9988 0.9991 0.9994 0.9996 0.9997 0.9988 0.9992 0.9994 0.9996 0.9997 0.9989 0.9992 0.9994 0.9996 0.9997 0.9989 0.9992 0.9994 0.9996 0.9997 0.9989 0.9992 0.9995 0.9996 0.9997 0.9990 0.9993 0.9995 0.9996 0.9997 0.9990 0.9993 0.9995 0.9997 0.9998 3.5 3.6 3.7 3.8 3.9 0.9998 0.9998 0.9999 0.9999 1.0000† 0.9998 0.9998 0.9999 0.9999 0.9998 0.9999 0.9999 0.9999 0.9998 0.9999 0.9999 0.9999 0.9998 0.9999 0.9999 0.9999 0.9998 0.9999 0.9999 0.9999 0.9998 0.9999 0.9999 0.9999 0.9998 0.9999 0.9999 0.9999 0.9998 0.9999 0.9999 0.9999 0.9998 0.9999 0.9999 0.9999 For z 3.90, the areas are 1.0000 to four decimal places www.downloadslide.com This page intentionally left blank www.downloadslide.com Indexes for Case Studies & Biographical Sketches Chapter 10 11 12 13 14 15 16 Case Study Biographical Sketch Top Films of All Time 1, 34 World’s Richest People 36, 91 The Beatles’ Song Length 93, 154 Texas Hold’em 156, 222 Aces Wild on the Sixth at Oak Hill 223, 261 Chest Sizes of Scottish Militiamen 262, 306 The Chesapeake and Ohio Freight Study 307, 329 Bank Robberies: A Statistical Analysis 331, 366 Gender and Sense of Direction 367, 437 Dexamethasone Therapy and IQ 438, 511 Speaker Woofer Driver Manufacturing 513, 543 Arrested Youths 544, 575 Eye and Hair Color 576, 617 Healthcare: Spending and Outcomes 618, 656 Shoe Size and Height 657, 696 Self-Perception and Physical Activity 698, 741 Florence Nightingale 34 Adolphe Quetelet 92 John Tukey 154 Andrei Kolmogorov 222 James Bernoulli 261 Carl Friedrich Gauss 306 Pierre-Simon Laplace 329 William Gosset 366 Jerzy Neyman 437 Gertrude Cox 511 W Edwards Deming 543 Abraham de Moivre 575 Karl Pearson 617 Adrien Legendre 656 Sir Francis Galton 697 Sir Ronald Fisher 741 www.downloadslide.com Index Adjacent values, 153 Alternative hypothesis, 390 choice of, 391 Analysis of residuals, 684 Analysis of variance, 720 one way, 723 ANOVA, see Analysis of variance Approximately normally distributed, 288 Assessing normality, 313 Associated variables, 612 Association, 611, 612 and causation, 624 hypothesis test for, 622 At least, 190 At most, 190 At random, 180 Back-to-back stem-and-leaf diagram, 494 Bar chart, 67 by computer, 71 procedure for constructing, 68 Bar graph segmented, 612 Basic counting rule, 230, 231 Basic principle of counting, see Basic counting rule Bayes’s rule, 223, 226 Bayes, Thomas, 223 Bell-shaped distribution, 99 Bernoulli trials, 261 and binomial coefficients, 263 Bernoulli, James biographical sketch, 283 Bias nonresponse, 39 response, 39 in sample surveys, 35 in samples, 39 undercoverage, 39 Biased estimator, 341, 355 Bimodal distribution, 98 Binomial coefficients, 260 and Bernoulli trials, 263 Binomial distribution, 261, 264, 319 as an approximation to the hypergeometric distribution, 268 by computer, 268 normal approximation to, 318 Poisson approximation to, 276 procedure for approximating by a normal distribution, 321 shape of, 266 Binomial probability formula, 264 procedure for finding, 264 Binomial probability tables, 266 Binomial random variable, 264 mean of, 267 standard deviation of, 267 Bins, 74 Bivariate data, 96, 201, 609 quantitative, 646 Bivariate quantitative data, 646 Bootstrap confidence intervals for one population mean, 384 Bootstrap distribution, 384 Box-and-whisker diagram, 153 Boxplot, 153 by computer, 157 Categorical variable, 59 Categories, 74 Cells of a contingency table, 202, 610 Census, 31 Census data, 100 Central limit theorem, 343 Certain event, 182 Chebyshev’s rule, 139 use in estimating relative standing, 171 χα2 , 537, 599 Chi-square curve, 536, 599 Chi-square curves basic properties of, 536, 599 Chi-square distribution, 536, 599 for a goodness-of-fit test, 602 for a homogeneity test, 629 for an independence test, 621 Chi-square goodness-of-fit test, 600, 603 by computer, 605 Chi-square homogeneity test, 628, 630 by computer, 624 Chi-square independence test, 619, 622 by computer, 624 concerning the assumptions for, 624 distribution of test statistic for, 621 χ -interval procedure for one population standard deviation, 543 Chi-square procedures, 598 Chi-square table use of, 537, 599 χ -test for one population standard deviation, 540 CI, 356 Class limits, 75 Class mark, 76 Class midpoint, 77 Class width, 76, 77 Classes, 74 choosing, 78, 95 cutpoint grouping, 76 limit grouping, 75 single-value, 74 Cluster sampling, 40 procedure for implementing, 40 Cochran, W G., 532, 534, 603, 622 Coefficient of determination, 