CHAPTER. Sampling Distributions. Parameters and statistics. In this chapter we cover...

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1 Gandee Vasan/Getty Images Sampling Distributions How much on the average do American households earn? The government s Current Population Survey contacted a sample of 113,146 households in March Their mean income in 2004 was x = $60, That $60,528 describes the sample, but we use it to estimate the mean income of all households. This is an example of statistical inference: we use information from a sample to infer something about a wider population. Because the results of random samples and randomized comparative experiments include an element of chance, we can t guarantee that our inferences are correct. What we can guarantee is that our methods usually give correct answers. We will see that the reasoning of statistical inference rests on asking, How often would this method give a correct answer if I used it very many times? If our data come from random sampling or randomized comparative experiments, the laws of probability answer the question What would happen if we did this many times? This chapter presents some facts about probability that help answer this question. CHAPTER 11 In this chapter we cover... Parameters and statistics Statistical estimation and the law of large numbers Sampling distributions The sampling distribution of x The central limit theorem Statistical process control x charts Thinking about process control Parameters and statistics As we begin to use sample data to draw conclusions about a wider population, we must take care to keep straight whether a number describes a sample or a population. Here is the vocabulary we use. 271

2 272 CHAPTER 11 Sampling Distributions PARAMETER, STATISTIC A parameter is a number that describes the population. In statistical practice, the value of a parameter is not known because we cannot examine the entire population. A statistic is a number that can be computed from the sample data without making use of any unknown parameters. In practice, we often use a statistic to estimate an unknown parameter. EXAMPLE 11.1 Household income The mean income of the sample of households contacted by the Current Population Survey was x = $60,528. The number $60,528 is a statistic because it describes this one Current Population Survey sample. The population that the poll wants to draw conclusions about is all 113 million U.S. households. The parameter of interest is the mean income of all of these households. We don t know the value of this parameter. population mean μ Remember: statistics come from samples, and parameters come from populations. As long as we were just doing data analysis, the distinction between population and sample was not important. Now, however, it is essential. The notation we use must reflect this distinction. We write μ (the Greek letter mu) for the mean of a population. This is a fixed parameter that is unknown when we use a sample for inference. The mean of the sample is the familiar x, the average of the observa- tions in the sample. This is a statistic that would almost certainly take a different value if we chose another sample from the same population. The sample mean x from a sample or an experiment is an estimate of the mean μ of the underlying population. sample mean x APPLY YOUR KNOWLEDGE Simon Marcus/CORBIS 11.1 Effects of caffeine. How does caffeine affect our bodies? In a matched pairs experiment, subjects pushed a button as quickly as they could after taking a caffeine pill and also after taking a placebo pill. The mean pushes per minute were 283 for the placebo and 311 for caffeine. Is each of the boldface numbers a parameter or a statistic? 11.2 Indianapolis voters. Voter registration records show that 68% of all voters in Indianapolis are registered as Republicans. To test a random digit dialing device, you use the device to call 150 randomly chosen residential telephones in Indianapolis. Of the registered voters contacted, 73% are registered Republicans. Is each of the boldface numbers a parameter or a statistic? 11.3 Inspecting bearings. A carload lot of bearings has mean diameter centimeters (cm). This is within the specifications for acceptance of the lot by the purchaser. By chance, an inspector chooses 100 bearings from the lot that have mean diameter cm. Because this is outside the specified limits, the lot is mistakenly rejected. Is each of the boldface numbers a parameter or a statistic?

3 Statistical estimation and the law of large numbers 273 Statistical estimation and the law of large numbers Statistical inference uses sample data to draw conclusions about the entire population. Because good samples are chosen randomly, statistics such as x are random variables. We can describe the behavior of a sample statistic by a probability model that answers the question What would happen if we did this many times? Here is an example that will lead us toward the probability ideas most important for statistical inference. EXAMPLE 11.2 Does this wine smell bad? Sulfur compounds such as dimethyl sulfide (DMS) are sometimes present in wine. DMS causes off-odors in wine, so winemakers want to know the odor threshold, the lowest concentration of DMS that the human nose can detect. Different people have different thresholds, so we start by asking about the mean threshold μ in the population of all adults. The number μ is a parameter that describes this population. To estimate μ, we present tasters with both natural wine and the same wine spiked with DMS at different concentrations to find the lowest concentration at which they identify the spiked wine. Here are the odor thresholds (measured in micrograms of DMS per liter of wine) for 10 randomly chosen subjects: The mean threshold for these subjects is x = It seems reasonable to use the sample result x = 27.4 to estimate the unknown μ. An SRS should fairly represent the population, so the mean x of the sample should be somewhere near the mean μ of the population. Of course, we don t expect x to be exactly equal to μ. We realize that if we choose another SRS, the luck of the draw will probably produce a different x. High-tech gambling There are more than 640,000 slot machines in the United States. Once upon a time, you put in a coin and pulled the lever to spin three wheels, each with 20 symbols. No longer. Now the machines are video games with flashy graphics and outcomes produced by random number generators. Machines can accept many coins at once, can pay off on a bewildering variety of outcomes, and can be networked to allow common jackpots. Gamblers still search for systems, but in the long run the law of large numbers guarantees the house its 5% profit. If x is rarely exactly right and varies from sample to sample, why is it nonetheless a reasonable estimate of the population mean μ? Here is one answer: if we keep on taking larger and larger samples, the statistic x is guaranteed to get closer and closer to the parameter μ. We have the comfort of knowing that if we can afford to keep on measuring more subjects, eventually we will estimate the mean odor threshold of all adults very accurately. This remarkable fact is called the law of large numbers. It is remarkable because it holds for any population, not just for some special class such as Normal distributions. LAW OF LARGE NUMBERS Draw observations at random from any population with finite mean μ. As the number of observations drawn increases, the mean x of the observed values gets closer and closer to the mean μ of the population.

