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+ Chapter 7: Sampling Distributions Section 7.1 What is a Sampling Distribution? The Practice of Statistics, 4th edition – For AP* STARNES, YATES, MOORE + Chapter 7 Sampling Distributions 7.1 What is a Sampling Distribution? 7.2 Sample Proportions 7.3 Sample Means + Section 7.1 What Is a Sampling Distribution? Learning Objectives After this section, you should be able to… DISTINGUISH between a parameter and a statistic DEFINE sampling distribution DISTINGUISH between population distribution, sampling distribution, and the distribution of sample data DETERMINE whether a statistic is an unbiased estimator of a population parameter DESCRIBE the relationship between sample size and the variability of an estimator + Introduction Different random samples yield different statistics. We need to be able to describe the sampling distribution of possible statistic values in order to perform statistical inference. We can think of a statistic as a random variable because it takes numerical values that describe the outcomes of the random sampling process. Therefore, we can examine its probability distribution using what we learned in Chapter 6. Population Sample Collect data from a representative Sample... Make an Inference about the Population. What Is a Sampling Distribution? The process of statistical inference involves using information from a sample to draw conclusions about a wider population. Definition: A parameter is a number that describes some characteristic of the population. In statistical practice, the value of a parameter is usually not known because we cannot examine the entire population. A statistic is a number that describes some characteristic of a sample. The value of a statistic can be computed directly from the sample data. We often use a statistic to estimate an unknown parameter. Remember s and p: statistics come from samples and parameters come from populations We write µ (the Greek letter mu) for the population mean and x ("x bar") for the sample mean. We use p to represent a population proportion. The sample proportion pˆ ("p - hat") is used to estimate the unknown parameter p. What Is a Sampling Distribution? As we begin to use sample data to draw conclusions about a wider population, we must be clear about whether a number describes a sample or a population. + Parameters and Statistics Problem: Identify the population, the parameter, the sample, and the statistic in each of the following settings. (a)A pediatrician wants to know the 75th percentile for the distribution of heights of 10-year-old boys so she takes a sample of 50 patients and calculates Q3 = 56 inches. (b) A Pew Research Center poll asked 1102 12- to 17-year-olds in the United States if they have a cell phone. Of the respondents, 71% said yes. http://www.pewinternet.org/Reports/2009/14--Teens-and-Mobile-Phones-Data-Memo.aspx Solution: (a) The population is all 10-year-old boys; the parameter of interest is the 75th percentile, or Q3. The sample is the 50 10-year-old boys included in the sample; the statistic is the sample Q3 = 56 inches. (b) The population is all 12- to 17-year-olds in the US; the parameter is the proportion with cell phones. The sample is the 1102 12- to 17-year-olds in the sample; the statistic is the sample proportion with a cell phone, = 0.71. + Example: Heights and Cell Phones This basic fact is called sampling variability: the value of a statistic varies in repeated random sampling. To make sense of sampling variability, we ask, “What would happen if we took many samples?” Sample Population Sample Sample Sample Sample Sample Sample Sample ? What Is a Sampling Distribution? How can x (sample) be an accurate estimate of µ (population)? After all, different random samples would produce different values of x . + Sampling Variability Sampling Distribution Applet Definition: 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. In practice, it’s difficult to take all possible samples of size n to obtain the actual sampling distribution of a statistic. Instead, we can use simulation to imitate the process of taking many, many samples. One of the uses of probability theory in statistics is to obtain sampling distributions without simulation. We’ll get to the theory later. What Is a Sampling Distribution? In the penny activity, we took several samples of 20 pennies. There are many, many possible SRSs of size 20 from a population of size 100. If we took every one of those possible samples, calculated the sample proportion for each, and graphed all of those values, we’d have a sampling distribution. + Sampling Distribution 1) The population distribution gives the values of the variable for all the individuals in the population. 2) The distribution of sample data shows the values of the variable for all the individuals in the sample. 