661 by computer, 664 interpretation of, 661 relation to linear correlation coefficient, 670 Combination, 234 Combinations rule, 235 Complement, 188 Complementation rule, 196 Completely randomized design, 49 Conditional distribution, 611, 680, 681 by computer, 614 Conditional mean, 680, 681 Conditional mean t-interval procedure, 702 Conditional probability, 207 definition of, 207 rule for, 210 Conditional probability distribution, 214 Conditional probability rule, 210 Conditional standard deviation, 681 Confidence interval, 356 length of, 364 relation to hypothesis testing, 417, 475 Confidence interval for a conditional mean in regression, 702 Confidence interval for the difference between two population means by computer for independent samples, and normal populations or large samples, 485 by computer for independent samples, normal populations or large samples, and equal but unknown standard deviations, 475 by computer for a paired sample, and normal differences or a large sample, 514 I-1 www.downloadslide.com I-2 INDEX Confidence interval for the difference between two population means (cont.) independent samples, and normal populations or large samples, 484 independent samples, normal populations or large samples, and equal but unknown standard deviations, 474 nonpooled t-interval procedure, 484 in one-way analysis of variance, 742 paired sample, and normal differences or large sample, 513 paired t-interval procedure, 513 pooled t-interval procedure, 474 Confidence interval for the difference between two population proportions, 588 by computer for large and independent samples, 590 two-proportions plus-four z-interval procedure, 589 Confidence interval for one population mean by computer in regression, 706 by computer when σ is known, 368 by computer when σ is unknown, 380 in one-way analysis of variance, 742 in regression, 702 σ known, 361 σ unknown, 377 Confidence interval for one population proportion, 570 by computer, 573 one-proportion plus-four z-interval procedure, 573, 578 Confidence interval for one population standard deviation, 543 by computer, 544 Confidence interval for the ratio of two population standard deviations, 555 by computer, 557 Confidence interval for the slope of a population regression line, 696 by computer, 697 Confidence intervals relation to hypotheses tests, 417 Confidence level, 356, 365 and accuracy, 365 family, 742 individual, 742 and margin of error, 365 Confidence-interval estimate, 355, 356 Contingency table, 96, 201, 609 by computer, 613 Continuous data, 60 Continuous variable, 59, 60 Control, 48 Control group, 48 Correction for continuity, 321 Correlation, 640 of events, 214 rank, 675 Correlation t-test, 710, 711 Count of a class, 64 Counting rules, 230 application to probability, 236 basic counting rule, 230, 231 combinations rule, 235 permutations rule, 233 special permutations rule, 234 Cox, Gertrude Mary biographical sketch, 533 Critical values, 400 obtaining, 401 use as a decision criterion in a hypothesis test, 400 Critical-value approach to hypothesis testing, 398 Cumulative frequency, 96 Cumulative probability, 266, 307 inverse, 308 Cumulative relative frequency, 96 Curves density, 285 Curvilinear regression, 654 Cutpoint grouping, 76 terms used in, 77 Data, 60 bivariate, 96, 201, 609 continuous, 60 discrete, 60 grouping of, 74 population, 100 qualitative, 60 quantitative, 60 sample, 100 univariate, 96, 201, 609 Data analysis a fundamental principle of, 362 Data classification and the choice of a statistical method, 61 Data set, 60 Deciles, 147 Degrees of freedom, 375 for an F-curve, 549, 721 Degrees of freedom for the denominator, 549, 721 Degrees of freedom for the numerator, 549, 721 Deming, W Edwards biographical sketch, 565 de Moivre, Abraham, 318 biographical sketch, 597 Density curves, 285 basic properties of, 285 Dependent events, 218 Descriptive measure resistant, 119 Descriptive measures, 115 of center, 116 of central tendency, 116 of spread, 127 of variation, 127 Descriptive statistics, 24, 25 Designed experiment, 27 Deviations from the mean, 128 Discrete data, 60 Discrete random variable, 246 mean of, 254 probability distribution of, 247 standard deviation of, 256 variance of, 256 Discrete random variables independence of, 259 Discrete variable, 59, 60 Distribution bell shaped, 98 bimodal, 98 conditional, 611, 680, 681 of a data set, 97 of the difference between the observed and predicted values of a response