Modeling Continuous Variables Lecture 19 Sections 6.1 - 6.3.1 Robb T. Koether
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Modeling Continuous Variables Robb T. Koether Homework Review Models Area And Probability Modeling Continuous Variables Lecture 19 Sections 6.1 - 6.3.1 Density Functions The Normal Distribution Robb T. Koether Examples Assignment Answers to Evennumbered Exercises Hampden-Sydney College Tue, Sep 29, 2009 Outline Modeling Continuous Variables Robb T. Koether Homework Review Models 1 Homework Review 2 Models 3 Area And Probability Density Functions 4 The Normal Distribution Examples 5 Assignment 6 Answers to Even-numbered Exercises Area And Probability Density Functions The Normal Distribution Examples Assignment Answers to Evennumbered Exercises Outline Modeling Continuous Variables Robb T. Koether Homework Review Models 1 Homework Review 2 Models 3 Area And Probability Density Functions 4 The Normal Distribution Examples 5 Assignment 6 Answers to Even-numbered Exercises Area And Probability Density Functions The Normal Distribution Examples Assignment Answers to Evennumbered Exercises Homework Review Modeling Continuous Variables Robb T. Koether Homework Review Models Exercise 5.20, page 336. Consider the following two frequency plots based on two sets of data with 10 observations: Frequency Plot 1 Frequency Plot 2 Area And Probability Density Functions The Normal Distribution Examples Assignment Answers to Evennumbered Exercises -2 -1 0 1 2 -2 -1 0 1 2 Note that Plot 1 can be converted to Plot 2 by moving the two blue X’s. Homework Review Modeling Continuous Variables Robb T. Koether Homework Review Models Area And Probability Density Functions The Normal Distribution Examples Assignment Answers to Evennumbered Exercises Exercise 5.20, page 336. (a) Do you think the range for Plot 1 is smaller than, equal to, or larger than the range for Plot 2? (b) Do you think the mean for Plot 1 is smaller than, equal to, or larger than the mean for Plot 2? (c) Do you think the standard deviation for Plot 1 is smaller than, equal to, or larger than the standard deviation for Plot 2? Explain. (d) Check your answers to parts (a), (b), and (c) by computing the range, mean, and standard deviation for each set of data. Homework Review Modeling Continuous Variables Robb T. Koether Homework Review Models Area And Probability Solution (a) The ranges are clearly equal. (b) Both distributions are symmetric, so the means are in the center, at 0. Thus, they are equal. Density Functions The Normal Distribution Examples Assignment Answers to Evennumbered Exercises (c) The two blue points in Plot 2 are closer to the mean than they are in Plot 1, so they have smaller deviations. All other points are the same between the two distributions. Thus, Plot 2 has a smaller standard deviation. Homework Review Modeling Continuous Variables Robb T. Koether Homework Review Models Area And Probability Density Functions The Normal Distribution Examples Assignment Answers to Evennumbered Exercises Solution (d) For Plot 1, 2 − (−2) = 4. 0 x = = 0. 10 SSX = 4 + 4 + 1 + 1 + 0 + 0 + 1 + 1 + 4 + 4 range = = r 20. 20 s = 9 = 1.491. Homework Review Modeling Continuous Variables Robb T. Koether Homework Review Models Area And Probability Density Functions The Normal Distribution Examples Assignment Answers to Evennumbered Exercises Solution (d) For Plot 2, 2 − (−2) = 4. 0 x = = 0. 10 SSX = 4 + 1 + 1 + 0 + 0 + 0 + 0 + 1 + 1 + 4 range = = r 12. 12 s = 9 = 1.155. Outline Modeling Continuous Variables Robb T. Koether Homework Review Models 1 Homework Review 2 Models 3 Area And Probability Density Functions 4 The Normal Distribution Examples 5 Assignment 6 Answers to Even-numbered Exercises Area And Probability Density Functions The Normal Distribution Examples Assignment Answers to Evennumbered Exercises Models Modeling Continuous Variables Robb T. Koether Homework Review Models Area And Probability Density Functions The Normal Distribution Examples Assignment Answers to Evennumbered Exercises Definition (Mathematical model) A mathematical model is a mathematical abstraction and, therefore, a simplification of a real situation, one that retains the essential features. Real situations are usually much too complicated to deal with in all their details. Example Modeling