Sampling Distributions

Transcription

1 CHAPTER 10 (Frank Pedrick/Index Stock Imagery/PictureQuest) Sampling Distributions What is the mean income of households in the United States? The government s Current Population Survey contacted a sample of 55,000 households in March Their mean income in 2001 was x = $58, That $58,208 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 do is guarantee that the methods we use 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. 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 Once 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. 249

2 250 CHAPTER 10 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 10.1 Household income The mean income of the sample of households contacted by the Current Population Survey was x = $58,208. The number $58,208 is a statistic because it describes this one Current Population Survey sample. The population that the poll wants to draw conclusions about is all 110 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 sample mean x 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 observations 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. APPLY YOUR KNOWLEDGE State whether each boldface number in Exercises 10.1 to 10.3 is a parameter or a statistic Apartment rents. Your local newspaper contains a large number of advertisements for unfurnished one-bedroom apartments. You choose 10 at random and calculate that their mean monthly rent is $540 and that the standard deviation of their rents is $ 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 Inspecting bearings. A carload lot of ball bearings has mean diameter centimeters (cm). This is within the specifications for acceptance of the lot by the purchaser. By chance, an inspector chooses

3 Statistical estimation and the law of large numbers bearings from the lot that have mean diameter cm. Because this is outside the specified limits, the lot is mistakenly rejected. 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 10.2 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, andwe 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 450,000 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 its5%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.

4 252 CHAPTER 10 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 9.1 (page 225) shows how proportions approach probability in one example. Here is an example of how sample means approach the population mean. EXAMPLE 10.3 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 10.1 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 10.2 had threshold 28, so the line in Figure 10.1 starts there. The mean for the first two subjects is x = = 34 2 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 Mean of first n observations ,000 Number of observations, n Figure 10.1 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 253 says that whatever path we get will always settle down at 25 as we draw more and more people. The Law of Large Numbers applet animates Figure 10.1 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 10.1, the mean of even 100 observations is not yet very close to µ.itisonlyin 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 APPLYYOURKNOWLEDGE 10.4 Means in action. Figure 10.1 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 10.1: find the mean of the first one, then 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 10.2 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:

6 254 CHAPTER 10 Sampling Distributions simulation 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. 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 10.4 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 10.2 with different subjects drawn at random from the population. Figure 10.2 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. 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 10.2 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.

7 Sampling distributions 255 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. 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 10.2, 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. APPLYYOURKNOWLEDGE 10.6 Generating a sampling distribution. Let us 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.

8 256 CHAPTER 10 Sampling Distributions (a) Find the mean of the 10 scores in the population. This is the population mean µ. (b) Use Table B to draw an SRS of size 4 from this population. Write the four scores in your sample and calculate the mean x of the sample scores. This statistic is an estimate of µ. (c) Repeat this process 10 times using different parts of Table B. Make a histogram of the 10 values of x. You are constructing the sampling distribution of x. Is the center of your histogram close to µ? 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 The sampling distribution of x Figure 10.2 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 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 mean of the sampling distribution of x is µ and its standard deviation is σ/ n. Both the mean and the standard deviation of the sampling distribution of x have important implications for statistical inference. The mean of the statistic x is always the same as 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. 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.

9 The sampling distribution of x 257 SAMPLING DISTRIBUTION OF A SAMPLE MEAN If individual observations have the N(µ, σ) distribution, then the sample mean x of n independent observations has the N(µ, σ/ n) distribution. EXAMPLE 10.5 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 σ = 7 = n 10 population distribution Figure 10.3 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 The distribution of sample means is less spread out. Means x of 10 subjects σ 10 = 2.21 Observations on 1 subject σ = Figure 10.3 The distribution of single observations compared with the distribution of the means x of 10 observations. Averages are less variable than individual observations.

10 258 CHAPTER 10 Sampling Distributions 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. Notice, however, that 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. APPLY YOUR KNOWLEDGE 10.7 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 Measuring blood cholesterol. The distribution of blood cholesterol level in the population of young men aged 20 to 34 years is close to Normal, with mean µ = 188 milligrams per deciliter (mg/dl) and standard deviation σ = 41 mg/dl. You measure the cholesterol level of 100 young men chosen at random and calculate the mean x. (a) If you did this many times, what would be the mean and standard deviation of the distribution of all the x-values? (b) What is the probability that your sample has mean x less than 180? 10.9 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. What is the probability that their mean score 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

11 The central limit theorem 259 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. CENTRAL LIMIT THEOREM Draw an SRS of size n from any population with mean µ and finite standard deviation σ.whenn is large, the sampling distribution of the sample mean x is approximately Normal: x is approximately N (µ, σ ) n More general versions of the central limit theorem say that the distribution of a sum or average of many small random quantities is close to Normal. This is true even if the quantities are not independent (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. EXAMPLE 10.6 The central limit theorem in action Figure 10.4 shows how the central limit theorem works for a very non-normal population. Figure 10.4(a) displays the density curve of a single observation, that is, of the population. The distribution is strongly 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 10.4(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 = 1andσ = 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.