3) The sampling distribution shows the statistic values from ALL the possible samples of the same size from the population. What Is a Sampling Distribution? There are actually three distinct distributions involved when we sample repeatedly and measure a variable of interest. + Population Distributions vs. Sampling Distributions p = proportion obtained in each p = actual proportion of in entire population sample The proportion of red chips (parameter p) is 1/3 Choose 12 chips and record the proportion of RED chips you have. Mark using p̂ on the graph. Population Distribution: What Is a Sampling Distribution? Activity: Counting Chips Given: 147 colored chips (1/3 red, 1/3 blue, 1/3 white) Distribution of Sample (n=12) Sampling Distribution + Example: Choosing Cards Imagine a deck of cards with aces and face cards removed so that only the cards 2 through 10 remain, shuffle the deck, randomly select 5 cards, and note the median value of the cards. For example, if the selected cards were 2, 2, 4, 5, and 9, the median would be 4. Record the value of the sample median on a dotplot going from 2 to 10. We used Fathom to simulate choosing 500 SRSs of size 5 from the deck of cards described above. The graph below shows the distribution of the sample median for these 500 samples. Problem: (a)Is this the sampling distribution of the sample median? Justify your answer. No, since the distribution didn’t use all possible samples of size 5, this isn’t the exact sampling distribution. (b) Describe the distribution. Are there any obvious outliers? Shape: The graph is roughly symmetric with a single peak at 6. Center: The mean of the distribution is about 6. Spread: The values fall mostly between 4 and 8, although there are values as low as 2 and as high as 10. Outliers: There don’t seem to be any outliers. + Example: Choosing Cards Imagine a deck of cards with aces and face cards removed so that only the cards 2 through 10 remain, shuffle the deck, randomly select 5 cards, and note the median value of the cards. For example, if the selected cards were 2, 2, 4, 5, and 9, the median would be 4. Record the value of the sample median on a dotplot going from 2 to 10. We used Fathom to simulate choosing 500 SRSs of size 5 from the deck of cards described above. The graph below shows the distribution of the sample median for these 500 samples. (c) Suppose that another student prepared a different deck of cards and claimed that it was exactly the same as the one used in the activity. However, when you took an SRS of size 5, the median was 4. Does this provide convincing evidence that the student’s deck is different? (c) Getting a sample median of 4 does not provide convincing evidence that the student’s deck is different. Getting a sample median of 4 or lower occurs fairly often just by chance when taking random samples of size 5 from a deck of cards with the aces and face cards removed. Center: Biased and unbiased estimators In the penny example, we collected many samples of size 20 and calculated the sample proportion of 21st century pennies. How well does the sample proportion estimate the true proportion of 21st century pennies, p = 0.47? Note that the center of the approximate sampling distribution is close to 0.47. In fact, if we took ALL possible samples of size 20 and found the mean of those sample proportions, we’d get exactly 0.47. Definition: A statistic used to estimate a parameter is an unbiased estimator if the mean of its sampling distribution is equal to the true value of the parameter being estimated. What Is a Sampling Distribution? The fact that statistics from random samples have definite sampling distributions allows us to answer the question, “How trustworthy is a statistic as an estimator of the parameter?” To get a complete answer, we consider the center, spread, and shape. + Describing Sampling Distributions + Describing Sampling Distributions To get a trustworthy estimate of an unknown population parameter, start by using a statistic that’s an unbiased estimator. This ensures that you won’t tend to overestimate or underestimate. Unfortunately, using an unbiased estimator doesn’t guarantee that the value of your statistic will be close to the actual parameter value. n=100 n=1000 Larger samples have a clear advantage over smaller samples. They are much more likely to produce an estimate close to the true value of the parameter. What Is a Sampling Distribution? Spread: Low variability is better! Variability of a Statistic The variability of a statistic is described by the spread of its sampling distribution. This spread is determined primarily by the size of the random sample. Larger samples give smaller spread. The spread of the sampling distribution does not depend on the size of the population, as long as the population is at least 10 times larger than the sample. We can think of the true value of the population parameter as the bull’s- eye on a target and of the sample statistic as