variable, 703 left skewed, 99 marginal, 612 mean, 681 multimodal, 98 normal, 284 of a population, 100 of the predicted value of a response variable, 701 rectangular, 98 reverse J shaped, 99 right skewed, 99 of a sample, 100 standard deviation, 681 symmetric, 98 triangular, 98 uniform, 98 unimodal, 98 of a variable, 100 Dotplot, 81 by computer, 87 procedure for constructing, 81 Double blinding, 53 Empirical rule, 140 for variables, 304 Equal-likelihood model, 182 Error, 648, 725 Error mean square, 725 Error sum of squares, 661 by computer, 664 computing formula for in regression, 663 in one-way analysis of variance, 725 in regression, 661 Estimator biased, 341, 355 unbiased, 341, 355 Event, 180, 186, 188 (A & B), 188 (A or B), 188 certain, 182 complement of, 188 given, 207 impossible, 182 (not E), 188 occurrence of, 187 www.downloadslide.com INDEX Events, 186 correlation of, 214 dependent, 218 exhaustive, 223 independent, 214, 217, 218, 222 mutually exclusive, 191 negatively correlated, 214 notation and graphical display for, 188 positively correlated, 214 relationships among, 188 Excel, 35 Exhaustive events, 223 Expectation, 254 Expected frequencies, 601 for a chi-square goodness-of-fit test, 602 for a chi-square homogeneity test, 629 for a chi-square independence test, 621 Expected utility, 258 Expected value, 254 Experiment, 180 Experimental design, 47 principles of, 48 Experimental units, 48 Experimentation, 31 Explanatory variable, 652 Exploratory data analysis, 58, 177 Exponential distribution, 348 Exponentially distributed variable, 348 Extrapolation, 652 Factor, 48, 723 Factorials, 232, 260 Failure, 261 Fα , 549, 722 Family confidence level, 742 F-curve basic properties of, 549, 721 F-distribution, 549, 552, 721 Finite-population correction factor, 341 F-interval procedure for two population standard deviations, 555 First quartile, 147, 148 Fisher, Ronald, 27, 549, 721 biographical sketch, 763 Five-number summary, 151 f /N rule, 180 Focus database, 56 Frequency, 64 cumulative, 96 Frequency distribution of qualitative data, 64 procedure for constructing, 64 Frequency histogram, 78 Frequentist interpretation of probability, 182 F-statistic, 725 for comparing two population standard deviations, 552 in one-way analysis of variance, 729 F-table use of, 549, 722 F-test for two population standard deviations, 553 by computer, 557 Functions utility, 258 Fundamental counting rule, see Basic counting rule Galton, Francis, 639 biographical sketch, 719 Gauss, Carl Friedrich, 678 biographical sketch, 328 General addition rule, 197 General multiplication rule, 215 Geometric distribution, 273 Given event, 207 Goodness of fit chi-square test for, 603 Gosset, William Sealy, 375 biographical sketch, 388 Graph improper scaling of, 107 truncated, 106 Grouped data formulas for the sample mean and sample standard deviation, 138 Grouping choosing the method, 77 by computer, 85 guidelines for, 74 single-value, 74 Heteroscedasticity, 681 Histogram, 78 by computer, 86 frequency, 78 percent, 78 probability, 247 procedure for constructing, 78 relative frequency, 78 Homogeneous, 628 Homoscedasticity, 681 Hypergeometric distribution, 268, 272 binomial approximation to, 268 Hypothesis, 390 Hypothesis test, 389, 390 choosing the hypotheses, 390 logic of, 392 possible conclusions for, 396 relation to confidence interval, 417, 475 Hypothesis test for association of two variables of a population, 622 Hypothesis test for one population mean by computer for σ known, 417 by computer for σ unknown, 426 σ known, 410 σ unknown, 424 Wilcoxon signed-rank test, 434 Hypothesis test for one population proportion, 579 by computer, 581 I-3 Hypothesis test for one population standard deviation, 541 by computer, 544 non-robustness of, 540 Hypothesis test for a population linear correlation coefficient, 711 by computer, 713 Hypothesis test for several population means Kruskal–Wallis test, 753 one-way ANOVA test, 731, 732 Hypothesis test for the slope of a population regression line, 694 by computer, 697 Hypothesis test for two population means by computer for independent samples, and normal populations or large samples, 485 by computer for independent samples, normal populations or large samples, and equal but unknown standard deviations, 475 by computer for a paired sample, and normal differences or a large sample, 514 independent samples, and normal populations or large samples, 481 independent samples, normal populations