4 274 CHAPTER 11 Sampling Distributions The law of large numbers can be proved mathematically starting from the basic laws of probability. The behavior of x is similar to the idea of probability. In the long run, the proportion of outcomes taking any value gets close to the probability of that value, and the average outcome gets close to the population mean. Figure 10.1 (page 248) shows how proportions approach probability in one example. Here is an example of how sample means approach the population mean. EXAMPLE 11.3 The law of large numbers in action In fact, the distribution of odor thresholds among all adults has mean 25. The mean μ = 25 is the true value of the parameter we seek to estimate. Figure 11.1 shows how the sample mean x of an SRS drawn from this population changes as we add more subjects to our sample. The first subject in Example 11.2 had threshold 28, so the line in Figure 11.1 starts there. The mean for the first two subjects is x = = 34 This is the second point on the graph. At first, the graph shows that the mean of the sample changes as we take more observations. Eventually, however, the mean of the observations gets close to the population mean μ = 25 and settles down at that value. If we started over, again choosing people at random from the population, we would get a different path from left to right in Figure The law of large numbers says that whatever path we get will always settle down at 25 as we draw more and more people. Mean of first n observations ,000 Number of observations, n FIGURE 11.1 The law of large numbers in action: as we take more observations, the sample mean x always approaches the mean μ of the population.

5 Sampling distributions 275 The Law of Large Numbers applet animates Figure 11.1 in a different setting. You can use the applet to watch x change as you average more observations until it eventually settles down at the mean μ. The law of large numbers is the foundation of such business enterprises as gambling casinos and insurance companies. The winnings (or losses) of a gambler on a few plays are uncertain that s why gambling is exciting. In Figure 11.1, the mean of even 100 observations is not yet very close to μ. It is only in the long run that the mean outcome is predictable. The house plays tens of thousands of times. So the house, unlike individual gamblers, can count on the long-run regularity described by the law of large numbers. The average winnings of the house on tens of thousands of plays will be very close to the mean of the distribution of winnings. Needless to say, this mean guarantees the house a profit. That s why gambling can be a business. APPLET APPLY YOUR KNOWLEDGE 11.4 Means in action. Figure 11.1 shows how the mean of n observations behaves as we keep adding more observations to those already in hand. The first 10 observations are given in Example Demonstrate that you grasp the idea of Figure 11.1: find the means of the first one, two, three, four, and five of these observations and plot the successive means against n. Verify that your plot agrees with the first part of the plot in Figure Insurance. The idea of insurance is that we all face risks that are unlikely but carry high cost. Think of a fire destroying your home. Insurance spreads the risk: we all pay a small amount, and the insurance policy pays a large amount to those few of us whose homes burn down. An insurance company looks at the records for millions of homeowners and sees that the mean loss from fire in a year is μ = $250 per person. (Most of us have no loss, but a few lose their homes. The $250 is the average loss.) The company plans to sell fire insurance for $250 plus enough to cover its costs and profit. Explain clearly why it would be unwise to sell only 12 policies. Then explain why selling thousands of such policies is a safe business. Sampling distributions The law of large numbers assures us that if we measure enough subjects, the statistic x will eventually get very close to the unknown parameter μ. But our study in Example 11.2 had just 10 subjects. What can we say about x from 10 subjects as an estimate of μ? We ask: What would happen if we took many samples of 10 subjects from this population? Here s how to answer this question: Take a large number of samples of size 10 from the population. Calculate the sample mean x for each sample. Make a histogram of the values of x. Examine the distribution displayed in the histogram for shape, center, and spread, as well as outliers or other deviations.

6 276 CHAPTER 11 Sampling Distributions Take many SRSs and collect their means x. The distribution of all the x's is close to Normal. SRS size 10 x = SRS size 10 x = SRS size 10 x = Population, mean μ = FIGURE 11.2 The idea of a sampling distribution: take many samples from the same population, collect the x s from all the samples, and display the distribution of the x s. The histogram shows the results of 1000 samples. simulation In practice it is too expensive to take many samples from a large population such as all adult U.S. residents. But we can imitate many samples by using software. Using software to imitate chance behavior is called simulation. EXAMPLE 11.4 What would happen in many samples? Extensive studies have found that the DMS odor threshold of adults follows roughly a Normal distribution with mean μ = 25 micrograms per liter and standard deviation σ = 7 micrograms per liter. With this information, we can simulate many repetitions of Example 11.2 with different subjects drawn at random from the population. Figure 11.2 illustrates the process of choosing many samples and finding the sample mean threshold x for each one. Follow the flow of the figure from the population at the left, to choosing an SRS and finding the x for this sample, to collecting together the x s from many samples. The first sample has x = The second sample contains a different 10 people, with x = 24.28, and so on. The histogram at the right of the figure shows the distribution of the values of x from 1000 separate SRSs of size 10. This histogram displays the sampling distribution of the statistic x. SAMPLING DISTRIBUTION The sampling distribution of a statistic is the distribution of values taken by the statistic in all possible samples of the same size from the same population.

7 Sampling distributions 277 Strictly speaking, the sampling distribution is the ideal pattern that would emerge if we looked at all possible samples of size 10 from our population. A distribution obtained from a fixed number of trials, like the 1000 trials in Figure 11.2, is only an approximation to the sampling distribution. One of the uses of probability theory in statistics is to obtain exact sampling distributions without simulation. The interpretation of a sampling distribution is the same, however, whether we obtain it by simulation or by the mathematics of probability. We can use the tools of data analysis to describe any distribution. Let s apply those tools to Figure What can we say about the shape, center, and spread of this distribution? Shape: It looks Normal! Detailed examination confirms that the distribution of x from many samples does have a distribution that is very close to Normal. Center: The mean of the 1000 x s is That is, the distribution is centered very close to the population mean μ = 25. Spread: The standard deviation of the 1000 x s is 2.217, notably smaller than the standard deviation σ = 7 of the population of individual subjects. Although these results describe just one simulation of a sampling distribution, they reflect facts that are true whenever we use random sampling. APPLY YOUR KNOWLEDGE 11.6 Generating a sampling distribution. Let s illustrate the idea of a sampling distribution in the case of a very small sample from a very small population. The population is the scores of 10 students on an exam: Student Score The parameter of interest is the mean score μ in this population. The sample is an SRS of size n = 4 drawn from the population. Because the students are labeled 0 to 9, a single random digit from Table B chooses one student for the sample. (a) Find the mean of the 10 scores in the population. This is the population mean μ. (b) Use the first digits in row 116 of Table B to draw an SRS of size 4 from this population. What are the four scores in your sample? What is their mean x? This statistic is an estimate of μ. (c) Repeat this process 9 more times, using the first digits in rows 117 to 125 of Table B. Make a histogram of the 10 values of x. You are