an arrow fired at the target. Both bias and variability describe what happens when we take many shots at the target. Bias means that our aim is off and we consistently miss the bull’s-eye in the same . Our sample values do not center on the population value. High variability means that repeated shots are widely scattered on the target. Repeated samples do not give very similar results. The lesson about center and spread is clear: given a choice of statistics to estimate an unknown parameter, choose one with no or low bias and minimum variability. What Is a Sampling Distribution? Bias, variability, and shape + Describing Sampling Distributions Sampling distributions can take on many shapes. The same statistic can have sampling distributions with different shapes depending on the population distribution and the sample size. Be sure to consider the shape of the sampling distribution before doing inference. Sampling distributions for different statistics used to estimate the number of tanks in the German Tank problem. The blue line represents the true number of tanks. Note the different shapes. Which statistic gives the best estimator? Why? What Is a Sampling Distribution? Bias, variability, and shape + Describing Sampling Distributions + Section 7.1 What Is a Sampling Distribution? Summary In this section, we learned that… A parameter is a number that describes a population. To estimate an unknown parameter, use a statistic calculated from a sample. The population distribution of a variable describes the values of the variable for all individuals in a population. The sampling distribution of a statistic describes the values of the statistic in all possible samples of the same size from the same population. A statistic can be an unbiased estimator or a biased estimator of a parameter. Bias means that the center (mean) of the sampling distribution is not equal to the true value of the parameter. The variability of a statistic is described by the spread of its sampling distribution. Larger samples give smaller spread. When trying to estimate a parameter, choose a statistic with low or no bias and minimum variability. Don’t forget to consider the shape of the sampling distribution before doing inference. + Chapter 7 Sampling Distributions 7.1 What is a Sampling Distribution? 7.2 Sample Proportions 7.3 Sample Means + Section 7.2 Sample Proportions Learning Objectives After this section, you should be able to… FIND the mean and standard deviation of the sampling distribution of a sample proportion DETERMINE whether or not it is appropriate to use the Normal approximation to calculate probabilities involving the sample proportion CALCULATE probabilities involving the sample proportion EVALUATE a claim about a population proportion using the sampling distribution of the sample proportion Sampling Distribution of pˆ Consider the approximate sampling distributions generated by a simulation in which SRSs of Reese’s Pieces are drawn from a population whose proportion of orange candies is either 0.45 or 0.15. What do you notice about the shape, center, and spread of each? Sample Proportions How good is the statistic pˆ as an estimate of the parameter p? The sampling distribution of pˆ answers this question. + The Sampling Distribution of pˆ Shape : In some cases, the sampling distribution of pˆ can be approximated by a Normal curve. This seems to depend on both the sample size n and the population proportion p. Center : The mean of the distribution is m pˆ = p. This makes sense because the sample proportion pˆ is an unbiased estimator of p. Spread : For a specific value of p , the standard deviation s pˆ gets smaller as n gets larger. The value of s pˆ depends on both n and p. pˆ = count of successes in sample X = size of sample n Sample Proportions What did you notice about the shape, center, and spread of each sampling distribution? + The Sampling Distribution of pˆ mX = np sX = np(1 - p) Since pˆ = X /n = (1/n) × X, we are just multiplying the random variable X by a constant (1/n) to get the random variable pˆ . Therefore, 1 m pˆ = (np) = p n 1 s pˆ = np(1- p) = n pˆ is an unbiased estimator of p np(1- p) = 2 n p(1- p) n As sample size increases, the spread decreases. Sample Proportions In Chapter 6, we learned that the mean and standard deviation of a binomial random variable X are + The Sampling Distribution of pˆ Sampling Distribution of a Sample Proportion Choose an SRS of size n from a population of size N with proportion p of successes. Let pˆ be the sample proportion of successes. Then: The mean of the sampling distribution of pˆ is m pˆ = p The standard deviation of the sampling distribution of pˆ is p(1- p) s pˆ = n as long as the 10% condition is satisfied : n £ (1/10)N. As n increases, the sampling distribution becomes approximately Normal. Before you perform Normal calculations, check that the Normal condition is satisfied: np ≥ 10 