Statistical Methods STAT 302 Chapter 4 Distributions of Random Variables Copyright 2025 by Aburweis All rights reserved No part of this work may be reproduced in any form without written permission Chapter 4 Learning Objectives 1 Describe the properties of a normal distribution 2 Find areas under the standard normal curve 3 Calculate probabilities for a normal distribution 4 Determine z values corresponding to given areas under the standard normal curve 5 Identify a binomial distribution and its characteristics 6 Calculate probabilities using the binomial distribution 7 Use the normal distribution as an approximation to calculate probabilities for a binomial distribution 2 Probability Distributions of Random Variables Normal Distribution Binomial Distribution Geometric Distribution Poisson Distribution 3 Normal Distributions are continuous distributions are bell shaped and continuous approximate the distributions of many different variables have an area under the curve equal to 1 are used in inferential procedures are distinguished from one another by their mean and standard deviation 4 Normal Distribution The normal distribution also known as the bell curve or Gaussian distribution is the most important continuous probability distribution in statistics Denoted as where mean of the distribution and standard deviation We read X as X is normally distributed with mean and standard deviation 5 Properties of the Normal Distribution 1 The mean median mode 2 The normal curve is bell shaped and is symmetric about its mean 3 The total area under the normal distribution curve is equal to 1 4 The normal curve extends infinitely in both directions approaching but never touching the x axis 6 Many Different Normal Distributions There are infinitely many different normal distributions and they depend on the values used for the mean and standard deviation Changing the mean of a normal distribution shifts it left or right while changing the standard deviation spreads the distribution out more We can explore more normal distributions here https www desmos com calculator 474pgdtvjd https en wikipedia org wiki Normal distribution 7 Standardizing with Z Score Since there are an infinite number of normal distributions we need a method to compare values across different distributions This method is called standardization or the Z score transformation A z score is a way to compare values across different distributions with the same shape https www istockphoto com vector apples and oranges gm165043575 1985272 How do we compare apples and oranges 8 Standardizing with Z Score Cont Z scores indicate how many standard deviations a value is away from the mean A negative z score indicates that the observed value x is below the mean while a positive z score indicates that x is above the mean Values that deviate more than two standard deviations from the mean i e Z 2 are considered unusual indicating a low probability of their occurrence 9 Example1 SAT and ACT Distributions SAT scores are distributed nearly normally with a mean of 1500 and a standard deviation of 300 ACT scores are distributed nearly normally with a mean of 21 and a standard deviation of 5 A college admissions officer wants to determine which of the two applicants scored better on their standardized test with respect to the other test takers Pam who earned 1800 on her SAT or Jim who scored a 24 on his ACT 10 Example1 SAT and ACT Distributions Solution Since we cannot directly compare these two raw scores we instead compare how many standard deviations each observation is beyond the mean using the z score Pam scored 1800 on her SAT so her z score would be 1 00 This 1800 1500 300 means that Pam was one standard deviation above the mean Jim scored 24 on his ACT so his z score would be 0 60 24 21 5 This means that Jim was 0 60 standard deviation above the mean Pam performed better 11 The Standard Normal Distribution The standard normal distribution is a normal distribution with a mean of 0 0 and a standard deviation of 1 1 0 1 Any normal distribution can be shifted and scaled to the standard normal distribution 0 1 by a z transformation Suppose then N 0 1 12 Standard Normal Table Z Table A Standard Normal Table available in the Canvas is a table that helps us determine the area or probability to the left of a specific z score on the standard normal distribution 13 Using the Z Table to Find Probability area under the normal curve Z table Description The values along the left and top margins represent induvial z scores while the values within the table indicate the corresponding areas under the standard normal curve for each z score A Use Z table to find To find the z score of 1 26 first locate the row of 1 2 and then the column of 0 06 since 1 2 0 06 1 26 The value at the cumulative area to the left of our z score of 1 26 which is 0 8962 intersection represents their 14 Using the Z Table to Find Probability Cont B Use Z table to find Area to the right of 1 26 15 Using the Z Table to Find Probability Cont C Use Z table to find 16 Using the Z Table to Find Probability Cont The systolic blood pressure of a certain population is normally distributed with a mean of 105 mmHg and a standard deviation of 20 mmHg A systolic blood pressure reading below 90 mmHg or above 130 mmHg is considered a cause for concern Determine the proportion of the population whose systolic blood pressure falls within this range of concern 90 575 1 90 105 20 15 20 0 75 2 130 105 20 1 25 25 20 0 75 0 2266 1 25 0 8944 90 575 0 75 1 1 25 0 2266 1 0 8944 17 Your Turn 1 Use Z Table to determine the following probabilities 1 Probability that z is less than 1 42 2 Probability is greater than 2 31 3 Probability that z is greater than 1 76 and less than 0 58 18 Your Turn 1 Solution 19 Your Turn 1 Solution Cont 20 Your Turn 1 Solution Cont 21 Example 2 Body Temperature Body temperatures of adults are normally distributed with a mean of 98 60 0F and a standard deviation of 0 73 0F What is the probability of a healthy adult having a body temperature less than 96 90 0F 22 Example 2 Body Temperature solution Body temperatures of adults are normally distributed with a mean of 98 60 0F and a standard deviation of 0 73 0F What is the probability of a healthy adult having a body temperature less than 96 90 0F We are interested in finding the probability of a person s body temperature is less than 96 90 0F 96 90 96 90 98 60 0 73 1 7 0 73 2 328767123 2 33 Note When calculating a z score be sure to
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