8 278 CHAPTER 11 Sampling Distributions constructing the sampling distribution of x. Is the center of your histogram close to μ? The sampling distribution of x Figure 11.2 suggests that when we choose many SRSs from a population, the sampling distribution of the sample means is centered at the mean of the original population and less spread out than the distribution of individual observations. Here are the facts. MEAN AND STANDARD DEVIATION OF A SAMPLE MEAN 2 Rigging the lottery We have all seen televised lottery drawings in which numbered balls bubble about and are randomly popped out by air pressure. How might we rig such a drawing? In 1980, when the Pennsylvania lottery used just three balls, a drawing was rigged by the host and several stagehands. They injected paint into all balls bearing 8 of the 10 digits. This weighed them down and guaranteed that all three balls for the winning number would have the remaining 2 digits. The perps then bet on all combinations of these digits. When popped out, they won $1.2 million. Yes, they were caught. unbiased estimator Suppose that x is the mean of an SRS of size n drawn from a large population with mean μ and standard deviation σ. Then the sampling distribution of x has mean μ and standard deviation σ/ n. These facts about the mean and the standard deviation of the sampling distribution of x are true for any population, not just for some special class such as Normal distributions. Both facts have important implications for statistical inference. The mean of the statistic x is always equal to the mean μ of the population. That is, the sampling distribution of x is centered at μ. In repeated sampling, x will sometimes fall above the true value of the parameter μ and sometimes below, but there is no systematic tendency to overestimate or underestimate the parameter. This makes the idea of lack of bias in the sense of no favoritism more precise. Because the mean of x is equal to μ, we say that the statistic x is an unbiased estimator of the parameter μ. An unbiased estimator is correct on the average in many samples. How close the estimator falls to the parameter in most samples is determined by the spread of the sampling distribution. If individual observations have standard deviation σ, then sample means x from samples of size n have standard deviation σ/ n. That is, averages are less variable than individual observations. We have described the center and spread of the sampling distribution of a sample mean x, but not its shape. The shape of the distribution of x depends on the shape of the population. Here is one important case: if measurements in the population follow a Normal distribution, then so does the sample mean. SAMPLING DISTRIBUTION OF A SAMPLE MEAN If individual observations have the N(μ, σ) distribution, then the sample mean x of an SRS of size n has the N(μ, σ/ n) distribution.

9 The sampling distribution of x 279 The distribution of sample means is less spread out. Means x of 10 subjects σ 10 = 2.21 Observations on 1 subject σ = FIGURE 11.3 The distribution of single observations compared with the distribution of the means x of 10 observations. Averages are less variable than individual observations. EXAMPLE 11.5 Population distribution, sampling distribution If we measure the DMS odor thresholds of individual adults, the values follow the Normal distribution with mean μ = 25 micrograms per liter and standard deviation σ = 7 micrograms per liter. We call this the population distribution because it shows how measurements vary within the population. Take many SRSs of size 10 from this population and find the sample mean x for each sample, as in Figure The sampling distribution describes how the values of x vary among samples. That sampling distribution is also Normal, with mean μ = 25 and standard deviation population distribution σ n = 7 10 = Figure 11.3 contrasts these two Normal distributions. Both are centered at the population mean, but sample means are much less variable than individual observations. Not only is the standard deviation of the distribution of x smaller than the standard deviation of individual observations, but it gets smaller as we take larger samples. The results of large samples are less variable than the results of small samples. If n is large, the standard deviation of x is small, and almost all samples will give values of x that lie very close to the true parameter μ. That is, the sample mean from a large sample can be trusted to estimate the population mean accurately. However, the standard deviation of the sampling distribution gets smaller only at the rate n. To cut the standard deviation of x in half, we must take four times as many observations, not just twice as many. CAUTION

10 280 CHAPTER 11 Sampling Distributions APPLY YOUR KNOWLEDGE 11.7 A sample of teens. A study of the health of teenagers plans to measure the blood cholesterol level of an SRS of youths aged 13 to 16. The researchers will report the mean x from their sample as an estimate of the mean cholesterol level μ in this population. (a) Explain to someone who knows no statistics what it means to say that x is an unbiased estimator of μ. (b) The sample result x is an unbiased estimator of the population truth μ no matter what size SRS the study uses. Explain to someone who knows no statistics why a large sample gives more trustworthy results than a small sample Measurements in the lab. Juan makes a measurement in a chemistry laboratory and records the result in his lab report. The standard deviation of students lab measurements is σ = 10 milligrams. Juan repeats the measurement 3 times and records the mean x of his 3 measurements. (a) What is the standard deviation of Juan s mean result? (That is, if Juan kept on making 3 measurements and averaging them, what would be the standard deviation of all his x s?) (b) How many times must Juan repeat the measurement to reduce the standard deviation of x to 5? Explain to someone who knows no statistics the advantage of reporting the average of several measurements rather than the result of a single measurement National math scores. The scores of 12th-grade students on the National Assessment of Educational Progress year 2000 mathematics test have a distribution that is approximately Normal with mean μ = 300 and standard deviation σ = 35. (a) Choose one 12th-grader at random. What is the probability that his or her score is higher than 300? Higher than 335? (b) Now choose an SRS of four 12th-graders and calculate their mean score x.if you did this many times, what would be the mean and standard deviation of all the x-values? (c) What is the probability that the mean score for your SRS is higher than 300? Higher than 335? The central limit theorem The facts about the mean and standard deviation of x are true no matter what the shape of the population distribution may be. But what is the shape of the sampling distribution when the population distribution is not Normal? It is a remarkable fact that as the sample size increases, the distribution of x changes shape: it looks less like that of the population and more like a Normal distribution. When the sample is large enough, the distribution of x is very close to Normal. This is true no matter what shape the population distribution has, as long as the population has a finite standard deviation σ. This famous fact of probability theory is called the central limit theorem. It is much more useful than the fact that the distribution of x is exactly Normal if the population is exactly Normal.

