23.3. Sampling Distributions. Engage Sampling Distributions. Learning Objective. Math Processes and Practices. Language Objective

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1 23.3 Sampling Distributions Essential Question: How is the mean of a sampling distribution related to the corresponding population mean or population proportion? Explore 1 Developing a Distribution of Means Resource Locker The tables provide the following data about the first 50 people to join a new gym: member ID number, age, and sex. ID Age Sex ID Age Sex ID Age Sex ID Age Sex ID Age Sex 1 30 M F F M M 2 48 M M M F F 3 52 M F F M F 4 25 F M M M M 5 63 F F M M F 6 50 F M F F F 7 18 F F F F F 8 28 F F M F M 9 72 M M F F F F F M F F Enter the age data into a graphing calculator and find the mean age μ and standard deviation σ for the population of the gym s first 50 members. Round each statistic to the nearest tenth. μ 41.2; σ 15.3 Use a graphing calculator s random number generator to choose a sample of 5 gym members. Find the mean age _ x for your sample. Round to the nearest tenth. Possible answer: x = 39.5 Sampling Distributions Learning Objective Students will use the sampling distribution of sample means and the sampling distribution of sample proportions to solve problems. Math Processes and Practices MPP6 Using Precise Mathematical Language Language Objective Work with a partner to compare and contrast the standard error of the mean and the standard error of the proportion. Online Resources Engage LESSON 23.3 An extra example for each Explain section is available online. Essential Question: How is the mean of a sampling distribution related to the corresponding population mean or population proportion? The mean of the sampling distribution of the sample mean is equal to the population mean. Similarly, the mean of the sampling distribution of the sample proportion is equal to the population proportion. Module Lesson 3 Preview: Lesson Performance Task Professional Development Learning Progressions In previous lessons, students constructed probability distributions, and they explored normal distributions in detail. In this lesson, students will construct distributions of data obtained from different samples of the same population. Students should understand that the mean and standard deviation for a sample usually will not match the statistics for the entire population or for a different sample of the same population. Students will learn how to use population statistics to determine the likelihood that a sample will have certain characteristics. In the next lesson they will build on this knowledge to make predictions about population parameters from sample statistics. View the Engage section online. Discuss the photo and note that the U.S. Census Bureau gathers data on the number of people in each household and the number of households that have each number of people, and that sampling could be used to find relevant probabilities. Then preview the Lesson Performance Task. Lesson

2 Explore 1 Developing a Distribution of Means Integrate Technology Students will use graphing calculators to first calculate the mean and standard deviation of a population, then randomly select samples from that population, and finally calculate statistics for the distribution of sample means. This simulation will enable them to see the effect of sample size on how closely the sample means approximate the population mean. How can you change the way a sample is selected to make the sample mean better match the population mean? Increase the sample size. As the sample size increases, the sample mean approaches the population mean. Explore 2 Developing a Distribution of Proportions What is the difference between this sampling distribution and the one in the first Explore activity? The first one was a distribution of sample means, while this is a distribution of sample proportions. Report your sample mean to your teacher. As other students report their sample means, create a class histogram. To do so, shade a square above the appropriate interval as each sample mean is reported. For sample means that lie on an interval boundary, shade a square on the interval to the right. For instance, if the sample mean is 39.5, shade a square on the interval from 39.5 to Make your own copy of the class histogram. Calculate the mean of the sample means, μ x, and the standard deviation of the sample means, σ x. Now use a graphing calculator s random number generator to choose a sample of 15 gym members. Find the mean for your sample. Round to the nearest tenth. Report your sample mean to your teacher. As other students report their sample means, create a class histogram and make your own copy of it. Calculate the mean of the sample means, μ x, and the standard deviation of the sample means, σ x. Reflect 1. The mean of the sample means is close to the population mean. 1. In the class histograms, how does the mean of the sample means compare with the population mean? 2. What happens to the standard deviation of the sample means as the sample size increases? See below. 3. What happens to the shape of the histogram as the sample size increases? The histogram gets closer to the shape of a normal distribution. Explore 2 Developing a Distribution of Proportions Use the tables of gym membership data from Explore 1. This time you will develop a sampling distribution based on a sample proportion rather than a sample mean. Find the proportion p of female gym members in the population. p = 0.6 Use a graphing calculator s random number generator to choose a sample of 5 gym members. Find the proportion p of female gym members for your sample. All sample proportions will be 0, 0.2, 0.4, 0.6, 0.8, or 1. Report your sample proportion to your teacher. As other students report their sample proportions, create a class histogram and make your own copy of it. Calculate the mean of the sample proportions, μ p, and the standard deviation of the sample proportions, σ p. Round to the nearest hundredth. Now use your calculator s random number generator to choose a sample of 10 gym members. Find the proportion of female members p for your sample. All sample proportions will be 0, 0.1, 0.2,, 0.9, or 1. Report your sample proportion to your teacher. As other students report their sample proportions, create a class histogram and make your own copy of it. 2. The standard deviation of the sample means decreases. Module Lesson 3 Integrate Math Processes and Practices Focus on Abstract and Quantitative Reasoning MPP2 When creating a histogram of sample proportions for a sample size of 5, ask students why there are only a few different values for the sample proportion, even when many samples are selected. Students should recognize that the only possible values for the proportion are 0, 0.2, 0.4, 0.6, 0.8 and Sampling Distributions