12 260 CHAPTER 10 Sampling Distributions 0 1 (a) 0 1 (b) 0 1 (c) 0 1 (d) Figure 10.4 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 one 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. 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. EXAMPLE 10.7 Maintaining air conditioners The time X that a technician requires to perform preventive maintenance on an airconditioning unit is governed by the exponential distribution whose density curve appears in Figure 10.4(a). The mean time is µ = 1 hour and the standard deviation is σ = 1 hour. Your company operates 70 of these units. What is the probability that their average maintenance time exceeds 50 minutes? The central limit theorem says that the sample mean time x (in hours) spent working on 70 units has approximately the Normal distribution with mean equal to the population mean µ = 1 hour and standard deviation σ = 1 = 0.12 hour The distribution of x is therefore approximately N(1, 0.12). This Normal curve is the solid curve in Figure 10.5.

13 The central limit theorem Figure 10.5 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 Because 50 minutes is 50/60 of an hour, or 0.83 hour, the probability we want is P (x > 0.83). A Normal distribution calculation gives this probability as This is the area to the right of 0.83 under the solid Normal curve in Figure Using more mathematics, we could 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 is the area under the dashed density curve. It is The central limit theorem Normal approximation is off by only about APPLYYOURKNOWLEDGE More on insurance. The insurance company of Exercise 10.5 sees that in the entire population of homeowners, the mean loss from fire is µ = $250 and the standard deviation of the loss is σ = $1000. The distribution of losses is strongly right-skewed: many policies have $0 loss, but a few have large losses. If the company sells 10,000 policies, what is the approximate probability that the average loss will be greater than $275? ACT scores. The scores of students on the ACT college entrance examination in 2001 had mean µ = 21.0 and standard deviation σ = 4.7. The distribution of scores is only roughly Normal. (a) What is the approximate probability that a single student randomly chosen from all those taking the test scores 23 or higher?

14 262 CHAPTER 10 Sampling Distributions (b) Now take an SRS of 50 students who took the test. What are the mean and standard deviation of the sample mean score x of these 50 students? What is the approximate probability that the mean score x of these students is 23 or higher? (c) Which of your two Normal probability calculations in (a) and (b) is more accurate? Why? Flaws in carpets. The number of flaws per square yard in a type of carpet material varies with mean 1.6 flaws per square yard and standard deviation 1.2 flaws per square yard. The population distribution cannot be Normal, because a count takes only whole-number values. An inspector samples 200 square yards of the material, records the number of flaws found in each square yard, and calculates x, the mean number of flaws per square yard inspected. Use the central limit theorem to find the approximate probability that the mean number of flaws exceeds 2 per square yard. 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 24.

15 x charts 263 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 long-term 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 will have to collect data from the process, establish control by uncovering and removing special causes, 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 10.8 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.

16 264 CHAPTER 10 Sampling Distributions TABLE 10.1 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 10.1 shows the samples and their means x s 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 10.6 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 the central limit theorem effect implies that sample means will be closer to 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

17 x charts The control limits mark the natural variation in the process. 350 UCL Sample mean LCL Sample number Figure 10.6 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.

18 266 CHAPTER 10 Sampling Distributions EXAMPLE 10.9 Interpreting x charts Figure 10.6 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 10.7 and 10.8 illustrate two ways in which the process can go out of control. In Figure 10.7, 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 10.8 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. 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. This point is out of control because it is above UCL UCL x Sample mean LCL Sample number Figure 10.7 This x chart is identical to that in Figure 10.6, 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.

19 x charts x x 350 UCL x Sample mean LCL Sample number Figure 10.8 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. APPLYYOURKNOWLEDGE 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 is raised 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 10.2 gives three sets of data, each representing x for 20 successive samples of n = 4tablets. 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.

20 268 CHAPTER 10 Sampling Distributions TABLE 10.2 Three sets of x s from 20 samples of size 4 Sample Data set A Data set B Data set C (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. (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? 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.

21 Chapter 10 Summary 269 In order to assess whether the process quality is satisfactory, we must observe the process operating in control free of breakdowns and other disturbances. 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. natural tolerances 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. APPLYYOURKNOWLEDGE 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 10 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.

22 270 CHAPTER 10 Sampling Distributions 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 samples of the same size from the same population. The mean of the sampling distribution is µ, sothatx 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. Chapter 10 EXERCISES Personal income. The government s Current Population Survey interviewed more than 131,000 people aged 25 or older in March The median income of the people with at least a bachelor s degree was $44,776. The median income of people with just a high school diploma was $25,303. Is each of the bold numbers a parameter or a statistic? 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? 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.