or large samples, and equal but unknown standard deviations, 470 Mann–Whitney test, 497 nonpooled t-test, 481 paired sample, and normal differences or a large sample, 510 paired t-test, 510 paired Wilcoxon signed-rank test, 521 pooled t-test, 470 Hypothesis test for two population proportions, 586 by computer for large and independent samples, 590 Hypothesis test for two population standard deviations, 553 by computer, 557 non-robustness of, 554 Hypothesis test for the utility of a regression, 694 Hypothesis testing critical-value approach to, 398 P-value approach to, 403 relation to confidence intervals, 417 Hypothesis tests critical-value approach to, 402 P-value approach to, 408 relation to confidence intervals, 417 Impossible event, 182 Improper scaling, 107 Inclusive, 190 Independence, 217 for three events, 222 Independent, 217, 218 Independent events, 217, 218, 222 special multiplication rule for, 218 versus mutually exclusive events, 219 www.downloadslide.com I-4 INDEX Independent random variables, 259 Independent samples, 461 Independent samples t-interval procedure, 484 Independent samples t-test, 481 pooled, 469 Independent simple random samples, 461 Indices, 120 Individual confidence level, 742 Inferences for two population means choosing between a pooled and a nonpooled t-procedure, 485 Inferential statistics, 24, 25 Influential observation, 652 Intercept, 643 Interquartile range, 150 Inverse cumulative probability, 308 IQR, 150 Joint percentage distribution, 206 Joint probability, 203 Joint probability distribution, 204 Kolmogorov, A N biographical sketch, 244 Kruskal–Wallis test, 750 alternate version of, 755 comparison with the one-way ANOVA test, 755 by computer, 755 method for dealing with ties, 750 procedure for, 753 test statistic for, 751 K -statistic, 751 Laplace, Pierre-Simon, 318 biographical sketch, 351 Law of averages, 255 Law of large numbers, 255 Leaf, 82 Least-squares criterion, 647, 648 Left-skewed distribution, 99 Left-tailed test, 391 rejection region for, 400 Legendre, Adrien-Marie biographical sketch, 678 Levels, 48, 723 Limit grouping, 75 terms used in, 76 Line, 641 Linear correlation coefficient, 667 and causation, 671 by computer, 672 computing formula for, 669 relation to coefficient of determination, 670 warning on the use of, 671 Linear equation with one independent variable, 641 Linear regression, 640 by computer, 654 warning on the use of, 654 Linearly correlated variables, 710 Linearly uncorrelated variables, 710 Lower class cutpoint, 76, 77 Lower class limit, 75, 76 Lower cutpoint of a class, 76, 77 Lower limit, 152 of a class, 75, 76 Mα , 494 Mann–Whitney table using, 494 Mann–Whitney test, 493, 497 alternate version of, 501 comparison with the pooled t-test, 500 by computer, 501 determining critical values for, 494 method for dealing with ties, 496 procedure for, 497 using a normal approximation, 506 Mann–Whitney–Wilcoxon test, 493 Margin of error, 357, 363–344 and confidence level, 365 for the estimate of μ, 364 for the estimate of p, 571 for the estimate of p1 − p2 , 594 for a one-mean t-interval, 380 and sample size, 366 Marginal distribution, 612 by computer, 614 Marginal probability, 203 Mark of a class, 76 Maximum error of the estimate, 370 Mean, 116 of a binomial random variable, 267 by computer, 121 conditional, 680 deviations from, 128 of a discrete random variable, 254 interpretation for random variables, 255 of a Poisson random variable, 275 of a population, see Population mean of a sample, see Sample mean trimmed, 119, 127 ¯ 336 of x, Mean of a random variable, 254 properties of, 259 Mean of a variable, 162 Measures of center, 116 comparison of, 118 Measures of central tendency, 116 Measures of spread, 127 Measures of variation, 127 Median, 117 by computer, 121 Minitab, 35 Modality, 98 Mode, 118 Modified boxplot, 153 Multimodal distributions, 98 Multiple comparisons Tukey method, 742 Multiple regression, 705 Multiplication rule, see Basic counting rule Multistage sampling, 45 Mutually exclusive events, 191 and the special addition rule, 195 versus independent events, 219 Negatively linearly correlated variables, 668, 710 Neyman, Jerzy biographical sketch, 459 Nightingale, Florence biographical sketch, 56 Nonhomogeneous, 628 Nonparametric methods, 379, 430 Nonpooled t-interval procedure, 484 Nonpooled t-test, 481 Nonrejection region, 400 Nonresponse, 39 Normal curve, 288 equation of, 288 parameters of, 288 standard, 