# BPS - 5th Ed. Chapter 111 Sampling Distributions.

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BPS - 5th Ed. Chapter 111 Sampling Distributions

BPS - 5th Ed. Chapter 112 Sampling Terminology u Parameter –fixed, unknown number that describes the population u Statistic –known value calculated from a sample –a statistic is often used to estimate a parameter u Variability –different samples from the same population may yield different values of the sample statistic u Sampling Distribution –tells what values a statistic takes and how often it takes those values in repeated sampling

BPS - 5th Ed. Chapter 113 Parameter vs. Statistic A properly chosen sample of 1600 people across the United States was asked if they regularly watch a certain television program, and 24% said yes. The parameter of interest here is the true proportion of all people in the U.S. who watch the program, while the statistic is the value 24% obtained from the sample of 1600 people.

BPS - 5th Ed. Chapter 114 Parameter vs. Statistic u The mean of a population is denoted by µ – this is a parameter. u The mean of a sample is denoted by – this is a statistic. is used to estimate µ. u The true proportion of a population with a certain trait is denoted by p – this is a parameter. u The proportion of a sample with a certain trait is denoted by (“p-hat”) – this is a statistic. is used to estimate p.

BPS - 5th Ed. Chapter 115 The Law of Large Numbers Consider sampling at random from a population with true mean µ. As the number of (independent) observations sampled increases, the mean of the sample gets closer and closer to the true mean of the population. ( gets closer to µ )

BPS - 5th Ed. Chapter 116 The Law of Large Numbers Gambling u The “house” in a gambling operation is not gambling at all –the games are defined so that the gambler has a negative expected gain per play (the true mean gain after all possible plays is negative) –each play is independent of previous plays, so the law of large numbers guarantees that the average winnings of a large number of customers will be close the the (negative) true average

BPS - 5th Ed. Chapter 117 Sampling Distribution u The sampling distribution of a statistic is the distribution of values taken by the statistic in all possible samples of the same size (n) from the same population –to describe a distribution we need to specify the shape, center, and spread –we will discuss the distribution of the sample mean (x-bar) in this chapter

BPS - 5th Ed. Chapter 118 Case Study Does This Wine Smell Bad? Dimethyl sulfide (DMS) is sometimes present in wine, causing “off-odors”. Winemakers want to know the odor threshold – the lowest concentration of DMS that the human nose can detect. Different people have different thresholds, and of interest is the mean threshold in the population of all adults.

BPS - 5th Ed. Chapter 119 Case Study Suppose the mean threshold of all adults is  =25 micrograms of DMS per liter of wine, with a standard deviation of  =7 micrograms per liter and the threshold values follow a bell-shaped (normal) curve. Does This Wine Smell Bad?

BPS - 5th Ed. Chapter 1110 Where should 95% of all individual threshold values fall? u mean plus or minus two standard deviations 25  2(7) = 11 25 + 2(7) = 39 u 95% should fall between 11 & 39 u What about the mean (average) of a sample of n adults? What values would be expected?

BPS - 5th Ed. Chapter 1111 Sampling Distribution u What about the mean (average) of a sample of n adults? What values would be expected? u Answer this by thinking: “What would happen if we took many samples of n subjects from this population?” (let’s say that n=10 subjects make up a sample) –take a large number of samples of n=10 subjects from the population –calculate the sample mean (x-bar) for each sample –make a histogram (or stemplot) of the values of x-bar –examine the graphical display for shape, center, spread

BPS - 5th Ed. Chapter 1112 Case Study Mean threshold of all adults is  =25 micrograms per liter, with a standard deviation of  =7 micrograms per liter and the threshold values follow a bell-shaped (normal) curve. Many (1000) repetitions of sampling n=10 adults from the population were simulated and the resulting histogram of the 1000 x-bar values is on the next slide. Does This Wine Smell Bad?

BPS - 5th Ed. Chapter 1113 Case Study Does This Wine Smell Bad?

BPS - 5th Ed. Chapter 1114 Mean and Standard Deviation of Sample Means If numerous samples of size n are taken from a population with mean  and standard deviation , then the mean of the sampling distribution of is  (the population mean) and the standard deviation is: (  is the population s.d.)

BPS - 5th Ed. Chapter 1115 Mean and Standard Deviation of Sample Means  Since the mean of is , we say that is an unbiased estimator of  u Individual observations have standard deviation , but sample means from samples of size n have standard deviation. Averages are less variable than individual observations.

BPS - 5th Ed. Chapter 1116 Sampling Distribution of Sample Means If individual observations have the N(µ,  ) distribution, then the sample mean of n independent observations has the N(µ,  / square root{n} ) distribution. “If measurements in the population follow a Normal distribution, then so does the sample mean.”