3 Calculate the mean of the sample proportions, μ p, and the standard deviation of the sample proportions, σ p. Round to the nearest hundredth. Reflect 4. In the class histograms, how does the mean of the sample proportions compare with the population proportion? The mean of the sample proportions is close to the population proportion. 5. What happens to the standard deviation of the sample proportions as the sample size increases? The standard deviation of the sample proportions decreases. Explain 1 Using the Sampling Distribution of the Mean The histograms that you made in the two Explores are sampling distributions. A sampling distribution shows how a particular statistic varies across all samples of n individuals from the same population. In Explore 1, you approximated sampling distributions of the sample mean, _ x, for samples of size 5 and 15. (The reason your sampling distributions are approximations is that you did not find all samples of a given size.) The mean of the sampling distribution of the sample mean is denoted µ _ x. The standard deviation of the sampling distribution of the sample mean is denoted σ _ x and is also called the standard error of the mean. In Explore 1, you may have discovered that µ _ x is close to _ x regardless of the sample size and that σ _ x decreases as the sample size n increases. You based these observations on simulations. When you consider all possible samples of n individuals, you arrive at one of the major theorems of statistics. Properties of the Sampling Distribution of the Mean If a random sample of size n is selected from a population with mean μ and standard deviation σ, then 1. µ _ x = µ, 2. σ _ x = σ_ _ n, and 3. The sampling distribution of the sample mean is normal if the population is normal; for all other populations, the sampling distribution of the mean approaches a normal distribution as n increases. EXPLAIN 1 Using the Sampling Distribution of the Mean If you create two sampling distributions by randomly selecting samples from the same population, with one distribution having samples of size 25 and the other having samples of size 100, how will the standard errors of the means of the two distributions compare? Explain. The standard error of the mean for samples of 100 will be half as large as the standard error for samples of 25. Standard error is inversely proportional to the square root of the sample size, so quadrupling the sample size reduces the standard error by a factor of 2. The third property stated above is known as the Central Limit Theorem. Example 1 Boxes of Cruncho cereal have a mean mass of 323 grams with a standard deviation of 20 grams. For random samples of 36 boxes, what interval centered on the mean of the sampling distribution captures 95% of the sample means? Write the given information about the population and a sample. µ = 323 σ = 20 n = 36 Find the mean of the sampling distribution of the sample mean and the standard error of the mean. µ _ x = µ = 323 σ _ x = σ _ _ n = 20 _ 36 = _ Module Lesson 3 Lesson