and n(1 – p) ≥ 10. Sample Proportions We can summarize the facts about the sampling distribution of pˆ as follows : + The A polling organization asks an SRS of 1500 first-year college students how far away their home is. Suppose that 35% of all first-year students actually attend college within 50 miles of home. What is the probability that the random sample of 1500 students will give a result within 2 percentage points of this true value? STATE: We want to find the probability that the sample proportion falls between 0.33 and 0.37 (within 2 percentage points, or 0.02, of 0.35). + Sample Proportions ˆ Using the Normal Approximation for p Inference about a population proportion p is based on the sampling distribution of pˆ . When the sample size is large enough for np and n(1- p) to both be at least 10 (the Normal condition), the sampling distribution of pˆ is approximately Normal. PLAN: We have an SRS of size n = 1500 drawn from a population in which the proportion p = 0.35 attend college within 50 miles of home. m pˆ = 0.35 s pˆ = (0.35)(0.65) = 0.0123 1500 DO: Since np = 1500(0.35) = 525 and n(1 – p) = 1500(0.65)=975 are both greater than 10, we’ll standardize and then use Table A to find the desired probability. z= 0.33 - 0.35 = -1.63 0.0123 z= 0.37 - 0.35 = 1.63 0.0123 P(0.33 £ pˆ £ 0.37) = P(-1.63 £ Z £1.63) = 0.9484 - 0.0516 = 0.8968 CONCLUDE: About 90% of all SRSs of size 1500 will give a result within 2 percentage points of the truth about the population. State: We want to find the probability that the proportion of middle school students who plan to attend a four-year college or university falls between ˆ ≤ 0.87). 73% and 87%. That is, P(0.73 ≤ p Sample Proportions The superintendent of a large school district wants to know what proportion of middle school students in her district are planning to attend a four-year college or university. Suppose that 80% of all middle school students in her district are planning to attend a four-year college or university. What is the probability that a SRS of size 125 will give a result within 7 percentage points of the true value? + Planning for College Plan: To calculate standard deviation of “p-hat”, we must satisfy the n ≤ (1/10)N rule. Because the school district is large, we can assume that there are more than 10(125) = 1250 middle school students so: 0.73 - 0.80 within Conclude: of (0.80)(0.20) all SRSs of size 125 will give a samplez proportion m pˆ = 0.80 About 95% = = -1.94 s pˆ = = 0.036 0.036 7 percentage points of the true proportion of middle school students who are planning 125 to attend a four-year college or university. We can consider the distribution of to be approximately 0.87 - 0.80 z= = 1.94 Normal since np = 125(0.80) = 100 ≥ 10 and 0.036 n(1 – p) = 125(0.20) = 25 ≥ 10. P(0.73 £ pˆ £ 0.87) = P(-1.94 £ Z £ 1.94) = 0.9738 - 0.0262 = 0.9476 + Assignment: Pg. 430 21 – 25 Pg. 439 28, 29, 33, 35, 37, 41 + Section 7.2 Sample Proportions Summary In this section, we learned that… When we want information about the population proportion p of successes, we often take an SRS and use the sample proportion pˆ to estimate the unknown parameter p. The sampling distribution of pˆ describes how the statistic varies in all possible samples from the population. The mean of the sampling distribution of pˆ is equal to the population proportion p. That is, pˆ is an unbiased estimator of p. p(1- p) The standard deviation of the sampling distribution of pˆ is s pˆ = for n an SRS of size n. This formula can be used if the population is at least 10 times as large as the sample (the 10% condition). The standard deviation of pˆ gets smaller as the sample size n gets larger. When the sample size n is larger, the sampling distribution of pˆ is close to a p(1- p) Normal distribution with mean p and standard deviation s pˆ = . n In practice, use this Normal approximation when both np ≥ 10 and n(1 - p) ≥ 10 (the Normal condition). + Chapter 7 Sampling Distributions 7.1 What is a Sampling Distribution? 7.2 Sample Proportions 7.3 Sample Means + Section 7.3 Sample Means Learning Objectives After this section, you should be able to… FIND the mean and standard deviation of the sampling distribution of a sample mean CALCULATE probabilities involving a sample mean when the population distribution is Normal EXPLAIN how the shape of the sampling distribution of sample means is related to the shape of the population distribution APPLY the central limit theorem to help find probabilities involving a sample mean Means Consider the mean household earnings for samples of size 100. Compare the population distribution on the left with the sampling distribution on the right. What do you notice about the shape, center, and spread of each? Sample Means Sample proportions arise most often when we are interested in categorical variables. When we record quantitative variables we are interested in other statistics such