# Key Concept. February 25, S6.5_3 The Central Limit Theorem. Chapter 6 Normal Probability Distributions. Central Limit Theorem

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1 MAT 155 Statistical Analysis Dr. Claude Moore Cape Fear Community College Chapter 6 Normal Probability Distributions 6 1 Review and Preview 6 2 The Standard Normal Distribution 6 3 Applications of Normal Distributions 6 4 Sampling Distributions and Estimators 6 5 The Central Limit Theorem 6 6 Normal as Approximation to Binomial 6 7 Assessing Normality See the Technology link on the Important Links webpage and click on Normal Distribution (xls) for the Excel program to calculate answers for most of these problems in this section. Use the Chapter Problem. Assume the population of weights of men is normally distributed with a mean of 172 lb and a standard deviation of 29 lb. Find the probability that 20 randomly selected men will have a mean weight that is greater than 175 lb (so that their total weight exceeds the safe capacity of 3500 pounds). Find this program in the Technology section on the Important Links webpage or go directly to Key Concept The Central Limit Theorem tells us that for a population with any distribution, the distribution of the sample means approaches a normal distribution as the sample size increases. The procedures in this section form the foundation for estimating population parameters and hypothesis testing. Given: Central Limit Theorem 1. The random variable x has a distribution (which may or may not be normal) with mean µ and standard deviation s. 2. Simple random samples all of size n are selected from the population. (The samples are selected so that all possible samples of the same size n have the same chance of being selected.) 1

2 Central Limit Theorem Conclusions: 1. The distribution of sample x will, as the sample size increases, approach a normal distribution. 2. The mean of the sample means is the population mean µ. 3. The standard deviation of all sample means is Practical Rules Commonly Used 1. For samples of size n larger than 30, the distribution of the sample means can be approximated reasonably well by a normal distribution. The approximation gets closer to a normal distribution as the sample size n becomes larger. 2. If the original population is normally distributed, then for any sample size n, the sample means will be normally distributed (not just the values of n larger than 30). Notation Example Normal Distribution the mean of the sample means the standard deviation of sample mean As we proceed from n = 1 to n = 50, we see that the distribution of sample means is approaching the shape of a normal distribution. (often called the standard error of the mean) 2

3 Example Uniform Distribution Example U Shaped Distribution As we proceed from n = 1 to n = 50, we see that the distribution of sample means is approaching the shape of a normal distribution. As we proceed from n = 1 to n = 50, we see that the distribution of sample means is approaching the shape of a normal distribution. Important Point As the sample size increases, the sampling distribution of sample means approaches a normal distribution. Example Water Taxi Safety Use the Chapter Problem. Assume the population of weights of men is normally distributed with a mean of 172 lb and a standard deviation of 29 lb. Find the probability that if an individual man is randomly selected, his weight is greater than 175 lb. Find the probability that 20 randomly selected men will have a mean weight that is greater than 175 lb (so that their total weight exceeds the safe capacity of 3500 pounds). 3

4 Example cont a) Find the probability that if an individual man is randomly selected, his weight is greater than 175 lb. Example cont b) Find the probability that 20 randomly selected men will have a mean weight that is greater than 175 lb (so that their total weight exceeds the safe capacity of 3500 pounds). Example cont a) Find the probability that if an individual man is randomly selected, his weight is greater than 175 lb. P(x > 175) = b) Find the probability that 20 randomly selected men will have a mean weight that is greater than 175 lb (so that their total weight exceeds the safe capacity of 3500 pounds). Interpretation of Results Given that the safe capacity of the water taxi is 3500 pounds, there is a fairly good chance (with probability ) that it will be overloaded with 20 randomly selected men. It is much easier for an individual to deviate from the mean than it is for a group of 20 to deviate from the mean. 4

5 Correction for a Finite Population When sampling without replacement and the sample size n is greater than 5% of the finite population of size N (that is, n > 0.05N ), adjust the standard deviation of sample means by multiplying it by the finite population correction factor: Recap In this section we have discussed: Central limit theorem. Practical rules. Effects of sample sizes. Correction for a finite population. Using the Central Limit Theorem. In Exercises 5 8, assume that SAT scores are normally distributed with mean μ = 1518 and standard deviation σ = 325 ( based on data from the College Board). 302/6. a. If 1 SAT score is randomly selected, find the probability that it is greater than b. If 64 SAT scores are randomly selected, find the probability that they have a mean greater than Using the Central Limit Theorem. In Exercises 5 8, assume that SAT scores are normally distributed with mean μ = 1518 and standard deviation σ = 325 ( based on data from the College Board). 302/8. a. If 1 SAT score is randomly selected, find the probability that it is between 1440 and b. If 16 SAT scores are randomly selected, find the probability that they have a mean between 1440 and c. Why can the central limit theorem be used in part (b), even though the sample size does not exceed 30? = normalcdf(1440, 1480, 1518, 325) = (b) P(1440 mean 1480) = normalcdf(1440, 1480, 1518, 325/ 16) = (c) We can use the Central Limit Theorem with sample size 16 < 30 because the original data is normally distributed. See page 294 in textbook: "If the original population is normally distributed, then for any same size n, the sample means will normally distributed." 5

7 304/18. Vending Machines Currently, quarters have weights that are normally distributed with a mean of g and a standard deviation of g. A vending machine is configured to accept only those quarters with weights between g and g. a. If 280 different quarters are inserted into the vending machine, what is the expected number of rejected quarters? b. If 280 different quarters are inserted into the vending machine, what is the probability that the mean falls between the limits of g and g? c. If you own the vending machine, which result would concern you more? The result from part (a) or the result from part (b)? Why? 304/20. Body Temperatures Assume that the population of human body temperatures has a mean of 98.6 F, as is commonly believed. Also assume that the population standard deviation is 0.62 F ( based on data from University of Maryland researchers). If a sample of size n = 106 is randomly selected, find the probability of getting a mean temperature of 98.2 F or lower. ( The value of 98.2 F was actually obtained; see the midnight temperatures for Day 2 in Data Set 2 of Appendix B.) Does that probability suggest that the mean body temperature is not 98.6 F? After further consideration, I changed the "no" to "yes" on this answer. 7

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