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Chapter 6 Notes - Chapter 6 CHAPTER 6 NORMAL PROBABILITY...

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Chapter 6 CHAPTER 6: NORMAL PROBABILITY DISTRIBUTIONS Complete this guide as you read. CHAPTER 6 6-2 Standard Normal Distributions 6-3 Normal Distribution Applications 6-4 Sampling Distributions and Estimators 6-5 Central Limit Theorem 6-6 Normal as Approximation to Binomial (next week) 6-7 Assessing Normality (next week) CHAPTER 6-2 STANDARD NORMAL DISTRIBUTIONS UNIFORM PROBABILITY DISTRIBUTIONS Let’s suppose we have the DISCRETE numbers, 1, 2, 3, 4, and 5. What is the probability that 1 is chosen? P(1 is chosen) = 1/5 = 0.2 = 20%. We can also say that P(4 is chosen) = 1/5. So, the probability that any one number is chosen is 1/5. Hence, we can make the probability distribution TABLE to the left. This UNIFORM distribution gives rise to the following HISTOGRAM: P(x) 0 1 2 3 4 5 To be more accurate, we will divide histogram into equal-sized segments, one for each possible choice. The area from 0-1 represents NUMBER 1; and the area from 1- 2 represents NUMBER 2, the area from 2 to 3 represents NUMBER 3 and so on…. P(x) 1/5 0 1 2 3 4 5 This histogram can also be used to demonstrate CONTINUITY. Consider numbers such as 0.001, 2.3, 3.666667 and 4.5. Can you fit these numbers into the histogram? Just point to, or put a dot, where the decimal numbers might be located. x P( x ) 1 1/5 2 1/5 3 1/5 4 1/5 5 1/5 1
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Chapter 6 We can use this histogram to also demonstrate the probability of selecting specific numbers or specific combinations of numbers. For example, the probability of choosing a 2 is 1/5. This is easy to see in the histogram below. P(2) 1/5 0 1 2 3 4 5 By looking at the histogram below, we know the probability of selecting 2 or 4 is P(2 or 4) = 2/5 = 0.4 or 40%. We can simply count the purple blocks to see the calculated probability is 2 out of 5. Conversely the probability of NOT selecting a 2 or 4 is 3/5 and we can count the clear blocks (1/5 + 1/5 + 1/5 = 3/5) to confirm this . P(2 or 4) 1/5 Now it’s your turn Class Notes Problems using the histograms below. 1 What is the probability of selecting 1, 3 or 5? 2 What is the probability of NOT selecting 1,3 or 5? 3 What is the probability of selecting 3 or 4? 4 What is the probability of selecting 3? 5 What is the probability of NOT selecting 3? P(1 or 3 or 5) 1/5 0 1 2 3 4 5 P(3 or 4) 1/5 0 1 2 3 4 5 P(3) 1/5 0 1 2 3 4 5 2
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Chapter 6 CONTINUOUS PROBABILITY DISTRIBUTIONS Let’s talk about the probability of choosing a number that is between 0 and 1. We know that P(1) = 1/5 and when we look at the area below one, we can also say that P(0 < x < 1) = 1/5 = 0.2 = 20%. The probability of choosing a number between 0 and 1 can be therefore written as: P(0 < x < 1) = 0.2 P(0 < x < 1) 0.2 0 1 2 3 4 5 It makes sense to say total probability that we can pick any number and that number is between 0 and 5 is 1. Or we can say P (0 < x < 5) = 1 since the numbers being considered are all between 0 and 5! NOTE: The sum total area of all the squares with in the rectangle is also equal to one!!!
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