BPS - 5th Ed. Chapter 1117 Case Study Mean threshold of all adults is  =25 with a standard deviation of  =7, and the threshold values follow a bell-shaped (normal) curve. Does This Wine Smell Bad? (Population distribution)

BPS - 5th Ed. Chapter 1118 Exercise 11.26: To estimate the mean height  of studets on your campus, you will meaure an SRS of students. From government data,we Know that the standard deviation of the heights Of young men is about 2.8 inches. Suppose that (unknown to you) he mean height of All male students is 70 inches. a)IF you choose a student at random, what is the probability that he is between 69 and 71 inches b) You measure 25 students. What is the sampling Distribution of their average height?

BPS - 5th Ed. Chapter 1119 c) What is the probability that the mean height of your sample is between 69 and 71 inches?

BPS - 5th Ed. Chapter 1120 Central Limit Theorem “No matter what distribution the population values follow, the sample mean will follow a Normal distribution if the sample size is large.” If a random sample of size n is selected from ANY population with mean  and standard deviation , then when n is large the sampling distribution of the sample mean is approximately Normal: is approximately N(µ,  / )

BPS - 5th Ed. Chapter 1121 Central Limit Theorem: Sample Size u How large must n be for the CLT to hold? –depends on how far the population distribution is from Normal v the further from Normal, the larger the sample size needed v a sample size of 25 or 30 is typically large enough for any population distribution encountered in practice v recall: if the population is Normal, any sample size will work (n≥1)

BPS - 5th Ed. Chapter 1122 Central Limit Theorem: Sample Size and Distribution of x-bar n=1 n=25 n=10 n=2

BPS - 5th Ed. Chapter 1123 11.31: The number of accidents per week at a Hazardous intersection varies with mean 2.2 And standard deviation 1.4. This distribution Takes only integer values, so it is certainly not Normal. a)Let x-bar be the mean number of accidents Per week at the intersection during the year (52 Weeks). What is the approx. distribution of x-bar According to the central limit theorem? b) What is the approximate probability that x-bar Is less than 2? c) What is the approx. prob.that there are fewer than 100 accidents at the inters. In a year?

BPS - 5th Ed. Chapter 1124 Statistical Process Control u Goal is to make a process stable over time and keep it stable unless there are planned changes u All processes have variation u Statistical description of stability over time: the pattern of variation remains stable (does not say that there is no variation)

BPS - 5th Ed. Chapter 1125 Statistical Process Control u A variable described by the same distribution over time is said to be in control u To see if a process has been disturbed and to signal when the process is out of control, control charts are used to monitor the process –distinguish natural variation in the process from additional variation that suggests a change –most common application: industrial processes

BPS - 5th Ed. Chapter 1126 Charts  There is a true mean  that describes the center or aim of the process u Monitor the process by plotting the means (x-bars) of small samples taken from the process at regular intervals over time u Process-monitoring conditions: –measure quantitative variable x that is Normal –process has been operating in control for a long period –know process mean  and standard deviation  that describe distribution of x when process is in control

BPS - 5th Ed. Chapter 1127 Control Charts u Plot the means (x-bars) of regular samples of size n against time  Draw a horizontal center line at   Draw horizontal control limits at  ± 3  / –almost all (99.7%) of the values of x-bar should be within the mean plus or minus 3 standard deviations u Any x-bar that does not fall between the control limits is evidence that the process is out of control

BPS - 5th Ed. Chapter 1128 Case Study Need to control the tension in millivolts (mV) on the mesh of fine wires behind the surface of the screen. –Proper tension is 275 mV (target mean  ) –When in control, the standard deviation of the tension readings is  =43 mV Making Computer Monitors

BPS - 5th Ed. Chapter 1129 Case Study Proper tension is 275 mV (target mean  ). When in control, the standard deviation of the tension readings is  =43 mV. Making Computer Monitors Take samples of n=4 screens and calculate the means of these samples –the control limits of the x-bar control chart would be

BPS - 5th Ed. Chapter 1130 Case Study Making Computer Monitors (data)

BPS - 5th Ed. Chapter 1131 Case Study Making Computer Monitors ( chart) (in control)

BPS - 5th Ed. Chapter 1132 Case Study Making Computer Monitors (examples of out of control processes)

BPS - 5th Ed. Chapter 1133 Natural Tolerances  For x-bar charts, the control limits for the mean of the process are  ± 3  / –almost all (99.7%) of the values of x-bar should be within the mean plus or minus 3 standard deviations  When monitoring a process, the natural tolerances for individual products are  ± 3  –almost all (99.7%) of the individual measurements should be within the mean plus or minus 3 standard deviations

u Exercise 11.34: Airline passengers average 190 pounds (including carry on luggage) with a standard deviation of 35 pounds. Weights are not Normally distributed but they are not very non-Normal. u A commuter plane carries 19 passengers. What is the approximate probability that the total weight of the passengers exceeds 400 pounds? BPS - 5th Ed. Chapter 1134