11 The central limit theorem 281 CENTRAL LIMIT THEOREM Draw an SRS of size n from any population with mean μ and finite standard deviation σ. When n is large, the sampling distribution of the sample mean x is approximately Normal: x is approximately N (μ, n σ ) The central limit theorem allows us to use Normal probability calculations to answer questions about sample means from many observations even when the population distribution is not Normal. More general versions of the central limit theorem say that the distribution of any sum or average of many small random quantities is close to Normal. This is true even if the quantities are correlated with each other (as long as they are not too highly correlated) and even if they have different distributions (as long as no one random quantity is so large that it dominates the others). The central limit theorem suggests why the Normal distributions are common models for observed data. Any variable that is a sum of many small influences will have approximately a Normal distribution. How large a sample size n is needed for x to be close to Normal depends on the population distribution. More observations are required if the shape of the population distribution is far from Normal. Here are two examples in which the population is far from Normal. EXAMPLE 11.6 The central limit theorem in action In March 2004, the Current Population Survey contacted 98,789 households. Figure 11.4(a) is a histogram of the earnings of the 62,101 households that had earned income greater than zero in As we expect, the distribution of earned incomes is strongly skewed to the right and very spread out. The right tail of the distribution is longer than the histogram shows because there are too few high incomes for their bars to be visible on this scale. In fact, we cut off the earnings scale at $300,000 to save space a few households earned even more than $300,000. The mean earnings for these 62,101 households was $57,085. Regard these 62,101 households as a population. Take an SRS of 100 households. The mean earnings in this sample is x = $48,600. That s less than the mean of the population. Take another SRS of size 100. The mean for this sample is x = $64,766. That s higher than the mean of the population. What would happen if we did this many times? Figure 11.4(b) is a histogram of the mean earnings for 500 samples, each of size 100. The scales in Figures 11.4(a) and 11.4(b) are the same, for easy comparison. Although the distribution of individual earnings is skewed and very spread out, the distribution of sample means is roughly symmetric and much less spread out. Figure 11.4(c) zooms in on the center part of the axis for another histogram of the same 500 values of x. Although n = 100 is not a very large sample size and the

12 Percent of households Earned income (thousands of dollars) (a) Percent of sample means Because the scales are the same, you can compare this distribution directly with Figure 11.4(a) Mean earned income in sample (thousands of dollars) (b) 40,000 45,000 50,000 55,000 60,000 65,000 70,000 75,000 Mean earned income in sample (dollars) (c) FIGURE 11.4 The central limit theorem in action. (a) The distribution of earned income in a population of 62,101 households. (b) The distribution of the mean earnings for 500 SRSs of 100 households each from this population. (c) The distribution of the sample means in more detail: the shape is close to Normal.

13 The central limit theorem 283 population distribution is extremely skewed, we can see that the distribution of sample means is close to Normal. Comparing Figures 11.4(a) and 11.4(b) illustrates the two most important ideas of this chapter. THINKING ABOUT SAMPLE MEANS Means of random samples are less variable than individual observations. Means of random samples are more Normal than individual observations. EXAMPLE 11.7 The central limit theorem in action The Central Limit Theorem applet allows you to watch the central limit theorem in action. Figure 11.5 presents snapshots from the applet. Figure 11.5(a) shows the density curve of a single observation, that is, of the population. The distribution is strongly APPLET 0 1 (a) 0 1 (b) 0 1 (c) 0 1 (d) FIGURE 11.5 The central limit theorem in action: the distribution of sample means x from a strongly non-normal population becomes more Normal as the sample size increases. (a) The distribution of 1 observation. (b) The distribution of x for 2 observations. (c) The distribution of x for 10 observations. (d) The distribution of x for 25 observations.

14 284 CHAPTER 11 Sampling Distributions right-skewed, and the most probable outcomes are near 0. The mean μ of this distribution is 1, and its standard deviation σ is also 1. This particular distribution is called an exponential distribution. Exponential distributions are used as models for the lifetime in service of electronic components and for the time required to serve a customer or repair a machine. Figures 11.5(b), (c), and (d) are the density curves of the sample means of 2, 10, and 25 observations from this population. As n increases, the shape becomes more Normal. The mean remains at μ = 1, and the standard deviation decreases, taking the value 1/ n. The density curve for 10 observations is still somewhat skewed to the right but already resembles a Normal curve having μ = 1 and σ = 1/ 10 = The density curve for n = 25 is yet more Normal. The contrast between the shapes of the population distribution and of the distribution of the mean of 10 or 25 observations is striking. Let s use Normal calculations based on the central limit theorem to answer a question about the very non-normal distribution in Figure 11.5(a). EXAMPLE 11.8 Maintaining air conditioners 4STEP STEP STATE: The time (in hours) that a technician requires to perform preventive maintenance on an air-conditioning unit is governed by the exponential distribution whose density curve appears in Figure 11.5(a). The mean time is μ = 1 hour and the standard deviation is σ = 1 hour. Your company has a contract to maintain 70 of these units in an apartment building. You must schedule technicians time for a visit to this building. Is it safe to budget an average of 1.1 hours for each unit? Or should you budget an average of 1.25 hours? FORMULATE: We can treat these 70 air conditioners as an SRS from all units of this type. What is the probability that the average maintenance time for 70 units exceeds 1.1 hours? That the average time exceeds 1.25 hours? SOLVE: The central limit theorem says that the sample mean time x spent working on 70 units has approximately the Normal distribution with mean equal to the population mean μ = 1 hour and standard deviation σ 70 = 1 70 = 0.12 hour The distribution of x is therefore approximately N(1, 0.12). This Normal curve is the solid curve in Figure Using this Normal distribution, the probabilities we want are P (x > 1.10 hours) = P (x > 1.25 hours) = (Software gives these probabilities immediately, or you can standardize and use Table A. Don t forget to use standard deviation 0.12 in your software or when you standardize x.) CONCLUDE: If you budget 1.1 hours per unit, there is a 20% chance that the technicians will not complete the work in the building within the budgeted time. This chance drops to 2% if you budget 1.25 hours. You therefore budget 1.25 hours per unit.