291 Normal differences, 509 Normal distribution, 284, 288 approximate, 288 as an approximation to the binomial distribution, 318 assessing using normal probability plots, 313 by computer, 307 standard, 291 Normal population, 362 Normal probability plots, 313 use in detecting outliers, 314 Normal scores, 312 Normally distributed population, 288 Normally distributed variable, 288 procedure for finding a range, 306 procedure for finding percentages for, 302 68-95-99.7 rule for, 304 standardized version of, 291 Not statistically significant, 396 Null hypothesis, 390 choice of, 390 Number of failures, 568 Number of successes, 568 Observation, 60 Observational study, 27 Observed frequencies, 601 Observed significance level, 406 Occurrence of an event, 187 Odds, 186 Ogive, 96 One-mean t-interval procedure, 377 www.downloadslide.com INDEX One-mean t-test, 421 procedure for, 424 One-mean z-interval procedure, 361 One-mean z-test, 409, 410 obtaining critical values for, 401 obtaining the P-value for, 406 One-median sign test, 443 One-proportion plus-four z-interval procedure, 573 One-proportion z-interval procedure, 569 One-proportion z-test, 579 One-sample sign test, 443 One-sample t-interval procedure, 377 One-sample t-test, 421, 424 One-sample Wilcoxon confidence-interval procedure, 379 One-sample Wilcoxon signed-rank test, 429 One-sample z-interval procedure, 361 for a population proportion, 570 One-sample z-interval procedure for a population proportion, 569 One-sample z-test, 409 for a population proportion, 579 One-sample z-test for a population proportion, 579 One-standard-deviation χ -interval procedure, 543 One-standard-deviation χ -test, 540 One-tailed test, 391 One-variable proportion interval procedure, 569 One-variable proportion test, 579 One-variable sign test, 443 One-variable t-interval procedure, 377 One-variable t-test, 421 One-variable Wilcoxon signed-rank test, 429 One-variable z-interval procedure, 361 One-variable z-test, 409 One-way analysis of variance, 723 assumptions for, 723 by computer, 736 distribution of test statistic for, 729 procedure for, 732 One-way ANOVA identity, 730 One-way ANOVA table, 730 One-way ANOVA test, 731, 732 comparison with the Kruskal–Wallis test, 755 Ordinal data, 63 measures of center for, 126 Outlier, 126, 151 detection of with normal probability plots, 314 effect on the standard deviation, 138 identification of, 152 in regression, 652 Paired difference, 508 Paired samples, 507 Paired sign test, 529 Paired t-interval procedure, 512, 513 Paired t-test, 510 comparison with the paired Wilcoxon signed-rank test, 524 Paired Wilcoxon signed-rank test, 521 comparison with the paired t-test, 524 procedure for, 521 Parameter, 165 Parameters of a normal curve, 288 Parametric methods, 379, 430 Pearson product moment correlation coefficient, see Linear correlation coefficient Pearson, Karl, 27, 719 biographical sketch, 639 Percent histogram, 78 Percentage and probability, 180 and relative frequency, 65 Percentage distribution joint, 206 Percentiles, 147 of a normally distributed variable, 312 Permutation, 232 Permutation distribution, 480 Permutation test for comparing two means, 479 Permutations rule, 233 special, 234 Pictogram, 107 Pie chart, 67 by computer, 69 procedure for constructing, 67 Plus-four confidence interval procedures, 573, 589 Point estimate, 354, 355 Poisson distribution, 273, 274 as an approximation to the binomial distribution, 276 by computer, 277 Poisson probability formula, 274 Poisson random variable, 274 mean of, 275 standard deviation of, 275 Poisson, Simeon, 273, 352 Pool, 469 Pooled independent samples t-interval procedure, 473 Pooled independent samples t-test, 469 Pooled sample proportion, 586 Pooled sample standard deviation, 469 Pooled t-interval procedure, 473, 474 Pooled t-test, 469, 470 comparison with the Mann–Whitney test, 500 Pooled two-variable t-interval procedure, 473 Pooled two-variable t-test, 469 Population, 25 distribution of, 100 normally distributed, 288 I-5 Population data, 100 Population distribution, 100 Population linear correlation coefficient, 710 Population mean, 162 Population median notation for, 168 Population proportion, 566–546 Population regression equation, 681 Population regression line, 681 Population standard deviation, 164 computing formula for, 164 confidence interval for, 543 hypothesis test for, 541 Population standard deviations confidence interval for the ratio of two, 