as the median or mean or standard deviation of the variable. Sample means are among the most common statistics. + Sample Sampling Distribution of x Mean and Standard Deviation of the Sampling Distribution of Sample Means Suppose that x is the mean of an SRS of size n drawn from a large population with mean m and standard deviation s . Then : The mean of the sampling distribution of x is mx = m The standard deviation of the sampling distribution of x is sx = s n as long as the 10% condition is satisfied: n ≤ (1/10)N. Note : These facts about the mean and standard deviation of x are true no matter what shape the population distribution has. Sample Means When we choose many SRSs from a population, the sampling distribution of the sample mean is centered at the population mean µ and is less spread out than the population distribution. Here are the facts. + The from a Normal Population In one important case, there is a simple relationship between the two distributions. If the population distribution is Normal, then so is the sampling distribution of x. This is true no matter what the sample size is. Sampling Distribution of a Sample Mean from a Normal Population Suppose that a population is Normally distributed with mean m and standard deviation s . Then the sampling distribution of x has the Normal distribution with mean m and standard deviation s / n, provided that the 10% condition is met. Sampling Applet Sample Means We have described the mean and standard deviation of the sampling distribution of the sample mean x but not its shape. That's because the shape of the distribution of x depends on the shape of the population distribution. + Sampling + Movie going students Suppose that the number of movies viewed in the last year by high school students has an average of 19.3 with a standard deviation of 15.8. Suppose we take an SRS of 100 high school students and calculate the mean number of movies viewed by the members of the sample. Problem: (a)What is the mean of the sampling distribution of x? (b) What is the standard deviation of the sampling distribution of Check whether the 10% condition is satisfied. s 15.8 sX = = = 1.58 n 100 x? 100 is less than 10% of the population of high school student. Example: Young Women’s Heights Find the probability that a randomly selected young woman is taller than 66.5 inches. Sample Means The height of young women follows a Normal distribution with mean µ = 64.5 inches and standard deviation σ = 2.5 inches. Let X = the height of a randomly selected young woman. X is N(64.5, 2.5) 66.5 - 64.5 P(X > 66.5) = P(Z > 0.80) =1- 0.7881 = 0.2119 z= = 0.80 2.5 The probability of choosing a young woman at random whose height exceeds 66.5 inches is about 0.21. Find the probability that the mean height of an SRS of 10 young women exceeds 66.5 inches. For an SRS of 10 young women, the sampling distribution of their sample mean height will have a mean and standard deviation s 2.5 mx = m = 64.5 sx = = = 0.79 n 10 Since the population distribution is Normal, the sampling distribution will follow an N(64.5, 0.79) distribution. P(x > 66.5) = P(Z > 2.53) 66.5 - 64.5 z= = 2.53 = 1- 0.9943 = 0.0057 0.79 It is very unlikely (less than a 1% chance) that we would choose an SRS of 10 young women whose average height exceeds 66.5 inches. Central Limit Theorem It is a remarkable fact that as the sample size increases, the distribution of sample means changes its shape: it looks less like that of the population and more like a Normal distribution! When the sample is large enough, the distribution of sample means is very close to Normal, no matter what shape the population distribution has, as long as the population has a finite standard deviation. Definition: Draw an SRS of size n from any population with mean m and finite standard deviation s . The central limit theorem (CLT) says that when n is large, the sampling distribution of the sample mean x is approximately Normal. Note: How large a sample size n is needed for the sampling distribution to be close to Normal depends on the shape of the population distribution. More observations are required if the population distribution is far from Normal. Sample Means Most population distributions are not Normal. What is the shape of the sampling distribution of sample means when the population distribution isn’t Normal? + The Central Limit Theorem Describe the shape of the sampling distributions as n increases. What do you notice? Sample Means Consider the strange population distribution from the Rice University sampling distribution applet. + The Normal Condition for Sample Means If the population distribution is Normal, then so is the sampling distribution of x. This is true no matter what the sample size n is. If the population distribution is not Normal, the central limit theorem tells us that the sampling distribution of x will be approximately Normal in most cases if n ³ 