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## Chapter 8

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**Chapter 8**Sampling Variability and Sampling Distributions**Suppose we are interested in finding the true mean (m) fat**content of quarter-pound hamburgers marketed by a national fast food chain. To learn something about m, we could obtain a sample of n = 50 hamburgers and determine the fat content of each one. To answer these questions, we will examine the sampling distribution, which describes the long-run behavior of sample statistic. Would the sample mean be a good estimate of m? How close is the sample mean to m? Recall that the sample mean is a statistic Will other samples of n = 50 have the same sample mean?**Statistic**• A number that that can be computed from sample data • Some statistics we will use include x – sample mean s – standard deviation p – sample proportion • The observed value of the statistic depends on the particular sample selected from the population and it will vary from sample to sample. This variability is called sampling variability**We caught fish with lengths 6.3 inches, 2.2 inches, and 13.3**inches. x = 7.27 inches 2nd sample - 8.5, 4.6, and 5.6 inches. x = 6.23 inches 3rd sample – 10.3, 8.9, and 13.4 inches. x = 10.87 inches The campus of Wolf City College has a fish pond. Suppose there are 20 fish in the pond. The lengths of the fish (in inches) are given below: This is a statistic! The true mean m = 8. Notice that some sample means are closer and some farther away; some above and some below the mean. Let’s catch two more samples and look at the sample means. Suppose we randomly catch a sample of 3 fish from this pond and measure their length. What would the mean length of the sample be? This is an example of sampling variability**There are 1140 (20C3) different possible samples of size 3**from this population. If we were to catch all those different samples and calculate the mean length of each sample, we would have a distribution of all possible x. This would be the sampling distribution of x. Fish Pond Continued . . .**In this case, the sample statistic is the sample mean x.**Sampling Distributions of x • The distribution that would be formed by considering the value of asample statisticforeverypossible differentsample of agiven sizefrom apopulation.**Fish Pond Revisited . . .**Suppose there are only 5 fish in the pond. The lengths of the fish (in inches) are given below: We will keep the population size small so that we can find ALL the possible samples. What is the mean and standard deviation of this population? mx = 7.84 sx = 3.262**These values determine the sampling distributionof x for**samples of size 2. mx = 7.84 sx = 1.998 Fish Pond Revisited . . . mx = 7.84 and sx = 3.262 Let’s find all the samples of size 2. How many samples of size 2 are possible? How do these values compare to the population mean and standard deviation? What is the mean and standard deviation of these sample means?**These values determine the sampling distributionof x for**samples of size 3. mx = 7.84 sx = 1.332 Fish Pond Revisited . . . mx = 7.84 and sx = 3.262 Now let’s find all the samples of size 3. How many samples of size 3 are possible? What is the mean and standard deviation of these sample means? How do these values compare to the population mean and standard deviation?**mx=m**sx What do you notice? • The mean of the sampling distribution EQUALS the mean of the population. • As the sample size increases, the standard deviation of the sampling distribution decreases. as n**General Properties of Sampling Distributions of x**Rule 1: Rule 2: This rule is exact if the population is infinite, and is approximately correct if the population is finite and no more than 10% of the population is included in the sample Note that in the previous fish pond examples this standard deviation formula was not correct because the sample sizes were more than 10% of the population.**The paper “Mean Platelet Volume in Patients with Metabolic**Syndrome and Its Relationship with Coronary Artery Disease” (Thrombosis Research, 2007) includes data that suggests that the distribution of platelet volume of patients who do not have metabolic syndrome is approximately normal with mean m = 8.25 and standard deviation s = 0.75. We can use Minitab to generate random samples from this population. We will generate 500 random samples of n = 5 and compute the sample mean for each.**Platelets Continued . . .**Similarly, we will generate 500 random samples of n = 10, n = 20, and n = 30. The density histograms below display the resulting 500 x for each of the given sample sizes. What do you notice about the standard deviation of these histograms? What do you notice about the shape of these histograms? What do you notice about the means of these histograms?**General Properties Continued . . .**Rule 3: When the population distribution is normal, the sampling distribution of x is also normal for any sample size n.**The paper “Is the Overtime Period in an NHL Game Long**Enough?” (American Statistician, 2008) gave data on the time (in minutes) from the start of the game to the first goal scored for the 281 regular season games from the 2005-2006 season that went into overtime. The density histogram for the data is shown below. Let’s consider these 281 values as a population. The distribution is strongly positively skewed with mean m = 13 minutes and with a median of 10 minutes. Using Minitab, we will generate 500 samples of the following sample sizes from this distribution: n = 5, n = 10, n = 20, n = 30.**What do you notice about the standard deviations of these**histograms? These are the density histograms for the 500 samples Are these histograms centered at approximately m = 13? What do you notice about the shape of these histograms?**General Properties Continued . . .**Rule 4:Central Limit Theorem When n is sufficiently large, the sampling distribution of x is well approximated by a normal curve, even when the population distribution is not itself normal. How large is “sufficiently large” anyway? CLT can safely be applied if n exceeds 30.