556 hypothesis test for comparing, 553 Population variance, 164 Positively linearly correlated variables, 668, 710 Posterior probability, 227 Potential outliers, 152 Power, 448 and sample size, 453 Power curve, 449 procedure for, 449 Practical significance versus statistical significance, 416 Predicted value t-interval procedure, 704 Prediction interval, 703 by computer, 706 procedure for, 704 relation to confidence interval, 703 Predictor variable, 652 Prior probability, 227 Probability application of counting rules to, 236 basic properties of, 182 conditional, 207 cumulative, 266, 307 equally-likely outcomes, 180 frequentist interpretation of, 182 inverse cumulative, 308 joint, 203 marginal, 203 model of, 182 notation for, 195 posterior, 227 prior, 227 rules of, 195 Probability distribution binomial, 264 conditional, 214 of a discrete random variable, 247 geometric, 273 hypergeometric, 268, 272 interpretation of, 250 joint, 204 Poisson, 273, 274 Probability histogram, 247 Probability model, 182 Probability sampling, 32 Probability theory, 178 www.downloadslide.com I-6 INDEX Proportion population, see Population proportion sample, see Sample proportion sampling distribution of, 568, 569 Proportional allocation, 43 P-value, 405 as the observed significance level, 406 determining, 406 general procedure for obtaining, 409 use in assessing the evidence against the null hypothesis, 408 use as a decision criterion in a hypothesis test, 406 qα , 742 q-curve, 742 q-distribution, 742 Qualitative data, 60 bar chart of, 67 frequency distribution of, 64 pie chart of, 67 relative-frequency distribution of, 65 Qualitative variable, 59, 60 Quantitative data, 60 bivariate, 646 choosing classes for, 95 dotplot of, 81 histogram of, 78 organizing, 74 stem-and-leaf diagram of, 82 using technology to organize, 85 Quantitative variable, 59, 60 Quartile first, 147, 148 second, 147, 148 third, 147, 148 Quartiles, 147, 148 of a normally distributed variable, 312 Quetelet, Adolphe biographical sketch, 114 Quintiles, 147 Random sample simple, 32 Random sampling, 32 with replacement, 32 without replacement, 32 simple, 33 systematic, 39 Random variable, 246 binomial, 264 discrete, see Discrete random variable interpretation of mean of, 255 notation for, 247 Poisson, 274 Random variables independence of, 259 Random-number generator, 35 Random-number table, 33 Randomization, 48 Randomized block design, 50 Range, 128 Rank correlation, 675 Rectangular distribution, 99 Regression multiple, 641, 705 simple linear, 705 Regression equation definition of, 649 determination of using the sample covariance, 660 formula for, 649 Regression identity, 662 Regression inferences assumptions for in simple linear regression, 681 Regression line, 649 criterion for finding, 654 definition of, 649 Regression model, 681 Regression sum of squares, 661 by computer, 664 computing formula for, 663 Regression t-interval procedure, 696 Regression t-test, 694 Rejection region, 400 Relative frequency, 65 cumulative, 96 and percentage, 65 Relative-frequency distribution procedure for constructing, 66 of qualitative data, 65 Relative-frequency histogram, 78 Relative-frequency polygon, 96 Relative standing and Chebyshev’s rule, 171 estimating, 171 Replication, 48 Representative sample, 32 Research hypothesis, 390 Residual, 684 in ANOVA, 723 Residual analysis in ANOVA, 723 Residual plot, 685 Residual standard deviation, 684 Resistant measure, 119 Response bias, 39 Response variable, 48, 735 in regression, 652 Reverse-J-shaped distribution, 99 Right skewed, 99 property of a χ -curve, 536, 599 property of an F-curve, 549, 721 Right-skewed distribution, 99 Right-tailed test, 391 rejection region for, 400 Robust, 362 Robust procedure, 362 Rounding error, 77 Roundoff error, 77 Rule of total probability, 223, 224 Rule of 24, 723 Same shape, 492 Same-shape populations, 496 Sample, 25 distribution of, 100 representative, 32 simple random, 32 size of, 121 stratified, 43 Sample covariance, 660 Sample data, 100 Sample distribution, 100 Sample mean, 121 as an estimate for a population mean, 163 formula for grouped data, 138 sampling distribution of, 331 standard error of, 339 Sample proportion, 567, 568 formula for, 568 pooled, 586 Sample size, 121 and accuracy, 366 for estimating the difference between two population proportions, 594 for estimating a population mean, 366 for estimating a population proportion, 572 and margin of error, 366 and