Continuous Variables Robb T. Koether Homework Review Models Area And Probability Density Functions The Normal Distribution Examples Assignment Answers to Evennumbered Exercises The “bell curve” is a model (an abstraction) of many populations. Real populations have all sorts of bumps and twists and irregularities. The bell curve is smooth and perfectly symmetric. In statistics, the bell curve is called the normal curve, or normal distribution. Models Modeling Continuous Variables Robb T. Koether Homework Review Models Area And Probability Density Functions The Normal Distribution Examples Assignment Answers to Evennumbered Exercises Our models will be models of distributions, presented either as histograms or as continuous distributions. Outline Modeling Continuous Variables Robb T. Koether Homework Review Models 1 Homework Review 2 Models 3 Area And Probability Density Functions 4 The Normal Distribution Examples 5 Assignment 6 Answers to Even-numbered Exercises Area And Probability Density Functions The Normal Distribution Examples Assignment Answers to Evennumbered Exercises Histograms and Area Modeling Continuous Variables Robb T. Koether Homework Review Models Area And Probability Density Functions The Normal Distribution Examples Assignment Answers to Evennumbered Exercises In a histogram, frequency is represented by area. Consider the following distribution of rainfall data. Rainfall (in) 0.00 - 2.49 2.50 - 4.99 5.00 - 7.49 7.50 - 9.99 10.00 - 12.49 12.50 - 14.99 Frequency 9 12 4 2 2 1 Histograms and Area Modeling Continuous Variables 12 Robb T. Koether Homework Review Area And Probability Density Functions The Normal Distribution 9 No. of Years Models 6 Examples Assignment Answers to Evennumbered Exercises 3 0 0.0 2.5 5.0 7.5 10.0 Rainfall (in) 12.5 15.0 Histograms and Area Modeling Continuous Variables Robb T. Koether Homework Review Models Area And Probability Density Functions The Normal Distribution Examples Assignment Answers to Evennumbered Exercises What is the total area of this histogram? Histograms and Area Modeling Continuous Variables 12 Robb T. Koether Homework Review Area And Probability Density Functions The Normal Distribution 9 No. of Years Models 6 30 22.5 Examples Assignment Answers to Evennumbered Exercises 3 10 5 5 0 0.0 2.5 5.0 7.5 10.0 Rainfall (in) 2.5 12.5 15.0 Histograms and Area Modeling Continuous Variables Robb T. Koether Homework Review Models The total area is Area And Probability Density Functions The Normal Distribution Examples Assignment Answers to Evennumbered Exercises 25 + 30 + 7.5 + 5 + 5 + 2.5 = 75. Histograms and Probability Modeling Continuous Variables Robb T. Koether If we select one year at random, what is the probability that September rainfall was between 5 inches and 10 inches? Homework Review 12 Models Area And Probability The Normal Distribution Examples Assignment Answers to Evennumbered Exercises 9 No. of Years Density Functions 6 30 22.5 3 10 5 5 0 0.0 2.5 5.0 7.5 10.0 Rainfall (in) 2.5 12.5 15.0 Histograms and Probability Modeling Continuous Variables Robb T. Koether Homework Review Models Area And Probability Density Functions The Normal Distribution Examples Assignment Answers to Evennumbered Exercises We will rescale the vertical axis so that the total area equals 1, representing 100%. This new scale is called the density. Histograms and Area Modeling Continuous Variables Robb T. Koether Homework Review Models Area And Probability Density Functions The Normal Distribution Examples Assignment Answers to Evennumbered Exercises Divide the frequencies by the original area to get the density. Rainfall (in) 0.00 - 2.49 2.50 - 4.99 5.00 - 7.49 7.50 - 9.99 10.00 - 12.49 12.50 - 14.99 Frequency 9 12 4 2 2 1 Density 0.1200 0.1600 0.0533 0.0267 0.0267 0.0133 Histograms and Area Modeling Continuous Variables 0.16 Robb T. Koether Homework Review 0.12 Area And Probability Density Functions The Normal Distribution Density Models 0.08 Examples Assignment Answers to Evennumbered Exercises 0.04 0.0 0.0 2.5 5.0 7.5 10.0 Rainfall (in) 12.5 15.0 Histograms and Area Modeling Continuous Variables 0.16 Robb T. Koether Homework Review 0.12 Area And Probability Density Functions The Normal Distribution Density Models 0.4000 0.08 0.3000 Examples Assignment Answers to Evennumbered Exercises 0.04 0.1333 0.0667 0.0667 