15 The central limit theorem 285 Exact density curve for x. Normal curve from the central limit theorem. FIGURE 11.6 The exact distribution (dashed) and the Normal approximation from the central limit theorem (solid) for the average time needed to maintain an air conditioner, for Example The probability we want is the area to the right of Using more mathematics, we can start with the exponential distribution and find the actual density curve of x for 70 observations. This is the dashed curve in Figure You can see that the solid Normal curve is a good approximation. The exactly correct probability for 1.1 hours is an area to the right of 1.1 under the dashed density curve. It is The central limit theorem Normal approximation is off by only about APPLY YOUR KNOWLEDGE What does the central limit theorem say? Asked what the central limit theorem says, a student replies, As you take larger and larger samples from a population, the histogram of the sample values looks more and more Normal. Is the student right? Explain your answer Detecting gypsy moths. The gypsy moth is a serious threat to oak and aspen trees. A state agriculture department places traps throughout the state to detect the moths. When traps are checked periodically, the mean number of moths trapped is only 0.5, but some traps have several moths. The distribution of moth counts is discrete and strongly skewed, with standard deviation 0.7. (a) What are the mean and standard deviation of the average number of moths x in 50 traps? (b) Use the central limit theorem to find the probability that the average number of moths in 50 traps is greater than 0.6. Bruce Coleman/Alamy

16 286 CHAPTER 11 Sampling Distributions 4STEP STEP SAT scores. The total SAT scores of high school seniors in recent years have mean μ = 1026 and standard deviation σ = 209. The distribution of SAT scores is roughly Normal. (a) Ramon scored If scores have a Normal distribution, what percentile of the distribution is this? (That is, what percent of scores are lower than Ramon s?) (b) Now consider the mean x of the scores of 70 randomly chosen students. If x = 1100, what percentile of the sampling distribution of x is this? (c) Which of your calculations, (a) or (b), is less accurate because SAT scores do not have an exactly Normal distribution? Explain your answer More on insurance. An insurance company knows that in the entire population of millions of homeowners, the mean annual loss from fire is μ = $250 and the standard deviation of the loss is σ = $1000. The distribution of losses is strongly right-skewed: most policies have $0 loss, but a few have large losses. If the company sells 10,000 policies, can it safely base its rates on the assumption that its average loss will be no greater than $275? Follow the four-step process in your answer. Statistical process control The sampling distribution of the sample mean x has an immediate application to statistical process control. The goal of statistical process control is to make a process stable over time and then keep it stable unless planned changes are made. You might want, for example, to keep your weight constant over time. A manufacturer of machine parts wants the critical dimensions to be the same for all parts. Constant over time and the same for all are not realistic requirements. They ignore the fact that all processes have variation. Your weight fluctuates from day to day; the critical dimension of a machined part varies a bit from item to item; the time to process a college admission application is not the same for all applications. Variation occurs in even the most precisely made product due to small changes in the raw material, the adjustment of the machine, the behavior of the operator, and even the temperature in the plant. Because variation is always present, we can t expect to hold a variable exactly constant over time. The statistical description of stability over time requires that the pattern of variation remain stable, not that there be no variation in the variable measured. STATISTICAL CONTROL A variable that continues to be described by the same distribution when observed over time is said to be in statistical control, or simply in control. Control charts are statistical tools that monitor a process and alert us when the process has been disturbed so that it is now out of control. This is a signal to find and correct the cause of the disturbance. *The rest of this chapter is optional. A more complete treatment of process control appears in Companion Chapter 27.

17 x charts 287 Control charts work by distinguishing the natural variation in the process from the additional variation that suggests that the process has changed. A control chart sounds an alarm when it sees too much variation. The most common application of control charts is to monitor the performance of an industrial process. The same methods, however, can be used to check the stability of quantities as varied as the ratings of a television show, the level of ozone in the atmosphere, and the gas mileage of your car. Control charts combine graphical and numerical descriptions of data with use of sampling distributions. They therefore provide a natural bridge between exploratory data analysis and formal statistical inference. x charts The population in the control chart setting is all items that would be produced by the process if it ran on forever in its present state. The items actually produced form samples from this population. We generally speak of the process rather than the population. Choose a quantitative variable, such as a diameter or a voltage, that is an important measure of the quality of an item. The process mean μ is the longterm average value of this variable; μ describes the center or aim of the process. The sample mean x of several items estimates μ and helps us judge whether the center of the process has moved away from its proper value. The most common control chart plots the means x of small samples taken from the process at regular intervals over time. When you first apply control charts to a process, the process may not be in control. Even if it is in control, you don t yet understand its behavior. You must collect data from the process, establish control by uncovering and removing the reasons for disturbances, and then set up control charts to maintain control. To quickly explain the main ideas, we ll assume that you know the usual behavior of the process from long experience. Here are the conditions we will work with. PROCESS-MONITORING CONDITIONS Measure a quantitative variable x that has a Normal distribution. The process has been operating in control for a long period, so that we know the process mean μ and the process standard deviation σ that describe the distribution of x as long as the process remains in control. EXAMPLE 11.9 Making computer monitors A manufacturer of computer monitors must control the tension on the mesh of fine wires that lies behind the surface of the viewing screen. Too much tension will tear the mesh, and too little will allow wrinkles. Tension is measured by an electrical device with output readings in millivolts (mv). The proper tension is 275 mv. Some variation is always present in the production process. When the process is operating properly, the standard deviation of the tension readings is σ = 43 mv.

18 288 CHAPTER 11 Sampling Distributions TABLE 11.1 Twenty control chart samples of mesh tension Sample Tension measurements x The operator measures the tension on a sample of 4 monitors each hour. The mean x of each sample estimates the mean tension μ for the process at the time of the sample. Table 11.1 shows the samples and their means for 20 consecutive hours of production. How can we use these data to keep the process in control? A time plot helps us see whether or not the process is stable. Figure 11.7 is a plot of the successive sample means against the order in which the samples were taken. We have plotted each sample mean from the table against its sample number. For example, the mean of the first sample is mv, and this is the value plotted for sample 1. Because the target value for the process mean is μ = 275 mv, we draw a center line at that level across the plot. How much variation about this center line do we expect to see? For example, are samples 13 and 19 so high that they suggest lack of control? The tension measurements are roughly Normal, and we know that sample means are more Normal than individual measurements. So the x-values from successive samples will follow a Normal distribution. If the standard deviation of the individual screens remains at σ = 43 mv, the standard deviation of x from 4 screens is σ n = 43 4 = 21.5 mv

19 x charts The control limits mark the natural variation in the process. 350 UCL Sample mean LCL Sample number FIGURE 11.7 x chart for the mesh tension data of Table The control limits are labeled UCL for upper control limit and LCL for lower control limit. No points lie outside the control limits. As long as the mean remains at its target value μ = 275 mv, the 99.7 part of the rule says that almost all values of x will lie between μ 3 σ n = 275 (3)(21.5) = μ + 3 σ n = (3)(21.5) = We therefore draw dashed control limits at these two levels on the plot. The control limits show the extent of the natural variation of x-values when the process is in control. We now have an x control chart. x CONTROL CHART To evaluate the control of a process with given standards μ and σ, make an x control chart as follows: Plot the means x of regular samples of size n against time. Draw a horizontal center line at μ. Draw horizontal control limits at μ ± 3σ/ n. Any x that does not fall between the control limits is evidence that the process is out of control.

