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Chapter 5: Continuous Probability Distribution Chapter 5: Continuous Probability Distribution Keith E. Emmert Department of Mathematics Tarleton State University June 16, 2011 Chapter 5: Continuous Probability Distribution Outline 1 Why Do We Need to Know Models? 2 Modeling Continuous Variables 3 Normal Distributions Chapter 5: Continuous Probability Distribution Why Do We Need to Know Models? Example Scenario I: A patient visits his doctor complaining of a number of symptoms. The doctor suspects the patient is suffering from some disease. The doctor performs a diagnostic test to check for this disease. High responses on the test support that the patient may have the disease. The patients test response is 200. What does this say? Model for healthy subjects 80 100 120 140 160 180 200 Test Response Based on this model, it is very unlikely that a test response of 200, or greater, would have occurred if the subject were actually healthy. Thus, either the patient has this disease or a very unlikely event has occurred. Chapter 5: Continuous Probability Distribution Why Do We Need to Know Models? Example Scenario II: Suppose we wish to compare two drugs, Drug A and Drug B, for relieving arthritis pain. Subjects suitable for the study are randomized to one of the two drug groups and are given instructions for dosage and how to measure their “time to relief.” Results of the study are summarized by presenting the models for the time to relief for the two drugs. Drug A Drug B t Time to relief Consider any point in time, say time t as indicated on the above axis. A higher proportion of subjects treated with Drug A have felt “relief” by this time point as compared to those treated with Drug B. If the study design was sound, then we might conclude that Drug A works quicker than Drug B. Chapter 5: Continuous Probability Distribution Modeling Continuous Variables Density Functions A density function is a (nonnegative) function or curve that describes the overall shape of a distribution. The total area under the entire curve is equal to 1, and proportions or probabilities are measured as areas under the density function. As a simple example, below is a density curve (the blue curve). The shaded area represents the probability that a random variable takes on values between 6 and 20. 6 20 Chapter 5: Continuous Probability Distribution Modeling Continuous Variables Let’s Do It! Using Density Functions Density Let the variable X represent the length of life, in years, for an electrical component. The following figure is the density curve for the distribution of X . 0.39 0 0.24 1 0.14 2 0.09 3 0.05 4 0.03 5 6 x (a) What proportion of electrical components lasts longer than 6 years? (b) What proportion of electrical components lasts longer than 1 year? (c) Describe the shape of the distribution. Chapter 5: Continuous Probability Distribution Normal Distributions The Normal Distribution The normal distribution has probability density function (or pdf) given by 1 2 2 e −(x−µ) /2σ . f (x) = √ 2 2πσ 2 X is N(µ, σ ) means that the variable X is normally distributed with mean µ, variance σ 2 , and standard deviation σ. ::::::: :::::::::: ::::::::::::::::::: Below is an example of a typical normal curve. Notice it has inflection points at x = µ ± σ and is symmetric about x = µ. y Μ-Σ Μ Μ+Σ x Chapter 5: Continuous Probability Distribution Normal Distributions The Normal Distribution Some Example Curves NHΜ1 , Σ2 L NHΜ2 , Σ1 L NHΜ1 , Σ1 L Μ1 , Μ1 Μ2 For N(µ1 , σ12 ) and N(µ2 , σ12 ), the mean is changed, which slides the curve left or right. (Here µ2 > µ1 .) For N(µ1 , σ12 ) and N(µ1 , σ22 ), the variance is changed, which forces the curve to change it’s height (the distribution becomes more or less “concentrated” as the variance decreases or increases, respectively). (Here σ2 < σ1 .) Chapter 5: Continuous Probability Distribution Normal Distributions The Normal Distribution Important Properties 1 A normal distribution is bell-shaped. 2 The mean, median, and mode are equal and are located at the center of the distribution. 3 A normal distribution curve is unimodal (i.e. it has one mode). 4 The curve is symmetric about the mean. 5 The curve is continuous, that is, there are no gaps or holes. For any x, there is a corresponding y . 6 The curve never touches the x-axis, but it does extend infinitely in either direction. 7 The total area under a normal curve is 1. (Trust me.) Chapter 5: Continuous Probability Distribution Normal Distributions The Empirical Rule For a random variable based upon the normal curve N(µ, σ 2 ), the following empirical rule holds Pr (µ − σ < X < µ + σ) ≈ 0.68: This means that the area under the curve within 1 standard deviation of the mean is approximately 68%. Pr (µ − 2σ < X < µ + 2σ) ≈ 0.95: This means that the area under the curve within 2 standard deviations of the mean is approximately 95%. Pr (µ − 3σ < X < µ + 3σ) ≈ 0.997: This means that the area under the curve within 3 standard deviation of the mean is approximately 99.7%. Chapter 5: Continuous Probability Distribution Normal Distributions Example Let the variable X represent IQ scores of 12-year-olds. Suppose that the distribution of X is normal with a mean of 100 and a standard deviation of 16that is, X is N(100, 162 ). Jessica is a 12-year-old and has an IQ score of 132.We would like to determine the proportion of 12-year olds that have IQ scores less than Jessicas score of 132. Since the area under the density curve corresponds to proportion, we want to find the area to the left of 132 under an N(100, 162 ) curve. Sketch this curve and show the corresponding area that represents this proportion. 