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CHAPTER 7 Sampling Distributions 7.3 Sample Means The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers 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. CHECK the 10% condition before calculating the standard deviation of a sample mean. EXPLAIN how the shape of the sampling distribution of a sample mean is affected by the shape of the population distribution and the sample size. If appropriate, use a Normal distribution to CALCULATE probabilities involving sample means. The Practice of Statistics, 5th Edition 2 The Sampling Distribution of x The Practice of Statistics, 5th Edition 3 Activity: Penny for Your Thoughts The Practice of Statistics, 5th Edition 4 Activity: Penny for Your Thoughts 5. Place a sticky for each of the five ages of your pennies on the ages dotplot. 6. Does this distribution of ages surprise you? 7. Now place a sticky for the mean age on the mean age dotplot. 8. What differences do you notice between the dotplots? The Practice of Statistics, 5th Edition 5 Note: 1. The histogram of the ages of the individual pennies is the graph of the population. 2. The histogram of the means of the five ages is the graph of the sampling distribution of the sample mean for samples of size 5. 3. These distributions WERE NOT THE SAME. Sampling distributions are DIFFERENT DISTRIBUTIONS than population distributions. The three facts listed above are the three facts that make up the Central Limit Theorem. But before we discuss that in more detail, let’s look at a what a larger class would have seen if they did the same activity. The Practice of Statistics, 5th Edition 6 Note: Suppose my class had 350 students in it, and each of them did the same activity. So there are 5 x 350 = 1750 pennies. And there are 350 means. We find the same skewed shape for the ages of pennies that we saw in the smaller class. But now there are enough means that we can see more clearly the shape of the distribution of means. The Practice of Statistics, 5th Edition 7 Note: Now we can notice several things about this graph of means. a. It is not nearly as spread out as the distribution of the individual ages. (Look carefully at the scale along the horizontal axis to see this.) – b. c. That means the standard deviation is smaller. The center of it is about the same as the center of the distribution of the individual ages. It is still skewed, but not nearly as skewed as the distribution of the individual ages. With this many observations, it is clear that it looks more normally distributed than the distribution of individual ages. The Practice of Statistics, 5th Edition 8 The Sampling Distribution of x This example illustrates an important fact that we will make precise in this section: averages are less variable than individual observations. The Practice of Statistics, 5th Edition 9 The Sampling Distribution of x 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. Sampling Distribution of a Sample Mean 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. The Practice of Statistics, 5th Edition 10 The Practice of Statistics, 5th Edition 11 Ex: This Wine Stinks The Practice of Statistics, 5th Edition 12 The Practice of Statistics, 5th Edition 13 Sampling From a Normal Population The Practice of Statistics, 5th Edition 14 In this Activity, you’ll use Professor Lane’s applet to explore the shape of the sampling distribution when the population is Normally distributed. 1. There are choices for the population distribution: Normal, uniform, skewed, and custom. The default is Normal. Click the “Animated” button. What happens? Click the button several more times. 2. What do the black boxes represent? What is the blue square that drops down onto the plot below? What does the red horizontal band under the population histogram tell us? 3. Click on “Clear lower 3” to start clean. Then click on the “10,000” button under “Sample:” so you simulate taking 10,000 SRSs of size n = 5 from the population. 4. Does the approximate sampling distribution (blue bars) have a recognizable shape? Click the box next to “Fit normal.” The Practice of Statistics, 5th Edition 15 The Practice of Statistics, 5th Edition 16 Sampling From a Normal Population Sampling Distribution of a Sample Mean from a Normal Population Suppose that a population is Normally distribute d with mean and standard deviation . Then the sampling distributi on of x has the Normal distributi on with mean and standard deviation n , provided that the 10% condition is met. The Practice of Statistics, 5th Edition 17 Ex: Young Women’s Heights PROBLEM: The height of young women follows a Normal distribution with mean μ = 64.5 inches and standard deviation σ = 2.5 inches. a. Find the probability that a randomly selected young woman is taller than 66.5 inches. Show your work. b. Find the probability that the mean height of an SRS of 10 young women exceeds 66.5 inches. Show your work. The Practice of Statistics, 5th Edition 18 On Your Own: The Practice of Statistics, 5th Edition 19 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? Let’s find out with the same Applet. The Practice of Statistics, 5th Edition 20 1. Select “Skewed” population. Set the bottom two graphs to display the mean—one for samples of size 2 and the other for samples of size 5. Click the Animated button a few times to be sure you see what’s happening. Then “Clear lower 3” and take 10,000 SRSs. Describe what you see. 2. Change the sample sizes to n = 10 and n = 16 and repeat Step 1. What do you notice? 3. Now change the sample sizes to n = 20 and n = 25 and take 10,000 more samples. Did this confirm what you saw in Step 2? The Practice of Statistics, 5th Edition 21 The 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. This famous fact of probability theory is called the central limit theorem (sometimes abbreviated as CLT). 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. The Practice of Statistics, 5th Edition 22 CAUTION: The Practice of Statistics, 5th Edition 23 The Central Limit Theorem Consider the strange population distribution from the Rice University sampling distribution applet. Describe the shape of the sampling distributions as n increases. What do you notice? The Practice of Statistics, 5th Edition 24 The Central Limit Theorem As the previous example illustrates, even when the population distribution is very non-Normal, the sampling distribution of the sample mean often looks approximately Normal with sample sizes as small as n = 25. Normal/Large 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. The central limit theorem allows us to use Normal probability calculations to answer questions about sample means from many observations even when the population distribution is not Normal. The Practice of Statistics, 5th Edition 25 Ex: Servicing Air Conditioners The Practice of Statistics, 5th Edition 26 The Sampling Distribution of The Practice of Statistics, 5th Edition x 27 Sample Means Section Summary In this section, we learned how to… FIND the mean and standard deviation of the sampling distribution of a sample mean. CHECK the 10% condition before calculating the standard deviation of a sample mean. EXPLAIN how the shape of the sampling distribution of a sample mean is affected by the shape of the population distribution and the sample size. If appropriate, use a Normal distribution to CALCULATE probabilities involving sample means. The Practice of Statistics, 5th Edition 28