20 290 CHAPTER 11 Sampling Distributions EXAMPLE Interpreting x charts Figure 11.7 is a typical x chart for a process in control. The means of the 20 samples do vary, but all lie within the range of variation marked out by the control limits. We are seeing the natural variation of a stable process. Figures 11.8 and 11.9 illustrate two ways in which the process can go out of control. In Figure 11.8, the process was disturbed sometime between sample 12 and sample 13. As a result, the mean tension for sample 13 falls above the upper control limit. It is common practice to mark all out-of-control points with an x to call attention to them. A search for the cause begins as soon as we see a point out of control. Investigation finds that the mounting of the tension-measuring device has slipped, resulting in readings that are too high. When the problem is corrected, samples 14 to 20 are again in control. Figure 11.9 shows the effect of a steady upward drift in the process center, starting at sample 11. You see that some time elapses before the x for sample 18 is out of control. The one-point-out signal works better for detecting sudden large disturbances than for detecting slow drifts in a process. This point is out of control because it is above UCL UCL x Sample mean LCL Sample number FIGURE 11.8 This x chart is identical to that in Figure 11.7, except that a disturbance has driven x for sample 13 above the upper control limit. The out-of-control point is marked with an x. x chart An x control chart is often called simply an x chart. Because a control chart is a warning device, it is not necessary that our probability calculations be exactly correct. Approximate Normality is good enough. In that same spirit, control charts use the approximate Normal probabilities given by the rule rather than more exact calculations using Table A.

21 x charts x x 350 UCL x Sample mean LCL Sample number FIGURE 11.9 The first 10 points on this x chart are as in Figure The process mean drifts upward after sample 10, and the sample means x reflect this drift. The points for samples 18, 19, and 20 are out of control. APPLY YOUR KNOWLEDGE Auto thermostats. A maker of auto air conditioners checks a sample of 4 thermostatic controls from each hour s production. The thermostats are set at 75 F and then placed in a chamber where the temperature rises gradually. The temperature at which the thermostat turns on the air conditioner is recorded. The process mean should be μ = 75. Past experience indicates that the response temperature of properly adjusted thermostats varies with σ = 0.5. The mean response temperature x for each hour s sample is plotted on an x control chart. Calculate the center line and control limits for this chart Tablet hardness. A pharmaceutical manufacturer forms tablets by compressing a granular material that contains the active ingredient and various fillers. The hardness of a sample from each lot of tablets is measured in order to control the compression process. The process has been operating in control with mean at the target value μ = 11.5 and estimated standard deviation σ = 0.2. Table 11.2 gives three sets of data, each representing x for 20 successive samples of n = 4 tablets. One set remains in control at the target value. In a second set, the process mean μ shifts suddenly to a new value. In a third, the process mean drifts gradually. (a) What are the center line and control limits for an x chart for this process? (b) Draw a separate x chart for each of the three data sets. Mark any points that are beyond the control limits.

22 292 CHAPTER 11 Sampling Distributions TABLE 11.2 Three sets of x s from 20 samples of size 4 Sample Data set A Data set B Data set C (c) Based on your work in (b) and the appearance of the control charts, which set of data comes from a process that is in control? In which case does the process mean shift suddenly, and at about which sample do you think that the mean changed? Finally, in which case does the mean drift gradually? CAUTION Thinking about process control The purpose of a control chart is not to ensure good quality by inspecting most of the items produced. Control charts focus on the process itself rather than on the individual products. By checking the process at regular intervals, we can detect disturbances and correct them quickly. Statistical process control achieves high quality at a lower cost than inspecting all of the products. Small samples of 4 or 5 items are usually adequate for process control. A process that is in control is stable over time, but stability alone does not guarantee good quality. The natural variation in the process may be so large that many of the products are unsatisfactory. Nonetheless, establishing control brings a number of advantages. In order to assess whether the process quality is satisfactory, we must observe the process when it is operating in control, free of breakdowns and other disturbances.

23 Chapter 11 Summary 293 A process in control is predictable. We can predict both the quantity and the quality of items produced. When a process is in control, we can easily see the effects of attempts to improve the process, which are not hidden by the unpredictable variation that characterizes lack of statistical control. A process in control is doing as well as it can in its present state. If the process is not capable of producing adequate quality even when undisturbed, we must make some major change in the process, such as installing new machines or retraining the operators. If the process is kept in control, we know what to expect in the finished product. The process mean μ and standard deviation σ remain stable over time, so (assuming Normal variation) the 99.7 part of the rule tells us that almost all measurements on individual products will lie in the range μ ± 3σ. These are sometimes called the natural tolerances for the product. Be careful to distinguish μ ± 3σ, the range we expect for individual measurements, from the x chart control limits μ ± 3σ/ n, which mark off the expected range of sample means. EXAMPLE Natural tolerances for mesh tension The process of setting the mesh tension on computer monitors has been operating in control. The x chart is based on μ = 275 mv and σ = 43 mv. We are therefore confident that almost all individual monitors will have mesh tension between μ ± 3σ = 275 ± (3)(43) = 275 ± 129 We expect mesh tension measurements to vary between 146 mv and 404 mv. You see that the spread of individual measurements is wider than the spread of sample means used for the control limits of the x chart. natural tolerances CAUTION APPLY YOUR KNOWLEDGE Auto thermostats. Exercise describes a process that produces auto thermostats. The temperature that turns on the thermostats has remained in control with mean μ = 75 F and standard deviation σ = 0.5. What are the natural tolerances for this temperature? What range covers the middle 95% of response temperatures? CHAPTER 11 SUMMARY When we want information about the population mean μ for some variable, we often take an SRS and use the sample mean x to estimate the unknown parameter μ. The law of large numbers states that the actually observed mean outcome x must approach the mean μ of the population as the number of observations increases. The sampling distribution of x describes how the statistic x varies in all possible SRSs of the same size from the same population.

