5.3 The Central Limit Theorem

5.3 The Central Limit Theorem
LEARNING GOAL Understand the basic idea behind the Central Limit Theorem and its important role in statistics. Page 215 Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc.
Suppose we roll one die 1,000 times and record the outcome of each roll, which can be the number 1, 2, 3, 4, 5, or 6. Figure 5.23 shows a histogram of outcomes. All six outcomes have roughly the same relative frequency, because the die is equally likely to land in each of the six possible ways. That is, the histogram shows a (nearly) uniform distribution (see Section 4.2). It turns out that the distribution in Figure 5.23 has a mean of 3.41 and a standard deviation of 1.73. Page 215 Figure 5.23 Frequency and relative frequency distribution of outcomes from rolling one die 1,000 times. Copyright © 2014 Pearson Education, Inc. Slide

Now suppose we roll two dice 1,000 times and record the mean of the two numbers that appear on each roll. To find the mean for a single roll, we add the two numbers and divide by 2. Figure 5.25a shows a typical result. The most common values in this distribution are the central values 3.0, 3.5, and 4.0. These values are common because they can occur in several ways. The mean and standard deviation for this distribution are 3.43 and 1.21, respectively. Pages Figure 5.25a Frequency and relative frequency distribution of sample means from rolling two dice 1,000 times. Copyright © 2014 Pearson Education, Inc. Slide

Suppose we roll five dice 1,000 times and record the mean of the five numbers on each roll. A histogram for this experiment is shown in Figure 5.25b. Once again we see that the central values around 3.5 occur most frequently, but the spread of the distribution is narrower than in the two previous cases. The mean and standard deviation are 3.46 and 0.74, respectively. Page 216 Figure 5.25b Frequency and relative frequency distribution of sample means from rolling five dice 1,000 times. Copyright © 2014 Pearson Education, Inc. Slide

If we further increase the number of dice to ten on each of 1,000 rolls, we find the histogram in Figure 5.25c, which is even narrower. In this case, the mean is 3.49 and standard deviation is 0.56. Page 216 Figure 5.25c Frequency and relative frequency distribution of sample means from rolling ten dice 1,000 times. Copyright © 2014 Pearson Education, Inc. Slide

Table 5.2 shows that as the sample size increases, the mean of the distribution of means approaches the value 3.5 and the standard deviation becomes smaller (making the distribution narrower). Page 216 More important, the distribution looks more and more like a normal distribution as the sample size increases. Copyright © 2014 Pearson Education, Inc. Slide

The Central Limit Theorem Suppose we take many random samples of size n for a variable with any distribution (not necessarily a normal distribution) and record the distribution of the means of each sample. Then, The distribution of means will be approximately a normal distribution for large sample sizes. 2. The mean of the distribution of means approaches the population mean, m, for large sample sizes. 3. The standard deviation of the distribution of means approaches for large sample sizes, where s is the standard deviation of the population. Page 217 σ/ n Copyright © 2014 Pearson Education, Inc. Slide

Be sure to note the very important adjustment, described by item 3 above, that must be made when working with samples or groups instead of individuals: The standard deviation of the distribution of sample means is not the standard deviation of the population, s, but rather , where n is the size of the samples. s/ n Page 217 Copyright © 2014 Pearson Education, Inc. Slide

TECHNICAL NOTE (1) For practical purposes, the distribution of means will be nearly normal if the sample size is larger than 30. (2) If the original population is normally distributed, then the sample means will be normally distributed for any sample size n. (3) In the ideal case, where the distribution of means is formed from all possible samples, the mean of the distribution of means equals μ and the standard deviation of the distribution of means equals Page 217 σ/ n Copyright © 2014 Pearson Education, Inc. Slide

TIME OUT TO THINK Confirm that the standard deviations of the distributions of means given in Table 5.2 (slide 6) for n = 2, 5, 10 agree with the prediction of the Central Limit Theorem, given that σ = 1.73 (the population standard deviation found in Figure 5.23). For example, with n = 2, = 1.22 ≈ 1.21. σ/ n Page Take note of the summarizing paragraph at the bottom of the page. Copyright © 2014 Pearson Education, Inc. Slide

