Using Models To Make Decisions

1
Chapter 6

• Using Models To Make Decisions

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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 are distributed.

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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 referencefor comparison, to determine
  if an observation is unusual or not

4
Blood Pressure

• What is a healthy BP?
• What is an unhealthy BP?
• Where did those statements come from?

5
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.

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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.

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Lets Get Normal

• The normal distribution is the symmetric bell
  shaped distribution.

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Abducted by an alien circus company,
Professor Arnold is forced to write statistics
equations in Center Ring.
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Lets Get Normal

• Normal distribution

Curve is bell shaped and symmetric
Density
Data Axis
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EX Heights of Adult Men and Women. Note that the
shape of the distribution is dependent on the
mean and standard deviation.

• Women
• µ 63.6
• ? 2.5

Men µ 69.0 ? 2.8
63.6
69.0
Height (inches)
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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, 2.8)
  Height of women is N(63.6, 2.5)

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Note

• Since we are modeling populations, the mean and
  standard deviation are the parameters given by m
  and s respectively.

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The Z score for Population Models
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• The Standard Normal Distribution

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ZScore

• The zscore tells you how many standard
  deviations an observed value falls from the mean.
  
• If z gt 0 then the value of x is above the mean.
• If z lt 0 then the value of x is below the mean.
• If z 0 then the value of x is equal to the mean.

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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?

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Using the TI-83 to Find Proportions in a Normal
Distribution (p. 331)

• STEP 1 Draw a picture and shade the area that
  represents the proportion to be found.
• STEP 2 Use NormalCDF.NormalCDf(lower, upper, m,
  s)
• Note -E99 is negative infinity and E99 is
  positive infinity

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Lets Do It

• Page 332 LDI 6.1
• Page 333 LDI 6.2

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68-95-99.7 Rule for N(m,s)

• 68 of the observations fall within one standard
  deviation of the mean m - s, m s
• 95 of the observations fall within two standard
  deviations of the meanm - 2s, m 2s
• 99.7 of the observations fall within three
  standard deviations of the mean m - 3s, m 3s

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Lets Do It

• Page 335 LDI 6.4, LDI 6.5

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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?

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Big Deal!

• Area Proportion
• Position Data Value

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Lets Do It

• Page 338 LDI 6.6 and LDI 6.7

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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

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Lets Do It

• Do a histogram and a normal quantile plot for the
  AGE, Left hand, and Right hand data. Assess
  normality for each of these distributions.
• Page 340 LDI 6.8 by shape.

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Homework 9

• Read Chapter 6
• LDI 6.16.7
• EX 6.16.25 odd

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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

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Lets Get Uniform

• The second most commonly used continuous
  distribution is the uniform distribution.

a
b
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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.

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Lets Do It

• Page 350 6.9, 6.10

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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

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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.

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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

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The Plot of the Mass Function
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Lets Do It

• Page 356 LDI 6.11
• Page 357 LDI 6.13
• Page 358 LDI 6.14

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Homework 10

• LDI, 6.10, 6.11, 6.13, 6.14
• EX 6.26, 6.29, 6.31, 6.37, 6.39, 6.44, 6.56,
  6.59, 6.61, 6.64, 6.77, 6.79