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Chapter 10 Sampling Distributions BPS - 3rd Ed. Chapter 10 1 Parameters and Statistics Parameter – Is a fixed number that describes the location or spread of a population – Its value is NOT known in statistical practice Statistic – A calculated number from data in the sample – Its value IS known in statistical practice – Some statistics are used to estimate parameters Sampling variability – Different samples or experiments from the same population yield different values of the statistic BPS - 3rd Ed. Chapter 10 2 Parameters and statistics mean of a population is called µ this is a parameter The mean of a sample is called “x-bar” this is a statistic Illustration: The – Average age of all SJSU students (µ) is 26.5 – A SRS of 10 San Jose State students yields a mean age (x-bar) of 22.3 – x-bar and µ are related but are not the same thing! BPS - 3rd Ed. Chapter 10 3 The Law of Large Numbers BPS - 3rd Ed. Chapter 10 4 Example 10.2 (p. 251) Does This Wine Smell Bad? Dimethyl sulfide (DMS) is sometimes present in wine causing off-odors Different people have different thresholds for smelling DMS Winemakers want to know the odor threshold that the human nose can detect. Population values: – Mean threshold of all adults is 25 µg/L wine – Standard deviation of all adults is 7 µg/L – Distribution is Normal BPS - 3rd Ed. Chapter 10 5 Simulation: Example 10.2 (cont.) Suppose you take a simple random sample (SRS) from the population and the first individual in sample has a value of 28 The second individual has a value of 40 The average of the first two individuals = (28+40)/2 = 34 Continue sampling individuals at random and calculating means Plot the means BPS - 3rd Ed. Chapter 10 6 Simulation: law of large numbers Fig 10.1 The sample mean gets close to population mean as we take more and more samples BPS - 3rd Ed. Chapter 10 7 Sampling distribution of xbar Key questions: What would happen if we took many samples or did the experiment many times? How would the statistics from these repeated samples vary? BPS - 3rd Ed. Chapter 10 8 Case Study Does This Wine Smell Bad? • Recall = 25 µg / L, = 7 µg / L and the distribution is Normal • Suppose you take 1,000 repetitions of samples, each of n =10 from this population • You calculate x-bar in each sample • You plot the x-bars as a histogram • You study the histogram (next slide) BPS - 3rd Ed. Chapter 10 9 Simulation: 1000 sample means Example 10.4 BPS - 3rd Ed. Chapter 10 10 Mean and Standard Deviation of “x-bar” BPS - 3rd Ed. Chapter 10 11 Mean and Standard Deviation of Sample Means the mean of X is , we say that X is an unbiased estimator of Since Individual observations have standard deviation , but sample means X from samples of size n have standard deviation n . Averages are less variable than individual observations. BPS - 3rd Ed. Chapter 10 12 Case Study Does This Wine Smell Bad? (Population distribution) BPS - 3rd Ed. Chapter 10 13 Illustration Exercise 10.8 (p. 258) Suppose blood cholesterol in a population of men is Normal with = 188 and = 41. You select 100 men at random (a) What is mean and standard deviation of the 100 x-bars from these samples? – x-barx-bar = 188 (same as µ) – sx-bar = 41 / sqrt(100) = 4.1 (one-tenth of ) (b) What is probability a given x-bar is less than 180? Pr(x-bar < 180) standardize z = (180 – 188) / 4.1 = -1.95 = Pr(Z < –1.95) TABLE A = .0256 BPS - 3rd Ed. Chapter 10 14 Central Limit Theorem No matter what the shape of the population, the sampling distribution of xbar, will follow a Normal distribution when the sample size is large. BPS - 3rd Ed. Chapter 10 15 Central Limit Theorem in Action Example 10.6 (p. 259) Data = time to perform an activity – NOT Normal (Fig a) – µ = 1 hour – σ = 1 hour Fig (a) is for single observations Fig (b) is for x-bars based on n = 2 Fig (c) is for x-bars based on n = 10 Fig (d) is for x-bars based on n = 25 Notice how distribution becomes increasingly Normal as n increases! BPS - 3rd Ed. Chapter 10 16 Example 10.7 (cont.) Sample n = 70 activities What is the distribution of x-bars? – x-barx-bar = 1 (same as µ) – sx-bar = 1 / sqrt(70) = 0.12 (by formula) – Normal (central limit theorem) What % of x-bars will be less than 0.83 hours? Pr(x-bar < 0.83) standardize z = (0.83 – 1) / 0.12 = -1.42 Pr(Z < - 1.42) TABLE A = 0.0778 Notice that if Pr(x-bar < 0.83 = 0.0778, then Pr(Z > 0.83) = 1 – 0.0778 = 0.9222 BPS - 3rd Ed. Chapter 10 17 BPS - 3rd Ed. Chapter 10 18 Statistical process control* Skip pp. 262 – 269 BPS - 3rd Ed. Chapter 10 19