Page 218 Figure 5.26 As the sample size increases (n = 5, 10, 30), the distribution of sample means approaches a normal distribution, regardless of the shape of the original distribution. The larger the sample size, the smaller is the standard deviation of the distribution of sample means. Copyright © 2014 Pearson Education, Inc. Slide

EXAMPLE 1 Predicting Test Scores You are a middle school principal and your 100 eighth-graders are about to take a national standardized test. The test is designed so that the mean score is m = 400 with a standard deviation of s = 70. Assume the scores are normally distributed. a. What is the likelihood that one of your eighth-graders, selected at random, will score below 375 on the exam? Solution: In dealing with an individual score, we use the method of standard scores discussed in Section 5.2. Given the mean of 400 and standard deviation of 70, a score of 375 has a standard score of z = = = Pages data value – mean standard deviation 375 – 400 70 Copyright © 2014 Pearson Education, Inc. Slide

EXAMPLE 1 Predicting Test Scores Solution: (cont.) According to Table 5.1, a standard score of corresponds to about the 36th percentile— that is, 36% of all students can be expected to score below 375. Thus, there is about a 0.36 chance that a randomly selected student will score below 375. Notice that we need to know that the scores have a normal distribution in order to make this calculation, because the table of standard scores applies only to normal distributions. Pages Table 5.1 is on page 211. Copyright © 2014 Pearson Education, Inc. Slide

EXAMPLE 1 Predicting Test Scores You are a middle school principal and your 100 eighth-graders are about to take a national standardized test. The test is designed so that the mean score is m = 400 with a standard deviation of s = 70. Assume the scores are normally distributed. b. Your performance as a principal depends on how well your entire group of eighth-graders scores on the exam. What is the likelihood that your group of 100 eighth-graders will have a mean score below 375? Solution: b. The question about the mean of a group of students must be handled with the Central Limit Theorem. According to this theorem, if we take random samples of size n = 100 students and compute the mean test score of each group, the distribution of means is approximately normal. Pages Copyright © 2014 Pearson Education, Inc. Slide

EXAMPLE 1 Predicting Test Scores Solution: (cont.) Moreover, the mean of this distribution is m = 400 and its standard deviation is = 70/ 100 = 7. With these values for the mean and standard deviation, the standard score for a mean test score of 375 is data value – mean standard deviation 375 – 400 7 z = = = Table 5.1 shows that a standard score of -3.5 corresponds to the 0.02th percentile, and the standard score in this case is even lower. In other words, fewer than 0.02% of all random samples of 100 students will have a mean score of less than 375. Pages Table 5.1 is on page 211. Copyright © 2014 Pearson Education, Inc. Slide

EXAMPLE 1 Predicting Test Scores Solution: (cont.) Therefore, the chance that a randomly selected group of 100 students will have a mean score below 375 is less than , or about 1 in 5,000. Notice that this calculation regarding the group mean did not depend on the individual scores’ having a normal distribution. This example has an important lesson. The likelihood of an individual scoring below 375 is more than 1 in 3 (36%), but the likelihood of a group of 100 students having a mean score below 375 is less than 1 in 5,000 (0.02%). In other words, there is much more variation in the scores of individuals than in the means of groups of individuals. Pages Copyright © 2014 Pearson Education, Inc. Slide

The Value of the Central Limit Theorem The Central Limit Theorem allows us to say something about the mean of a group if we know the mean, m, and the standard deviation, s, of the entire population. This can be useful, but it turns out that the opposite application is far more important. Two major activities of statistics are making estimates of population means and testing claims about population means. Is it possible to make a good estimate of the population mean knowing only the mean of a much smaller sample? As you can probably guess, being able to answer this type of question lies at the heart of statistical sampling, especially in polls and surveys. The Central Limit Theorem provides the key to answering such questions. Page 220 Copyright © 2014 Pearson Education, Inc. Slide

5.3 The Central Limit Theorem

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