30. Example: Servicing Air Conditioners Your company will service an SRS of 70 air conditioners. You have budgeted 1.1 hours per unit. Will this be enough? Sample Means Based on service records from the past year, the time (in hours) that a technician requires to complete preventative maintenance on an air conditioner follows the distribution that is strongly right-skewed, and whose most likely outcomes are close to 0. The mean time is µ = 1 hour and the standard deviation is σ = 1 Since the 10% condition is met (there are more than 10(70)=700 air conditioners in the population), the sampling distribution of the mean time spent working on the 70 units has s 1 sx = = = 0.12 mx = m =1 n 70 The sampling distribution of the mean time spent working is approximately N(1, 0.12) since n = 70 ≥ 30. We need to find P(mean time > 1.1 hours) z= 1.1 -1 = 0.83 0.12 P(x > 1.1) = P(Z > 0.83) = 1- 0.7967 = 0.2033 If you budget 1.1 hours per unit, there is a 20% chance the technicians will not complete the work within the budgeted time. Buy Me Some Peanuts and Sample Means Problem: At the P. Nutty Peanut Company, dry roasted, shelled peanuts are placed in jars by a machine. The distribution of weights in the bottles is approximately Normal, with a mean of 16.1 ounces and a standard deviation of 0.15 ounces. (a)Without doing any calculations, explain which outcome is more likely, randomly selecting a single jar and finding the contents to weigh less than 16 ounces or randomly selecting 10 jars and finding the average contents to weigh less than 16 ounces. Averages are less variable than individual measurements, thus, it is more likely that a single jar would weigh less than 16 ounces than the average of 10 jars to be less than 16 ounces. (b) Find the probability of each event described above. Let X = weight of the contents of a randomly selected jar of peanuts. X is N(16.1, 0.15). P(X < 16) OR z= 16 -16.1 = -.67 0.15 P(X<16)=0.2514 normalcdf(–100, 16, 16.1, 0.15) = 0.2525 (b) Find the probability of each event described above. Let X = weight of the contents of a randomly selected jar of peanuts. X is N(16.1, 0.15). P(X < 16) OR z= 16 -16.1 = -.67 0.15 P(X<16)=0.2514 normalcdf(–100, 16, 16.1, 0.15) = 0.2525 Let X = average weight of the contents of a random sample of 10 jars. X is N(16.1, 0.15 ). Find P( X < 16) 10 z= 16 -16.1 = -2.13 0.047 P(z < 16) = 0.0166 OR normalcdf(–100, 16, 16.1, 0.047) = 0.0166 Therefore, there is a better chance of choosing single jar containing less than 16 oz. then there is of choosing an SRS of 10 jars with a mean less than 16 oz. Mean Texts Suppose that the number of texts sent during a typical day by a randomly selected high school student follows a right-skewed distribution with a mean of 15 and a standard deviation of 35. Assuming that students at your school are typical texters, how likely is it that a random sample of 50 students will have sent more than a total of 1000 texts in the last 24 hours? State: What is the probability that the total number of texts in the last 24 hours is greater than 1000 for a random sample of 50 high school students? Plan: A total of 1000 texts among 50 students is the same as an average number of texts of 1000/50 = 20 We want to find P( X > 20), where n is large (50 > 30), X = sample mean number of texts. X is approximately N(15, 35/√50 ). Do: P( X > 20) normalcdf (20, 9999,15, 4.95) = 0.1562. Conclude: There is about a 16% chance that a random sample of 50 high school students will send more than 1000 texts in a day. + Section 7.3 Sample Means Summary In this section, we learned that… When we want information about the population mean m for some variable, we often take an SRS and use the sample mean x to estimate the unknown parameter m. The sampling distribution of x describes how the statistic varies in all possible samples of the same size from the population. The mean of the sampling distribution is m, so that x is an unbiased estimator of m. The standard deviation of the sampling distribution of x is s / n for an SRS of size n if the population has standard deviation s . This formula can be used if the population is at least 10 times as large as the sample (10% condition). + Section 7.3 Sample Means Summary In this section, we learned that… Choose an SRS of size n from a population with mean m and standard deviation s . If the population is Normal, then so is the sampling distribution of the sample mean x. If the population distribtution is not Normal, the central limit theorem (CLT) states that when n is large, the sampling distribution of x is approximately Normal. We can use a Normal distribution to calculate approximate probabilities for events involving x whenever the Normal condition is met : If the population distribution is Normal, so is the sampling distribution of x . If n ³ 30, the CLT tells us that the sampling distribution of x will be approximately Normal in most cases.