100 132 Chapter 5: Continuous Probability Distribution Normal Distributions The Normal Standardizing Scores to Compare Observations from Different Normal Distributions If X is N(µ, σ 2 ), then the standardized normal variable X −µ defined by Z = is N(0, 1). σ Given a particular observation, x, of a random variable X that is N(µ, σ 2 ), the z-score or standard score for an observed value tells us how many standard deviations the observed value is from the mean that is, it tells us how far the observed value is from the mean in standard-deviation units. It is calculated by: x −µ z= . σ Chapter 5: Continuous Probability Distribution Normal Distributions Example Comparing IQ Scores from Different Age Groups Recall that the IQ scores for 12-year olds is N(100, 162 ). Jessica had a score of 132. Compute Jessica’s standardized score. Suppose Jessica has an older brother, Mike, who is 20 years old and has an IQ score of 144. It wouldnt make sense to directly compare Mikes score of 144 to Jessicas score of 132. The two scores come from different distributions due to the age difference. Assume that the distribution of IQ scores for 20-year-olds is normal with a mean of 120 and a standard deviation of 20, i.e. N(120, 202 ). Compute Mikes standardized score. Relative to their respective age group, who had the higher IQ score - Jessica or Mike? Chapter 5: Continuous Probability Distribution Normal Distributions The Normal Distribution Calculating Probability/Proportions One way to calculate proportions is to standardize and then use a table. Another way is to use technology, such as the TI-84 or SPSS. We’ll take this route. Chapter 5: Continuous Probability Distribution Normal Distributions The Normal Distribution Calculating Probability/Proportions As an example, using the standard normal, N(0, 1), we calculate Pr (Z < 1.35) = 0.9114919. Notice that you do not type the mean and standard deviation when using standard normal, the calculator assumes it for you. You should type: normalcdf(−E 99, 1.35) or normalcdf(−E 99, 1.35, 0, 1) VARS 2nd 2 (-) , Opens the DISTR menu Selects the normalcdf(lower, upper, µ, σ) function , 9 9 -E99 is minus 2nd infinity 1 . 3 5 ) ENTER Chapter 5: Continuous Probability Distribution Normal Distributions The Normal Distribution Calculating Probability/Proportions As an example, using the standard normal, N(0, 1), we calculate Pr (Z > 1.35) = 0.0885081. Type normalcdf(1.35, E 99) or normalcdf(1.35, E 99, 0, 1) 2nd 2 , 2nd VARS Opens the DISTR menu Selects the normalcdf(lower, upper, µ, σ) function E99 is plus in. 3 5 1 finity , 9 9 ) ENTER P(Z > 1.35) = 1 − Pr (Z ≤ 1.35) = 1 − Pr (Z < 1.35) = 1 − 0.9114919 = 0.0885081. Chapter 5: Continuous Probability Distribution Normal Distributions The Normal Distribution Calculating Probability/Proportions As an example, using a normal distribution, N(100, 162 ), we calculate Pr (X < 132) = 0.9772499. Type normalcdf(−E 99, 132, 100, 16) VARS 2nd 2 (-) Opens the DISTR menu Selects the normalcdf(lower, upper, µ, σ) function , 9 9 -E99 is minus 2nd infinity , 1 , 100 3 2 , 16 ) ENTER Chapter 5: Continuous Probability Distribution Normal Distributions Let’s Do It! Standard Normal Areas Find the area under the standard normal distribution between z = 0 and z = 1.22. Sketch the area and use your calculator to find the area. Find the area under the standard normal distribution between z = −1.22 and z = 1.22. Sketch the area and use your calculator to find the area. Find the area under the standard normal distribution to the left of z = −1.22. Sketch the area and use your calculator to find the area. 0 0 0 Chapter 5: Continuous Probability Distribution Normal Distributions Let’s Do It! IQ Scores We will continue with the model for IQ score of 12-year-olds. Recall that X = IQ score of 12-year olds has a N(100, 162 ) distribution. What proportion of the 12-year olds have IQ scores below 84? What proportion of the 12-year olds have IQ scores 84 or more? What proportion of the 12-year olds have IQ scores between 84 and 116? 100 100 100 Chapter 5: Continuous Probability Distribution Normal Distributions Example Top 1% of the IQ Distribution Recall the N(100, 162 ) model for IQ scores of 12-year olds. What IQ score must a 12-year old have to place in the top 1% of the distribution of IQ scores? Type invNorm(0.99, 100, 10) VARS 2nd 3 100 Opens the DISTR menu Selects the invNorm(area, µ, σ) function 0 , , 100 This yields an answer of 137.221566. 9 9 , 16 ) ENTER Chapter 5: Continuous Probability Distribution Normal Distributions Let’s Do It! Freestyle Swim Times The finishing times for 11-12-year-old male swimmers performing the 50-yard freestyle are normally distributed with a mean of 35 seconds and a standard deviation of 2 seconds. (a) The sponsors of a swim meet decide to give certificates to all 11-12-year-old male swimmers who finish their 50-yard race in under 32 seconds. If there are 50 such swimmers entered in the 50-yard freestyle event, approximately how many certificates will be needed? (b) In what amount of time must a swimmer finish to be in the “top” fastest 2% of the distribution of finishing times? 35 35 Chapter 5: Continuous Probability Distribution Normal Distributions Let’s Do It! Hours Per Week According to a study, men in the US devote an average of 16 hours per week to house work. Assume that the number of hours men devote to house work is normally distributed with a standard deviation of 3.5. (a) Suppose that the lower 10% of men on the distribution devote fewer than x hours per week. Find the value of x. (b) Suppose the upper 5% of men on the distribution devote more than x hrs per week. Find the value of x. 16 16 Chapter 5: Continuous Probability Distribution Normal Distributions Let’s Do It! Middle Portion of the Normal If one-person household spends an average of $40 per month on medications and doctor visits, find the maximum and minimum dollar amounts spent per month for the middle 50% of one-person household. Assume that the standard deviation is $5 and that the amount spent is normally distributed. 40 Chapter 5: Continuous Probability Distribution Normal Distributions Homework HW page 154: 1-14 all, 17, 18, 27, 28