0.0 0.0 2.5 5.0 7.5 10.0 Rainfall (in) 0.0333 12.5 15.0 Histograms and Area Modeling Continuous Variables Robb T. Koether If we select one year at random, what is the probability that September rainfall was between 5 inches and 10 inches? Homework Review 0.16 Models Area And Probability 0.12 The Normal Distribution Examples Density Density Functions Assignment Answers to Evennumbered Exercises 0.4000 0.08 0.3000 0.04 0.1333 0.0667 0.0667 0.0 0.0 2.5 5.0 7.5 10.0 Rainfall (in) 0.0333 12.5 15.0 Outline Modeling Continuous Variables Robb T. Koether Homework Review Models 1 Homework Review 2 Models 3 Area And Probability Density Functions 4 The Normal Distribution Examples 5 Assignment 6 Answers to Even-numbered Exercises Area And Probability Density Functions The Normal Distribution Examples Assignment Answers to Evennumbered Exercises Density Functions Modeling Continuous Variables Robb T. Koether Homework Review AREA = PROPORTION = PROBABILITY Models Area And Probability Density Functions The Normal Distribution Examples Assignment Answers to Evennumbered Exercises This is the fundamental property that connects the graph of a continuous model to the population that it represents, namely: The area under the graph between two points on the x-axis represents the proportion of the population that lies between those two points. Density Functions Modeling Continuous Variables Robb T. Koether Consider an arbitrary continuous distribution. Homework Review Area And Probability Density Functions Density Models The Normal Distribution Examples Assignment Answers to Evennumbered Exercises 0 1 2 3 4 5 6 7 Density Functions Modeling Continuous Variables Robb T. Koether The area under the curve between 2 and 5 is the proportion of the values of x that lie between 2 and 5. Homework Review Area And Probability Density Functions Density Models The Normal Distribution Examples Assignment Answers to Evennumbered Exercises 0 1 2 3 4 5 6 7 Density Functions Modeling Continuous Variables Robb T. Koether The area under the curve between 2 and 5 is the proportion of the values of x that lie between 2 and 5. Homework Review Area And Probability Density Functions Density Models The Normal Distribution Examples Assignment Answers to Evennumbered Exercises 0 1 2 3 4 5 6 7 Density Functions Modeling Continuous Variables Robb T. Koether The area under the curve between 2 and 5 is the proportion of the values of x that lie between 2 and 5. Homework Review Area And Probability Density Functions Density Models The Normal Distribution Examples Assignment Answers to Evennumbered Exercises 0 1 2 3 4 5 6 7 Density Functions Modeling Continuous Variables Robb T. Koether The area under the curve between 2 and 5 is the proportion of the values of x that lie between 2 and 5. Homework Review Area And Probability Density Functions Density Models The Normal Distribution Area = Proportion Examples Assignment Answers to Evennumbered Exercises 0 1 2 3 4 5 6 7 Outline Modeling Continuous Variables Robb T. Koether Homework Review Models 1 Homework Review 2 Models 3 Area And Probability Density Functions 4 The Normal Distribution Examples 5 Assignment 6 Answers to Even-numbered Exercises Area And Probability Density Functions The Normal Distribution Examples Assignment Answers to Evennumbered Exercises The Normal Distribution Modeling Continuous Variables Robb T. Koether Definition (Normal distribution) Homework Review Models The normal distribution is the statistician’s name for the bell curve. Area And Probability Density Functions The Normal Distribution Examples Assignment Answers to Evennumbered Exercises It is a density function in the shape of a “bell.” Symmetric. Unimodal. Extends over the entire real line (no endpoints). “Main part” lies within ±3σ of the mean. The Normal Distribution Modeling Continuous Variables Robb T. Koether Homework Review Models Area And Probability Density Functions The Normal Distribution Examples Assignment Answers to Evennumbered Exercises The curve has a bell shape, with infinitely long tails in both directions. The Normal Distribution Modeling Continuous Variables Robb T. Koether The mean µ is located in the center, at the peak. Homework Review Models Area And Probability Density Functions The Normal Distribution Examples Assignment Answers to Evennumbered Exercises µ The Normal Distribution Modeling Continuous Variables Robb T. Koether Homework Review The width of the “main” part of the curve is 6 standard deviations wide (3 standard deviations each way from the mean). Models Area And Probability Density Functions The Normal Distribution Examples σ Assignment Answers to Evennumbered Exercises µ − 3σ µ µ + 3σ The Normal Distribution Modeling Continuous Variables Robb T. Koether The area under the entire curve is 1. (The area outside of µ ± 3σ is approx. 