1. + The Practice of Statistics, 4 th edition – For AP* STARNES, YATES, MOORE Chapter 7: Sampling Distributions Section 7.1 What is a Sampling Distribution?

2. + Chapter 7 Sampling Distributions 7.1What is a Sampling Distribution? 7.2Sample Proportions 7.3Sample Means

3. + Section 7.1 What Is a Sampling Distribution? 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 Learning Objectives

4. + What Is a Sampling Distribution? IntroductionThe process of statistical inference involves using information from a sample to draw conclusions about a wider population. 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 samplingprocess. Therefore, we can examine its probability distribution usingwhat we learned in Chapter 6. Population Sample Collect data from a representative Sample... Make an Inference about the Population.

5. + Parameters and StatisticsAs we begin to use sample data to draw conclusions about a wider population, we must be clear about whether a numberdescribes a sample or a population. What Is a Sampling Distribution? 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

6. + Example: Heights and Cell Phones Problem: Identify the population, the parameter, the sample, and the statistic in each of the following settings. (a)A pediatrician wants to know the 75 th percentile for the distribution of heights of 10-year-old boys so she takes a sample of 50 patients and calculates Q 3 = 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 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 75 th percentile, or Q 3. The sample is the 50 10-year-old boys included in the sample; the statistic is the sample Q 3 = 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.

7. + Sampling VariabilityThis 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? ” What Is a Sampling Distribution? Population Sample ? ? µ

8. + Sampling Distribution Sampling Distribution Applet Sampling Distribution Applet In the penny activity, we took several samples of 20 pennies. There are many, many possible SRSs of size 20 from a population of size100. If we took every one of those possible samples, calculated thesample proportion for each, and graphed all of those values, we ’ d have a sampling distribution. What Is a Sampling Distribution? 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.

9. + Population Distributions vs. Sampling DistributionsThere are actually three distinct distributions involved when we sample repeatedly and measure a variable ofinterest. 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 fromthe population. What Is a Sampling Distribution? p = actual proportion of in entire population

10. Activity: Counting Chips Given: 147 colored chips (1/3 red, 1/3 blue, 1/3 white) What Is a Sampling Distribution? The proportion of red chips (parameter p) is 1/3 Choose 12 chips and record the proportion of RED chips you have. Mark using on the graph. Population Distribution:

11. Distribution of Sample (n=12) Sampling Distribution

12. + 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. 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. Example: Choosing Cards 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.

13. + 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. 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. Example: Choosing Cards (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. (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?

14. + Describing Sampling DistributionsThe 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. What Is a Sampling Distribution? 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. Center: Biased and unbiased estimators In the penny example, we collected many samples of size 20 and calculated the sample proportion of 21 st century pennies. How well does the sample proportion estimate the true proportion of 21 st 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.

15. + Describing Sampling Distributions What Is a Sampling Distribution? Spread: Low variability is better! 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. 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. 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. Variability of a Statistic n=100n=1000

16. + Describing Sampling Distributions What Is a Sampling Distribution? Bias, variability, and shape 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. 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.

17. + Describing Sampling Distributions What Is a Sampling Distribution? Bias, variability, and shape 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?

18. + Section 7.1 What Is a Sampling Distribution? 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. Summary

19. + Chapter 7 Sampling Distributions 7.1What is a Sampling Distribution? 7.2Sample Proportions 7.3Sample Means

20. + Section 7.2 Sample Proportions 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 Learning Objectives

21. + Sample Proportions The Sampling Distribution of 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. Reese ’ s Pieces What do you notice about the shape, center, and spread of each?

22. + Sample Proportions The Sampling Distribution of What did you notice about the shape, center, and spread of each sampling distribution?

23. + Sample Proportions The Sampling Distribution of In Chapter 6, we learned that the mean and standard deviation of a binomial random variable X are As sample size increases, the spread decreases.