power, 453 and sampling error, 334, 339 Sample space, 187, 188 Sample standard deviation, 128 computing formula for, 131, 132 defining formula for, 130, 131 as an estimate of a population standard deviation, 164 formula for grouped data, 138 pooled, 469 Sample variance, 130 Samples independent, 461 number possible, 236 paired, 507 Sampling, 31 cluster, 40 multistage, 45 with replacement, 268 without replacement, 268 simple random, 32 stratified, 43 systematic random, 39 Sampling distribution, 331 of the difference between two sample means, 466 of the difference between two sample proportions, 585 of the sample proportion, 568, 569 of the sample standard deviation, 540 of the slope of the regression line, 693 Sampling distribution of the sample mean, 331 for a normally distributed variable, 342 Sampling error, 330 and sample size, 334, 339 www.downloadslide.com INDEX Scatter diagram, 646 Scatterplot, 646 by computer, 654 Second quartile, 147, 148 Segmented bar graph, 612 Sensitivity, 229 Sign test for one median, 443 for two medians, 529 Significance level, 395 Simple linear regression, 705 Simple random paired sample, 507 Simple random sample, 32 by computer, 35 Simple random samples independent, 461 Simple random sampling, 32 with replacement, 32 without replacement, 32 Single-value classes, 74 Single-value grouping, 74 histograms for, 78 68-95-99.7 rule, 141, 304 Skewness, 99 Slope, 643 graphical interpretation of, 644 Spearman rank correlation coefficient, 675 Spearman, Charles, 675 Special addition rule, 195 Special multiplication rule, 218 Special permutations rule, 234 Specificity, 229 Squared deviations sum of, 129 Standard deviation of a binomial random variable, 267 of a discrete random variable, 256 of a Poisson random variable, 275 of a population, see Population standard deviation of a sample, see Sample standard deviation sampling distribution of, 540 ¯ 338 of x, Standard deviation of a random variable, 256 computing formula for, 256 properties of, 259 Standard deviation of a variable, 164 Standard error, 339 Standard error of the estimate, 683, 684 by computer, 687 Standard error of the sample mean, 339 Standard normal curve, 291 areas under, 297 basic properties of, 296 finding the z-score(s) for a specified area, 299 Standard normal distribution, 291 Standard-normal table use of, 297 Standard score, 167 Standardized variable, 166 Standardized version of a variable, 166 ¯ 374 of x, Statistic, 165 sampling distribution of, 331 test, 393 Statistical independence, 217 see also Independence Statistical significance versus practical significance, 416 Statistically dependent variables, 612 Statistically independent variables, 612 Statistically significant, 396 Statistics descriptive, 24, 25 inferential, 24, 25 Stat plots turning off, 87 Stem, 82 Stem-and-leaf diagram, 82 back-to-back, 494 procedure for constructing, 83 using more than one line per stem, 84 Stemplot, 82 Straight line, 641 Strata, 43 Stratified sampling, 43 proportional allocation in, 43 Stratified sampling theorem, 224 Stratified sampling with proportional allocation, 43 procedure for implementing, 43 Student’s t-distribution, see t-distribution Studentized range distribution, 742 ¯ 374 Studentized version of x, distribution of, 421 Subject, 48 Subscripts, 120 Success, 261 Success probability, 261 Sum of squared deviations, 129 Summation notation, 120 Symmetric, 98 property of a t-curve, 376 property of the standard normal curve, 296 Symmetric distribution, 98 assumption for the Wilcoxon signed-rank test, 430 Symmetric population, 433 Symmetry, 98 Systematic random sampling, 39 procedure for implementing, 39 tα , 376 t-curve, 375 basic properties of, 376 t-distribution, 375, 421, 469, 480, 509, 693, 702, 704, 710 Technology Center, 35 Test statistic, 393 Third quartile, 147, 148 I-7 TI-83/84 Plus, 35 Time series, 660 t-interval procedure, 377 Total sum of squares, 660 by computer, 664 in one-way analysis of variance, 730 in regression, 661 Transformations, 506 Treatment, 48, 725 Treatment group, 48 Treatment mean square in one-way analysis of variance, 725 Treatment sum of squares in one-way analysis of variance, 725 Tree diagram, 216 Trial, 260 Triangular distribution, 99 Trimmed mean, 119, 127 Truncated graph, 106 t-test, 421 comparison with the Wilcoxon signed-rank test, 438 Tukey multiple-comparison method by computer, 745 in one-way ANOVA, 742 procedure for, 743 Tukey, John, 153, 491 biographical