4 Integrate Technology Review how to use a graphing calculator to find the probability that a value in a normal distribution is between two specified values. Find the cumulative distribution function by going to the DISTR menu and selecting 2:normalcdf(, and entering the given information in the correct order (smaller value, larger value, mean, standard deviation) to find the probability. To find the probability that a quantity is less than a given value, enter 1 EE 99 for the lower value. To find the probability that a quantity is greater than a given value, enter 1 EE 99 for the upper value. The sampling distribution of the sample mean is approximately normal. In a normal distribution, 95% of the data fall within 2 standard deviations of the mean. µ _ x - 2 σ _ x = (3.3) = µ _ x + 2 σ _ x = (3.3) = So, for random samples of 36 boxes, 95% of the sample means fall between grams and grams. What is the probability that a random sample of 25 boxes has a mean mass of at most 325 grams? Write the given information about the population and the sample. µ = 323 σ = 20 n = 25 Find the mean of the sampling distribution of the sample mean and the standard error of the mean. µ x _ = µ = σ x _ = σ n = 20 _ = _ = The sampling distribution of the sample mean is approximately normal. Use a graphing calculator to find P ( _ x 325). P ( x _ 325) = normalcdf (-1e99, 325, 323, 4 ) 0.69 Avoid Common Errors Remind students that when using a calculator to find the probability that the mean of a sample is between two given values (or above or below a given value), they must first calculate σ _ x, the standard error of the mean, then use that value as the standard deviation that they enter on the calculator. If, instead, they use the standard deviation of the population, they will obtain an incorrect result. Explain 2 Using the Sampling Distribution of the Proportion How are the mean and standard deviation of the sampling distribution of a sample proportion similar to and different from the mean and standard deviation of the sampling distribution of a sample mean? In both cases, the mean of the sampling distribution is equal to the corresponding value (mean or proportion) for the population. However, the standard deviation is calculated somewhat differently for sample means and sample proportions. So, the probability that the random sample has a mean mass of at most 325 grams is about Your Turn Boxes of Cruncho cereal have a mean mass of 323 grams with a standard deviation of 20 grams. 6. For random samples of 50 boxes, what interval centered on the mean of the sampling distribution captures 99.7% of the sample means? between grams and grams 7. What is the probability that a random sample of 100 boxes has a mean mass of at least 320 grams? about 0.93 Explain 2 Using the Sampling Distribution of the Proportion When you work with the sampling distribution of a sample proportion, p represents the proportion of individuals in the population that have a particular characteristic (that is, the proportion of successes ) and p is the proportion of successes in a sample. The mean of the sampling distribution of the sample proportion is denoted µ p. The standard deviation of the sampling distribution of the sample proportion is denoted σ p and is also called the standard error of the proportion. Properties of the Sampling Distribution of the Proportion If a random sample of size n is selected from a population with proportion p of successes, then 1. µ p = p, 2. σ p = p (1 - _ p) n, and 3. if both np and n (1 p) are at least 10, then the sampling distribution of the sample proportion is approximately normal. Module Lesson 3 Collaborative Learning Peer-to-Peer Activity Have students work in pairs. Give each pair the mean and standard deviation for a population. Have each student choose a sample size, determine the standard error of the mean for that sample size, and identify an interval that contains either 68%, 95%, or 99.7% of the sample means. Next, have students tell their partners the sample size they used and the upper and lower bounds of the interval they found. Have each student calculate the percent of sample means that fall within the partner s interval. Have partners check each other s results. 840 Sampling Distributions