24. + The Sampling Distribution of 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. Sampling Distribution of a Sample Proportion Sample Proportions

25. + Using the Normal Approximation for Sample Proportions 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). 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. 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. CONCLUDE: About 90% of all SRSs of size 1500 will give a result within 2 percentage points of the truth about the population.

26. + Sample Proportions Planning for College 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? 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 73% and 87%. That is, P(0.73 ≤ ≤ 0.87). 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: We can consider the distribution of to be approximately Normal since np = 125(0.80) = 100 ≥ 10 and n(1 – p) = 125(0.20) = 25 ≥ 10. Conclude: About 95% of all SRSs of size 125 will give a sample proportion within 7 percentage points of the true proportion of middle school students who are planning to attend a four-year college or university.

27. + Assignment: Pg. 430 21 – 25 Pg. 439 28, 29, 33, 35, 37, 41

28. + Section 7.2 Sample Proportions In this section, we learned that… In practice, use this Normal approximation when both np ≥ 10 and n(1 - p) ≥ 10 (the Normal condition). Summary

29. + Chapter 7 Sampling Distributions 7.1What is a Sampling Distribution? 7.2Sample Proportions 7.3Sample Means

30. + Section 7.3 Sample Means 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 Learning Objectives

31. + 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. Consider the mean household earnings for samples of size 100. Compare the population distribution on the left with the samplingdistribution on the right. What do you notice about the shape, center,and spread of each?

32. + The Sampling Distribution of 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. as long as the 10% condition is satisfied: n ≤ (1/10)N. Mean and Standard Deviation of the Sampling Distribution of Sample Means Sample Means

33. + Sampling from a Normal Population Sampling Distribution of a Sample Mean from a Normal Population Sampling Applet

34. + 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 ? (b) What is the standard deviation of the sampling distribution of ? Check whether the 10% condition is satisfied. 100 is less than 10% of the population of high school student.

35. Example: Young Women’s Heights The height of young women follows a Normal distribution with mean µ = 64.5 inches and standard deviation σ = 2.5 inches. Find the probability that a randomly selected young woman is taller than 66.5 inches. Let X = the height of a randomly selected young woman. X is N(64.5, 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 Since the population distribution is Normal, the sampling distribution will follow an N(64.5, 0.79) distribution. 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. Sample Means

36. + The Central Limit Theorem Most population distributions are not Normal. What is the shape of the sampling distribution of sample means when the population distribution isn ’ t Normal? 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 populationand more like a Normal distribution! When the sample is large enough, thedistribution 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. Sample Means 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. Definition:

37. + The Central Limit Theorem Consider the strange population distribution from the Rice University sampling distribution applet. Sample Means Describe the shape of the sampling distributions as n increases. What do you notice? Normal Condition for Sample Means

38. Example: Servicing Air Conditioners 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 Your company will service an SRS of 70 air conditioners. You have budgeted 1.1 hours per unit. Will this be enough? 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 Sample Means The sampling distribution of the mean time spent working is approximately N(1, 0.12) since n = 70 ≥ 30. If you budget 1.1 hours per unit, there is a 20% chance the technicians will not complete the work within the budgeted time. We need to find P(mean time > 1.1 hours)

39. 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. (b) Find the probability of each event described above. 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. Let X = weight of the contents of a randomly selected jar of peanuts. X is N(16.1, 0.15). P(X < 16) normalcdf(–100, 16, 16.1, 0.15) = 0.2525 P(X<16)=0.2514 OR

40. (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) normalcdf(–100, 16, 16.1, 0.15) = 0.2525 P(X<16)=0.2514 OR Let = average weight of the contents of a random sample of 10 jars. is N(16.1, ). Find P( < 16) normalcdf(–100, 16, 16.1, 0.047) = 0.0166 OR 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.

41. 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( > 20), where = sample mean number of texts. n is large (50 > 30), is approximately N(15, 35/√50 ). Do: P( > 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.

42. + Section 7.3 Sample Means In this section, we learned that… Summary

43. + Section 7.3 Sample Means In this section, we learned that… Summary