sketch, 176 Tukey’s quick test, 492 Two-means z-interval procedure, 466 Two-means z-test, 466 Two-proportions plus-four z-interval procedure, 589 Two-proportions z-interval procedure, 588 Two-proportions z-test, 586 Two-sample F-interval procedure, 555 Two-sample F-test, 553 Two-sample t-interval procedure, 484 with equal variances assumed, 473 Two-sample t-test, 481 with equal variances assumed, 469 Two-sample z-interval procedure, 466 for two population proportions, 589 Two-sample z-test, 466 for two population proportions, 587 Two-standard-deviations F-interval procedure, 555 Two-standard-deviations F-test, 553 Two-tailed test, 391 Two-variable proportions interval procedure, 589 Two-variable proportions test, 587 Two-variable t-interval procedure, 484 pooled, 473 Two-variable t-test, 481 pooled, 469 Two-variable z-interval procedure, 466 Two-variable z-test, 466 Two-way table, 201, 609 Type I error, 393 probability of, 395 Type II error, 393 probability of, 395 www.downloadslide.com I-8 INDEX Type II error probabilities calculation of, 444 procedure for, 447 Unbiased estimator, 341, 355 Undercoverage, 39 Uniform distribution, 99, 351 Uniformly distributed variable, 351 Unimodal distribution, 98 Univariate data, 96, 201, 609 Upper class cutpoint, 76, 77 Upper class limit, 75, 76 Upper cutpoint of a class, 76, 77 Upper limit, 152 of a class, 75, 76 Utility expected, 258 Utility functions, 258 Variable, 59, 60 approximately normally distributed, 288 assessing normality, 313 categorical, 59 continuous, 59, 60 discrete, 59, 60 distribution of, 100 exponentially distributed, 348 mean of, 162 normally distributed, 288 qualitative, 59, 60 quantitative, 59, 60 standard deviation of, 164 standardized, 166 standardized version of, 166 uniformly distributed, 351 variance of, 164 Variance of a discrete random variable, 256 of a population, see Population variance of a random variable, 256 of a sample, see Sample variance of a variable, 164 Venn diagrams, 188 Venn, John, 188 Wα , 432 WeissStats Resource Site, 35 WeissStats site, 35 Whiskers, 153 Wilcoxon rank-sum test, 493 Wilcoxon signed-rank table using the, 432 Wilcoxon signed-rank test, 429 comparison with the t-test, 438 by computer, 438 dealing with ties, 436 determining critical values for, 432 observations equal to the null mean, 436 for paired samples, 521 procedure for, 434 testing a median with, 438 using a normal approximation, 443 x¯ critical value, 445 XLSTAT, 35 Y = functions turning off, 87 y-intercept, 643 z α , 360 z-curve, 296 see also Standard normal curve z-interval procedure, 361 for a population proportion, 569 z-score, 167 as a measure of relative standing, 168 z-test, 409 for a population proportion, 579 www.downloadslide.com This page intentionally left blank www.downloadslide.com Photo Credits About the Author Chapter p 5, Carol Weiss p 353, Folio/Alamy p 388 (top), Folio/Alamy p 388 (bottom), Pearson Education Chapter p 23, United Archives GmbH/Alamy p 24, TSN/ZUMAPRESS/Newscom p 25, Bettmann/Corbis p 27, Sports Illustrated/Getty Images p 56 (top), United Archives GmbH/Alamy p 56 (bottom), Library of Congress Chapter Chapter p 460, Steve Lovegrove/Fotolia p 533 (top), Steve 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Wavebreakmedia/Shutterstock p 678 (bottom), Pearson Education p 284, Holmes Garden Photos/Alamy p 328 (top), Holmes Garden Photos/Alamy p 328 (bottom), Library of Congress Prints and Photographs Division Chapter 15 Chapter Chapter 16 p 329, S oleg/ Shutterstock p 351 (top), S oleg/Shutterstock p 351 (bottom), Pearson Education p 720, Gareth Boden/Pearson Education p 763 (top), Gareth Boden/Pearson Education p 763 (bottom), Pearson Education p 679, Warren Goldswain/Fotolia p 718, Warren Goldswain/Fotolia p 719, Pearson Education C-1 ... 317 9 3 915 26 59 4503 27 50 10 32 4800 4660 2 911 20 69 18 22 3843 3570 3605 3056 4093 526 5 15 98 29 48 25 50 22 85 24 67 26 05 14 21 6 31 1478 23 53 3643 19 10 4550 955 420 0 2 816 514 5 5069 27 73 514 314 6 5 51. .. in the form point estimate ± margin of error 29 .80 7.40 14 .86 14 .86 21 2 17 5 19 5 16 6 21 7 21 8 20 8 20 7 21 3 16 1 18 9 15 2 17 9 2 21 20 8 17 2 18 4 18 1 23 7 22 0 Obtain and interpret a 95% confidence interval... 0 .10 t 0.05 t 0. 025 t 0. 01 t 0.005 df · · · 12 13 14 15 · · · · · · 1. 356 1. 350 1. 345 1. 3 41 · · · · · · 1. 7 82 1. 7 71 1.7 61 1.753 · · · · · · 2 .17 9 2 .16 0 2 .14 5 2 .13 1 · · · · · · 2. 6 81 2. 650 2. 624