24 294 CHAPTER 11 Sampling Distributions The mean of the sampling distribution is μ, so that x is an unbiased estimator of μ. The standard deviation of the sampling distribution of x is σ/ n for an SRS of size n if the population has standard deviation σ. That is, averages are less variable than individual observations. If the population has a Normal distribution, so does x. The central limit theorem states that for large n the sampling distribution of x is approximately Normal for any population with finite standard deviation σ. That is, averages are more Normal than individual observations. We can use the N(μ, σ/ n) distribution to calculate approximate probabilities for events involving x. All processes have variation. If the pattern of variation is stable over time, the process is in statistical control. Control charts are statistical plots intended to warn when a process is out of control. An x control chart plots the means x of samples from a process against the time order in which the samples were taken. If the process has been in control with mean μ and standard deviation σ, control limits at μ ± 3σ/ n mark off the range of variation we expect to see in the x-values. Values outside the control limits suggest that the process has been disturbed. CHECK YOUR SKILLS The Bureau of Labor Statistics announces that last month it interviewed all members of the labor force in a sample of 60,000 households; 4.9% of the people interviewed were unemployed. The boldface number is a (a) sampling distribution. (b) parameter. (c) statistic A study of voting chose 663 registered voters at random shortly after an election. Of these, 72% said they had voted in the election. Election records show that only 56% of registered voters voted in the election. The boldface number is a (a) sampling distribution. (b) parameter. (c) statistic Annual returns on the more than 5000 common stocks available to investors vary a lot. In a recent year, the mean return was 8.3% and the standard deviation of returns was 28.5%. The law of large numbers says that (a) you can get an average return higher than the mean 8.3% by investing in a large number of stocks. (b) as you invest in more and more stocks chosen at random, your average return on these stocks gets ever closer to 8.3%. (c) if you invest in a large number of stocks chosen at random, your average return will have approximately a Normal distribution Scores on the SAT college entrance test in a recent year were roughly Normal with mean 1026 and standard deviation 209. You choose an SRS of 100 students and average their SAT scores. If you do this many times, the mean of the average scores you get will be close to (a) (b) 1026/100 = (c) 1026/ 100 =

25 Chapter 11 Exercises Scores on the SAT college entrance test in a recent year were roughly Normal with mean 1026 and standard deviation 209. You choose an SRS of 100 students and average their SAT scores. If you do this many times, the standard deviation of the average scores you get will be close to (a) 209. (b) 100/ 209 = (c) 209/ 100 = A newborn baby has extremely low birth weight (ELBW) if it weighs less than 1000 grams. A study of the health of such children in later years examined a random sample of 219 children. Their mean weight at birth was x = 810 grams. This sample mean is an unbiased estimator of the mean weight μ in the population of all ELBW babies. This means that (a) in many samples from this population, the mean of the many values of x will be equal to μ. (b) as we take larger and larger samples from this population, x will get closer and closer to μ. (c) in many samples from this population, the many values of x will have a distribution that is close to Normal The number of hours a light bulb burns before failing varies from bulb to bulb. The distribution of burnout times is strongly skewed to the right. The central limit theorem says that (a) as we look at more and more bulbs, their average burnout time gets ever closer to the mean μ for all bulbs of this type. (b) the average burnout time of a large number of bulbs has a distribution of the same shape (strongly skewed) as the distribution for individual bulbs. (c) the average burnout time of a large number of bulbs has a distribution that is close to Normal A machine manufactures parts whose diameters vary according to the Normal distribution with mean μ = millimeters (mm) and standard deviation σ = mm. An inspector measures a random sample of 4 parts. The probability that the average diameter of these 4 parts is less than mm is about (a) (b) (c) CHAPTER 11 EXERCISES Women s heights. A random sample of female college students has a mean height of 65 inches, which is greater than the 64-inch mean height of all young women. Is each of the bold numbers a parameter or a statistic? Explain your answer Small classes in school. The Tennessee STAR experiment randomly assigned children to regular or small classes during their first four years of school. When these children reached high school, 40.2% of blacks from small classes took the ACT or SAT college entrance exams. Only 31.7% of blacks from regular classes took one of these exams. Is each of the boldface numbers a parameter or a statistic? Explain your answer.

26 296 CHAPTER 11 Sampling Distributions Matthias Kulka/CORBIS Gandee Vasan/Getty Images APPLET Playing the numbers. The numbers racket is a well-entrenched illegal gambling operation in most large cities. One version works as follows: you choose one of the 1000 three-digit numbers 000 to 999 and pay your local numbers runner a dollar to enter your bet. Each day, one three-digit number is chosen at random and pays off $600. The mean payoff for the population of thousands of bets is μ = 60 cents. Joe makes one bet every day for many years. Explain what the law of large numbers says about Joe s results as he keeps on betting Roulette. A roulette wheel has 38 slots, of which 18 are black, 18 are red, and 2 are green. When the wheel is spun, the ball is equally likely to come to rest in any of the slots. One of the simplest wagers chooses red or black. A bet of $1 on red returns $2 if the ball lands in a red slot. Otherwise, the player loses his dollar. When gamblers bet on red or black, the two green slots belong to the house. Because the probability of winning $2 is 18/38, the mean payoff from a $1 bet is twice 18/38, or 94.7 cents. Explain what the law of large numbers tells us about what will happen if a gambler makes very many bets on red The law of large numbers. Suppose that you roll two balanced dice and look at the spots on the up-faces. There are 36 possible outcomes, displayed in Figure 10.2 (page 251). Because the dice are balanced, all 36 outcomes are equally likely. The average number of spots is 7. This is the population mean μ for the idealized population that contains the results of rolling two dice forever. The law of large numbers says that the average x from a finite number of rolls gets closer and closer to 7 as we do more and more rolls. (a) Click More dice once in the Law of Large Numbers applet to get two dice. Click Show mean to see the mean 7 on the graph. Leaving the number of rolls at 1, click Roll dice three times. How many spots did each roll produce? What is the average for the three rolls? You see that the graph displays at each point the average number of spots for all rolls up to the last one. Now you understand the display. (b) Set the number of rolls to 100 and click Roll dice. The applet rolls the two dice 100 times. The graph shows how the average count of spots changes as we make more rolls. That is, the graph shows x as we continue to roll the dice. Make a rough sketch of the final graph. (c) Repeat your work from (b). Click Reset to start over, then roll two dice 100 times. Make a sketch of the final graph of the mean x against the number of rolls. Your two graphs will often look very different. What they have in common is that the average eventually gets close to the population mean μ = 7. The law of large numbers says that this will always happen if you keep on rolling the dice What s the mean? Suppose that you roll three balanced dice. We wonder what the mean number of spots on the up-faces of the three dice is. The law of large numbers says that we can find out by experience: roll three dice many times, and the average number of spots will eventually approach the true mean. Set up the Law of Large Numbers applet to roll three dice. Don t click Show mean yet. Roll the dice until you are confident you know the mean quite closely, then click Show mean to verify your discovery. What is the mean? Make a rough sketch of the path the averages x followed as you kept adding more rolls Lightning strikes. The number of lightning strikes on a square kilometer of open ground in a year has mean 6 and standard deviation 2.4. (These values are typical of much of the United States.) The National Lightning Detection