0.0027.) Homework Review Models Area And Probability Density Functions The Normal Distribution Area = 1 Examples Assignment Answers to Evennumbered Exercises µ − 3σ µ µ + 3σ The Normal Distribution Modeling Continuous Variables Robb T. Koether Homework Review Models Area And Probability Density Functions The Normal Distribution Examples Assignment Answers to Evennumbered Exercises The normal distribution with mean µ and standard deviation σ is denoted N (µ, σ). For example, if X is a variable whose distribution is normal with mean 30 and standard deviation 5, then we say that “X is N (30, 5).” Outline Modeling Continuous Variables Robb T. Koether Homework Review Models 1 Homework Review 2 Models 3 Area And Probability Density Functions 4 The Normal Distribution Examples 5 Assignment 6 Answers to Even-numbered Exercises Area And Probability Density Functions The Normal Distribution Examples Assignment Answers to Evennumbered Exercises The Normal Distribution Modeling Continuous Variables Robb T. Koether If X is N (30, 5), then the distribution of X looks like this: 0.08 Homework Review Models 0.06 Area And Probability Density Functions The Normal Distribution 0.04 Examples Assignment 0.02 Answers to Evennumbered Exercises 20 25 30 35 40 45 Some Normal Distributions Modeling Continuous Variables Some other examples: N (3, 1). 0.8 Robb T. Koether Homework Review 0.6 Models Area And Probability Density Functions 0.4 The Normal Distribution Examples Assignment 0.2 Answers to Evennumbered Exercises 0 2 4 6 8 Some Normal Distributions Modeling Continuous Variables Some other examples: N (5, 1). 0.8 Robb T. Koether Homework Review 0.6 Models Area And Probability Density Functions 0.4 The Normal Distribution Examples Assignment 0.2 Answers to Evennumbered Exercises 0 2 4 6 8 Some Normal Distributions Modeling Continuous Variables Some other examples: N 2, 21 . 0.8 Robb T. Koether Homework Review 0.6 Models Area And Probability Density Functions 0.4 The Normal Distribution Examples Assignment 0.2 Answers to Evennumbered Exercises 0 2 4 6 8 Some Normal Distributions Modeling Continuous Variables Some other examples: N 3 12 , 1 21 . 0.8 Robb T. Koether Homework Review 0.6 Models Area And Probability Density Functions 0.4 The Normal Distribution Examples Assignment 0.2 Answers to Evennumbered Exercises 0 2 4 6 8 Outline Modeling Continuous Variables Robb T. Koether Homework Review Models 1 Homework Review 2 Models 3 Area And Probability Density Functions 4 The Normal Distribution Examples 5 Assignment 6 Answers to Even-numbered Exercises Area And Probability Density Functions The Normal Distribution Examples Assignment Answers to Evennumbered Exercises Assignment Modeling Continuous Variables Robb T. Koether Homework Review Models Area And Probability Density Functions The Normal Distribution Examples Assignment Answers to Evennumbered Exercises Homework Read Section 6.1 - 6.3.1, pages 357 - 362. Let’s Do It! 6.1. Exercises 1, 2, 3, 32, page 376. Outline Modeling Continuous Variables Robb T. Koether Homework Review Models 1 Homework Review 2 Models 3 Area And Probability Density Functions 4 The Normal Distribution Examples 5 Assignment 6 Answers to Even-numbered Exercises Area And Probability Density Functions The Normal Distribution Examples Assignment Answers to Evennumbered Exercises Answers to Even-numbered Exercises Modeling Continuous Variables Robb T. Koether Homework Review Models Area And Probability Density Functions The Normal Distribution Examples Assignment Answers to Evennumbered Exercises Page 376, Exercises 2, 32 6.2 (a) 14 . (b) Less than 1. Clearly, more than half the area is to the left of 1. Therefore, to divide the area in half, the median must be to the left of 1. 6.32 (a) (b) (c) (d) (e) (f) 1 3. Symmetric. 1 4. 1 α = 36 . β = 0.65. H0 .