5 Example 2 40% of the students at a university live off campus. When sampling from this population, consider successes to be students who live off campus. For random samples of 50 students, what interval centered on the mean of the sampling distribution captures 95% of the sample proportions? Write the given information about the population and a sample. p = 0.4 n = 50 Find the mean of the sampling distribution of the sample proportion and the standard error of the proportion. = 0.4 (1-0.4) μ p = p = 0.4 σ p = p (1 - p) n Check that np and n (1 - p) are both at least 10. np = = 20 n (1 - p) = = Since np and n (1 - p) are both greater than 10, the sampling distribution of the sample proportion is approximately normal. In a normal distribution, 95% of the data fall within 2 standard deviations of the mean. μ p - 2 σ p = (0.069) = μ p + 2 σ p = (0.069) = So, for random samples of 50 students, 95% of the sample proportions fall between 26.2% and 53.8%. What is the probability that a random sample of 25 students has a sample proportion of at most 37%? Write the given information about the population and the sample, where a success is a student who lives off campus. p = 0.4 n = 25 Integrate Math Processes and Practices Focus on Using and Evaluating Logical Reasoning MPP3 Discuss with students the criteria for determining whether a sampling distribution is approximately normal. If the population is normally distributed, the sampling distribution of the sample mean will also be normal. Also, the sampling distribution of a sample mean approaches a normal distribution as the sample size n increases. The distribution is approximately normal when n is very large, but we do not have precise definitions of approximately normal or very large. For the sampling distribution of a sample proportion, how close the distribution is to normal depends on both n and the proportion p. If p is very large or very small, n must be especially large for the distribution to be approximately normal. We can use the rule of thumb that says that if both np and n (1 - p) are at least 10, we can consider the distribution approximately normal. Find the mean of the sampling distribution of the sample proportion and the standard error of the proportion. μ p = p = σ p = p (1 - p) n = 0.4 (1-0.4 ) 0.4 = Check that np and n (1 - p) are both at least 10. np = = 10 n (1 - p) = = 15 Since np and n (1 - p) are greater than or equal to 10, the sampling distribution of the sample proportion is approximately normal. Use a graphing calculator to find P (p 0.37). P (p 0.37) = normalcdf (-1e99, 0.37, 0.4, ) 0.38 So, the probability that the random sample has a sample proportion of at most 37% is about Module Lesson 3 Language Support Communicate Math Have students work with a partner to compare and contrast the standard error of the mean and the standard error of the proportion. Monitor students use of terminology to learn what words and concepts are difficult for them to understand or to communicate. Lesson

6 Elaborate Integrate Math Processes and Practices Focus on Abstract and Quantitative Reasoning MPP2 Ask students to describe different methods for determining the probability that a sample mean or proportion falls between two given values. They should understand that if the given values are multiples of a standard deviation away from the mean of the sampling distribution, they can use the known percentages of data that fall within 0, 1, 2, or 3 standard deviations of the mean. If not, they can calculate z-scores for the given values and look up the corresponding probabilities on a standard normal table. Alternatively, they can use the cumulative distribution function on a graphing calculator. Summarize The Lesson Given the mean and standard deviation of a population, how can you determine the mean and standard deviation of a sampling distribution of sample means? The mean of the sampling distribution of the sample mean is equal to the mean of the population. To find the standard deviation, divide the population standard deviation by the square root of the sample size. Your Turn 40% of the students at a university live off campus. When sampling from this population, consider successes to be students who live off campus. 8. For random samples of 80 students, what interval centered on the mean of the sampling distribution captures 68% of the sample proportions? between 34.5% and 45.5% 9. What is the probability that a random sample of 60 students includes more than 18 students who live off campus? about 0.94 Elaborate 10. A sampling distribution is a distribution that shows how a particular statistic varies across all samples of n individuals 10. What is a sampling distribution? from the same population. 11. What allows you to conclude that 95% of the sample means in a sampling distribution are within 2 standard deviations of the population mean? See margin. 12. When finding a sample mean or a sample proportion, why is using the greatest sample size possible (given constraints on the cost and time of sampling) a desirable thing to do? See margin. 13. Essential Question Check-In When you repeatedly take random samples of the same size from a population, what does the mean of the samples approximate? The mean of the samples approximates the population mean (if the data are numerical) or the population proportion (if the data are categorical). Evaluate: Homework and Practice 1. The general manager of a multiplex theater took random samples of size 10 from the audiences attending the opening weekend of a new movie. From each sample, the manager obtained the mean age and the proportion of those who said they liked the movie. The sample means and sample proportions are listed in the tables. number mean (age) proportion (liked the movie) number mean (age) proportion (liked the movie) Module Lesson 3 Answers 11. The Central Limit Theorem says that the sampling distribution of the sample mean is normal or approximately normal, so 95% of the sample means will fall within 2 standard deviations of the mean of the sampling distribution, but the mean of the sampling distribution is equal to the population mean, so 95% of the sample means will fall within 2 standard deviations of the population mean. 12. Increasing the sample size decreases the variation in the sampling distribution, which in turn means that the sample mean or sample proportion will be more accurate because it is more likely to fall closer to the population mean or the population proportion. 842 Sampling Distributions

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