27 Chapter 11 Exercises 297 Network uses automatic sensors to watch for lightning in a sample of 10 square kilometers. What are the mean and standard deviation of x, the mean number of strikes per square kilometer? Heights of male students. To estimate the mean height μ of male students on your campus, you will measure an SRS of students. You know from government data that the standard deviation of the heights of young men is about 2.8 inches. How large an SRS must you take to reduce the standard deviation of the sample mean to one-half inch? Use the four-step process to outline your work Heights of male students, continued. To estimate the mean height μ of male students on your campus, you will measure an SRS of students. You know from government data that heights of young men are approximately Normal with standard deviation about 2.8 inches. You want your sample mean x to estimate μ with an error of no more than one-half inch in either direction. (a) What standard deviation must x have so that 99.7% of all samples give an x within one-half inch of μ? (Use the rule.) (b) How large an SRS do you need to reduce the standard deviation of x to the value you found in part (a)? More on heights of male students. In Exercise 11.32, you decided to measure n male students. Suppose that the distribution of heights of all male students on your campus is Normal with mean 70 inches and standard deviation 2.8 inches. (a) If you choose one student at random, what is the probability that he is between 69 and 71 inches tall? (b) What is the probability that the mean height of your sample is between 69 and 71 inches? Durable press fabrics. Durable press cotton fabrics are treated to improve their recovery from wrinkles after washing. Unfortunately, the treatment also reduces the strength of the fabric. The breaking strength of untreated fabric is Normally distributed with mean 58 pounds and standard deviation 2.3 pounds. The same type of fabric after treatment has Normally distributed breaking strength with mean 30 pounds and standard deviation 1.6 pounds. 3 A clothing manufacturer tests an SRS of 5 specimens of each fabric. (a) What is the probability that the mean breaking strength of the 5 untreated specimens exceeds 50 pounds? (b) What is the probability that the mean breaking strength of the 5 treated specimens exceeds 50 pounds? Glucose testing. Shelia s doctor is concerned that she may suffer from gestational diabetes (high blood glucose levels during pregnancy). There is variation both in the actual glucose level and in the blood test that measures the level. A patient is classified as having gestational diabetes if the glucose level is above 140 milligrams per deciliter (mg/dl) one hour after a sugary drink. Shelia s measured glucose level one hour after the sugary drink varies according to the Normal distribution with μ = 125 mg/dl and σ = 10 mg/dl. (a) If a single glucose measurement is made, what is the probability that Shelia is diagnosed as having gestational diabetes? (b) If measurements are made on 4 separate days and the mean result is compared with the criterion 140 mg/dl, what is the probability that Shelia is diagnosed as having gestational diabetes? 4STEP STEP

28 298 CHAPTER 11 Sampling Distributions Alan Hicks/Getty Images 4STEP STEP 4STEP STEP Pollutants in auto exhausts. The level of nitrogen oxides (NOX) in the exhaust of cars of a particular model varies Normally with mean 0.2 grams per mile (g/mi) and standard deviation 0.05 g/mi. Government regulations call for NOX emissions no higher than 0.3 g/mi. (a) What is the probability that a single car of this model fails to meet the NOX requirement? (b) A company has 25 cars of this model in its fleet. What is the probability that the average NOX level x of these cars is above the 0.3 g/mi limit? Glucose testing, continued. Shelia s measured glucose level one hour after a sugary drink varies according to the Normal distribution with μ = 125 mg/dl and σ = 10 mg/dl. What is the level L such that there is probability only 0.05 that the mean glucose level of 4 test results falls above L?(Hint: This requires a backward Normal calculation. See page 81 in Chapter 3 if you need to review.) Pollutants in auto exhausts, continued. The level of nitrogen oxides (NOX) in the exhaust of cars of a particular model varies Normally with mean 0.2 g/mi and standard deviation 0.05 g/mi. A company has 25 cars of this model in its fleet. What is the level L such that the probability that the average NOX level x for the fleet is greater than L is only 0.01? (Hint: This requires a backward Normal calculation. See page 81 in Chapter 3 if you need to review.) Returns on stocks. Andrew plans to retire in 40 years. He is thinking of investing his retirement funds in stocks, so he seeks out information on past returns. He learns that over the 101 years from 1900 to 2000, the real (that is, adjusted for inflation) returns on U.S. common stocks had mean 8.7% and standard deviation 20.2%. 4 The distribution of annual returns on common stocks is roughly symmetric, so the mean return over even a moderate number of years is close to Normal. What is the probability (assuming that the past pattern of variation continues) that the mean annual return on common stocks over the next 40 years will exceed 10%? What is the probability that the mean return will be less than 5%? Follow the four-step process in your answer Auto accidents. The number of accidents per week at a hazardous intersection varies with mean 2.2 and standard deviation 1.4. This distribution takes only whole-number values, so it is certainly not Normal. (a) Let x be the mean number of accidents per week at the intersection during a year (52 weeks). What is the approximate distribution of x according to the central limit theorem? (b) What is the approximate probability that x is less than 2? (c) What is the approximate probability that there are fewer than 100 accidents at the intersection in a year? (Hint: Restate this event in terms of x.) Airline passengers get heavier. In response to the increasing weight of airline passengers, the Federal Aviation Administration in 2003 told airlines to assume that passengers average 190 pounds in the summer, including clothing and carry-on baggage. But passengers vary, and the FAA did not specify a standard deviation. A reasonable standard deviation is 35 pounds. Weights are not Normally distributed, especially when the population includes both men and women, but they are not very non-normal. A commuter plane carries 19 passengers. What is the approximate probability that the total weight of the passengers exceeds 4000 pounds? Use the four-step process to guide your work.