23 Chapter 10 Exercises 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 Samples of incomes. In March 2000 the Bureau of Labor Statistics recorded the incomes of 55,899 people between the ages of 25 and 65 who had worked but whose main work was not in agriculture. We will treat these 55,899 people as a population. As is usually the case, the distribution of incomes in this population is strongly skewed to the right. To estimate the mean income in this population, we can select an SRS and use the sample mean x to estimate the unknown population mean. How will the sample mean behave when we take many samples? We used software to choose 1000 SRSs of size 25 and another 1000 SRSs of size 100. Figure 10.9 shows histograms of the two sets of 1000 sample means, using the same classes and drawn to the same scale for easy comparison. (a) Which distribution is closer in shape to the bell curve of a Normal distribution? What important fact about sampling distributions does this comparison illustrate? (b) About what is the range (from smallest to largest) of the sample Count of samples, out of Count of samples, out of Sample mean income ($1000) Sample mean income ($1000) Figure 10.9 The distribution of sample means x for 1000 SRSs of size 25 (left)andof size 100 (right) from the same population, for Exercise

24 272 CHAPTER 10 Sampling Distributions means for samples of size 25? For samples of size 100? What important fact about sampling distributions does this comparison illustrate? (c) Based on the sample means for samples of size 100, about what is the value of the mean income for this entire population? (Peter Turnley/CORBIS) The cost of Internet access. The amount that households pay service providers for access to the Internet varies quite a bit, but the mean monthly fee is $28 and the standard deviation is $10. The distribution is not Normal: many households pay about $10 for limited dial-up access or about $25 for unlimited dial-up access, but some pay much more for faster connections. A sample survey asks an SRS of 500 households with Internet access how much they pay. What is the probability that the average fee paid by the sample households exceeds $29? Dust in coal mines. A laboratory weighs filters from a coal mine to measure the amount of dust in the mine atmosphere. Repeated measurements of the weight of dust on the same filter vary Normally with standard deviation σ = 0.08 milligram (mg) because the weighing is not perfectly precise. The dust on a particular filter actually weighs 123 mg. Repeated weighings will then have the Normal distribution with mean 123 mg and standard deviation 0.08 mg. (a) The laboratory reports the mean of 3 weighings. What is the distribution of this mean? (b) What is the probability that the laboratory reports a weight of 124 mg or higher for this filter? Making auto parts. An automatic grinding machine in an auto parts plant prepares axles with a target diameter µ = millimeters (mm). The machine has some variability, so the standard deviation of the diameters is σ = mm. A sample of 4 axles is inspected each hour for process control purposes, and records are kept of the sample mean diameter. What will be the mean and standard deviation of the numbers recorded? 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 one hour after a sugary drink is ingested. Shelia s measured glucose level one hour after ingesting 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 instead on 4 separate days and the mean

25 Chapter 10 Exercises 273 result is compared with the criterion 140 mg/dl, what is the probability that Shelia is diagnosed as having gestational diabetes? Glucose testing, continued. Shelia s measured glucose level one hour after ingesting the sugary drink varies according to the Normal distribution with µ = 125 mg/dl and σ = 10 mg/dl. Find the level L such that there is probability only 0.05 that the mean glucose level of 4 test results falls above L for Shelia s glucose level distribution. What is the value of L? (Hint: This requires a backward Normal calculation. See page 72 in Chapter 3 if you need to review.) Pollutants in auto exhausts. The level of nitrogen oxides (NOX) in the exhaust of a particular car model varies with mean 0.9 grams per mile (g/mi) and standard deviation 0.15 g/mi. A company has 125 cars of this model in its fleet. (a) What is the approximate distribution of the mean NOX emission level x for these cars? (b) What is the level L such that the probability that x is greater than L is only 0.01? (Hint: This requires a backward Normal calculation. See page 72 in Chapter 3 if you need to review.) 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.) How many people in a car? A study of rush-hour traffic in San Francisco counts the number of people in each car entering a freeway at a suburban interchange. Suppose that this count has mean 1.5 and standard deviation 0.75 in the population of all cars that enter at this interchange during rush hours. (a) Could the exact distribution of the count be Normal? Why or why not? (b) Traffic engineers estimate that the capacity of the interchange is 700 cars per hour. According to the central limit theorem, what is the approximate distribution of the mean number of persons x in 700 randomly selected cars at this interchange? (c) What is the probability that 700 cars will carry more than 1075 people? (Hint: Restate this event in terms of the mean number of people x per car.) (Kyle Krause/Inndex Stock Imagery/PictureQuest)