Chapter 6 Using Models To Make Decisions
Homework 10 • Read Chapter 6, pages 357-396 Ignore reference to Table II • LDI: 6.1–6.7 • Exercises: (page 376) 6.1, 6.2, 6.5, 6.6, 6.9, 6.10, 6.14, 6.27, 6.34
Model • A model is a representation of a real-world object or phenomenon. • In statistics we want to model populations. In particular we want to model how the values of the variable of interest in the population are distributed.
Purpose • A well crafted model of a population will help us make sound decisions between competing theories. • Statistical models bring order and understanding to the overwhelming flow of data. Models serve as a frame of reference–for comparison, to determine if an observation is unusual or not.
Blood Pressure • What is a healthy BP? • What is an unhealthy BP? • Where did those statements come from?
Modeling Continuous Variables • We need to have a model of what we think the distribution of the null looks like in order for us to decided if the observed data would be unusual to be seen under the assumption the null is true. Hence, we need to model populations. Since we will be discussing models for populations, the mean and standard deviation for a density curve or model will be represented by (mu) and (sigma), respectively.
Density Function • 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 are measured as areas under the density curve/function. • Big Deal: area = proportion in a continuous model.
Let’s Get Normal • The normal distribution is the symmetric bell shaped distribution.
Abducted by an alien circus company, Professor Arnold is forced to write statistics equations in Center Ring.
Let’s Get Normal • Normal distribution Curve is bell shaped and symmetric Density Data Axis
EX: Heights of Adult Men and Women (According to the National Center for Health Statistics). Note that the shape of the distribution is dependent on the mean and standard deviation. Women: µ = 63.6” = 2.5” Men: µ = 69.2” = 2.8” 63.6” 69.2” Height (inches)
Normal Notation • The notation “X is N(m, s)” means that the variable X is normally distributed with mean m and standard deviation s. • For example: Height of men is N(69.2, 2.8)Height of women is N(63.6, 2.5)
Using the TI to Find Proportions in a Normal Distribution • STEP 1: Draw a picture and shade the area that represents the proportion to be found. • STEP 2: Use NormalCDF (2nd-VARS).NormalCDf(lower, upper, m, s) • Note: -E99 is negative infinity and E99 is positive infinity
What Percentage? • Given the models from the NCHS, answer the following: • The percent of males more than 69.2 inches is ____ • The percent of females more than 69.2 inches is ____ • The percent of males less than 63.6 inches is ____
Keep in Mind • Since we are modeling populations, the mean and standard deviation are the parameters given by and respectively.
Let's Do It! 6.2 Standard Normal Areas • (a) 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. • (b) Find the area under the standard normal distribution to the left of z = -2.55. Sketch the area and use your calculator to find the area. • (c) 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.
Z–Score • The z–score tells you how many standard deviations an observed value falls from the mean. • If z > 0 then the value of x is above the mean. • If z < 0 then the value of x is below the mean. • If z = 0 then the value of x is equal to the mean.
If your height was 74.5 inches, find your z-score. • If your height was 54.5 inches find your z-score. Women: µ = 63.6” = 2.5” Men: µ = 69.2” = 2.8” 63.6” 69.2” Height (inches)
Finding Proportions and z-scores in the Normal Distribution • If you scored a 15 on the second statistics quiz, what would your z-score be? Assume the distribution of scores was normal with N(22,4). • What proportion of the class scored higher than you? lower?
Let's Do It! 6.3 IQ Scores • X = IQ score (12-year-olds) has an distribution. • (a) What proportion of the 12-year-olds has IQ scores below 84? z-score = _________, Sketch it. • (b) What proportion of the 12-year-olds has IQ scores 84 or more? z-score = ___________, Sketch it. • (c) What proportion of the 12-year-olds has IQ scores between 84 and 116?z-score (for 84) = ____________ z-score (for 116) = ____________, Sketch it.
68-95-99.7 Rule for • 68% of the observations fall within one standard deviation of the mean: • 95% of the observations fall within two standard deviations of the mean: • 99.7% of the observations fall within three standard deviations of the mean:
Let's Do It! 6.4 Using the Empirical Rule • Let Y represent the lifetime of a computer component. Suppose an appropriate model for Y is N (15, 2) in years. • (a) Draw the density curve for this distribution, and include a scale on the horizontal axis. • (b) What percent of components are expected to last longer than 19 years? • (c) Between what two lifetimes do the middle 95% of components fall? • (d) What percent of components are expected to last less than 9 years?
Let's Do It!6.5 Pine Needles • Different species of pine trees are grown at a Christmas-tree farm. It is known that the length of needles on a Species A pine tree follows a normal distribution. About 68% of such needles have lengths centered around the mean between 5.9 and 6.9 inches. • (a) What are the mean and standard deviation of the model for Species A pine-needle lengths? • (b) A 5.2-inch pine needle is found that looks like a Species A needle, but is somewhat shorter than expected. Is it likely that this needle is from a Species A pine tree? • Hint: Calculate the proportion of such pine needles 5.2 inches or shorter.
Finding Percentiles for a Normal Distribution • Assume that IQ scores for 12 year olds is well modeled by N(100,16). What IQ score must a 12 year old score to be placed in the top 5% of the distribution of IQ scores?
Big Deal! Area = Proportion Position = Data Value
Let’s Do It • LDI 6.6 • LDI 6.7
Assessing Normality • The best way to assess normality is with a normal quantile plot. If points on a normal quantile plot lie close to a straight line, the plot indicates that the data are normal. Systematic deviations from a straight line indicate a nonnormal distribution. Outliers appear as points that are far away from the overall pattern of the plot
Let’s Do It • Do a histogram and a normal quantile plot for the AGE, Left hand, and Right hand data respectively. Assess normality for each of these distributions using these two plots. • LDI 6.8 by shape.
Finding a z - score when given a proportion • Use InvNorm(area, mu, sigma). This will always give the area in the left tail. • Find the z score for 95% • Find the z score for 15% • Find the z scores that correspond to Q1 and Q3
Homework 12 • LDI, 6.10, 6.11, 6.13, 6.14
Let’s Get Uniform • The second most commonly used continuous distribution is the uniform distribution. a b
Notation • If a variable X is uniformally distributed we will say “X is U(a,b)” where a and b are the endpoints of the range of values. That is a is the minimum and b is the maximum.
Let’s Do It • LDI 6.9 • LDI 6.10
Models of Discrete Variables • If the variable of interest is countable, then the distribution will be discrete. • For example, the number of car models recalled by a certain manufacturer will be countable and finite. The values that the variable can take on
Mass Function • A mass function is used as a model for a discrete variable. For each possible value, the mass function gives the proportion of units in the population having that value. Thus, the values of the mass function must be between 0 and 1 and add up to 1. Proportions are measured directly as the values of the function, not as areas under the function.
Example • Number of books in a backpack. Let X be the number of books a student at CR carries in their backpack. The model that describes this variable is given by
Let’s Do It • LDI 6.11 • LDI 6.13 • LDI 6.14