**A soft-drink bottler claims that, on average, cans contain**12 oz of soda. Let x denote the actual volume of soda in a randomly selected can. Suppose that x is normally distributed with s = .16 oz. Sixteen cans are randomly selected, and the soda volume is determined for each one. Let x = the resulting sample mean soda. If the bottler’s claim is correct, then the sampling distribution of x is normally distributed with:**To standardized these endpoints, use**P(11.96 < x < 12.08) = Soda Problem Continued . . . Look these up in the table and subtract the probabilities. What is the probability that the sample mean soda volume is between 11.96 ounces and 12.08 ounces? .9772 - .1587 = .8185**So the distribution of x is approximately normal with**A hot dog manufacturer asserts that one of its brands of hot dogs has a average fat content of 18 grams per hot dog with standard deviation of 1 gram. Consumers of this brand would probably not be disturbed if the mean was less than 18 grams, but would be unhappy if it exceeded 18 grams. An independent testing organization is asked to analyze a random sample of 36 hot dogs. Suppose the resulting sample mean is 18.4 grams. Does this result suggest that the manufacturer’s claim is incorrect? Since the sample size is greater than 30, the Central Limit Theorem applies.**Values of x at least as large as 18.4 would be observed only**about .82% of the time. The sample mean of 18.4 is large enough to cause us to doubt that the manufacturer’s claim is correct. P(x> 18.4) = Hot Dogs Continued . . . Suppose the resulting sample mean is 18.4 grams. Does this result suggest that the manufacturer’s claim is incorrect? 1 - .9918 = .0082**Let’s explore what happens with in distributions of sample**proportions (p). Have students perform the following experiment. • Toss a penny 20 times and record the number of heads. • Calculate the proportion of heads and mark it on the dot plot on the board. What shape do you think the dot plot will have? This is a statistic! The dotplot is a partial graph of the sampling distribution of all sample proportions of sample size 20. What would happen to the dotplot if we flipped the penny 50 times and recorded the proportion of heads?**We will use:**p for the population proportion and p for the sample proportion In this case, we will use Sampling Distribution of p The distribution that would be formed by considering the value of asample statisticforeverypossible differentsample of agiven sizefrom apopulation.**Suppose we have a population of six students: Alice, Ben,**Charles, Denise, Edward, & Frank We are interested in the proportion of females.This is called What is the proportion of females? Let’s select samples of two from this population. How many different samples are possible? We will keep the population small so that we can find ALL the possible samples of a given size. the parameter of interest 1/3 6C2 =15**Alice & Ben .5**Alice & Charles .5 Alice & Denise 1 Alice & Edward .5 Alice & Frank .5 Ben & Charles 0 Ben & Denise .5 Ben & Edward 0 Ben & Frank 0 Charles & Denise .5 Charles & Edward 0 Charles & Frank 0 Denise & Edward .5 Denise & Frank .5 Edward & Frank 0 Find the 15 different samples that are possible and find the sample proportion of the number of females in each sample. How does the mean of the sampling distribution compare to the population parameter (p)? Find the mean and standard deviation of these sample proportions.**General Properties for Sampling Distributions of p**Rule 1: Rule 2: This rule is exact if the population is infinite, and is approximately correct if the population is finite and no more than 10% of the population is included in the sample Note that in the previous student example this standard deviation formula was not correct because the sample size was more than 10% of the population.**In the fall of 2008, there were 18,516 students enrolled at**California Polytechnic State University, San Luis Obispo. Of these students, 8091 (43.7%) were female. We will use a statistical software package to simulate sampling from this Cal Poly population. We will generate 500 samples of each of the following sample sizes: n = 10, n = 25, n = 50, n = 100 and compute the proportion of females for each sample. The following histograms display the distributions of the sample proportions for the 500 samples of each sample size.**What do you notice about the standard deviation of these**distributions? What do you notice about the shape of these distributions? Are these histograms centered around the true proportion p = .437?**The development of viral hepatitis after a blood**transfusion can cause serious complications for a patient. The article “Lack of Awareness Results in Poor Autologous Blood Transfusions” (Health Care Management, May 15, 2003) reported that hepatitis occurs in 7% of patients who receive blood transfusions during heart surgery. We will simulate sampling from a population of blood recipients. We will generate 500 samples of each of the following sample sizes: n = 10, n = 25, n = 50, n = 100 and compute the proportion of people who contract hepatitis for each sample. The following histograms display the distributions of the sample proportions for the 500 samples of each sample size.**Are these histograms centered around the true proportion p =**.07? What happens to the shape of these histograms as the sample size increases?**A conservative rule of thumb:**Ifnp> 10andn (1 – p) > 10, then a normal distribution provides a reasonable approximation to the sampling distribution of p. General Properties Continued . . . Rule 3:When n is large and p is nottoo near 0 or 1, the sampling distribution of p is approximately normal. The farther the value of p is from 0.5, the largern must be for the sampling distribution of p to be approximately normal.**To answer this question, we must consider the sampling**distribution of p. Blood Transfusions Revisited . . . Let p = proportion of patients who contract hepatitis after a blood transfusion p = .07 Suppose a new blood screening procedure is believed to reduce the incident rate of hepatitis. Blood screened using this procedure is given to n = 200 blood recipients. Only 6 of the 200 patients contract hepatitis. Does this result indicate that the true proportion of patients who contract hepatitis when the new screening is used is less than 7%?**Blood Transfusions Revisited . . .**Let p = .07 p = 6/200 = .03 Is the sampling distribution approximately normal? np = 200(.07) = 14 > 10 n(1-p) = 200(.93) = 186 > 10 What is the mean and standard deviation of the sampling distribution? Yes, we can use a normal approximation.**P(p < .03) =**Blood Transfusions Revisited . . . Let p = .07 p = 6/200 = .03 Does this result indicate that the true proportion of patients who contract hepatitis when the new screening is used is less than 7%? This small probability tells us that it is unlikely that a sample proportion of .03 or smaller would be observed if the screening procedure was ineffective. This new screening procedure appears to yield a smaller incidence rate for hepatitis. .0132 Assume the screening procedure is not effective and p = .07.