Examples of continuous probability distributions:

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Examples of continuous probability distributions

• The normal and standard normal

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The Normal Distribution
f(X)
Changing µ shifts the distribution left or right.
Changing s increases or decreases the spread.
s
X
µ
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The Normal Distributionas mathematical function
(pdf)
Note constants ?3.14159 e2.71828
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The Normal PDF

• Its a probability function, so no matter what
  the values of ? and ?, must integrate to 1!

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Normal distribution is defined by its mean and
standard dev.

• E(X)?
  
• Var(X)?2
  
• Standard Deviation(X)?

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The beauty of the normal curve
No matter what ? and ? are, the area between ?-?
and ?? is about 68 the area between ?-2? and
?2? is about 95 and the area between ?-3? and
?3? is about 99.7. Almost all values fall
within 3 standard deviations.
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68-95-99.7 Rule
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68-95-99.7 Rulein Math terms
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How good is rule for real data?

• Check some example data
• The mean of the weight of the women 127.8
• The standard deviation (SD) 15.5

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68 of 120 .68x120 82 runners In fact, 79
runners fall within 1-SD (15.5 lbs) of the mean.
127.8
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95 of 120 .95 x 120 114 runners In fact,
115 runners fall within 2-SDs of the mean.
127.8
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99.7 of 120 .997 x 120 119.6 runners In
fact, all 120 runners fall within 3-SDs of the
mean.
127.8
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Example

• Suppose SAT scores roughly follows a normal
  distribution in the U.S. population of
  college-bound students (with range restricted to
  200-800), and the average math SAT is 500 with a
  standard deviation of 50, then
• 68 of students will have scores between 450 and
  550
• 95 will be between 400 and 600
• 99.7 will be between 350 and 650

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Example

• BUT
• What if you wanted to know the math SAT score
  corresponding to the 90th percentile (90 of
  students are lower)?
• P(XQ) .90 ?

Solve for Q?.Yikes!
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The Standard Normal (Z)Universal Currency

• The formula for the standardized normal
  probability density function is

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The Standard Normal Distribution (Z)

• All normal distributions can be converted into
  the standard normal curve by subtracting the mean
  and dividing by the standard deviation

Somebody calculated all the integrals for the
standard normal and put them in a table! So we
never have to integrate! Even better, computers
now do all the integration.
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Comparing X and Z units
100
200
X
(? 100, ? 50)
Z
2.0
0
(? 0, ? 1)
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Example

• For example Whats the probability of getting a
  math SAT score of 575 or less, ?500 and ?50?

• i.e., A score of 575 is 1.5 standard deviations
  above the mean

Yikes! But to look up Z 1.5 in standard normal
chart (or enter into SAS)? no problem! .9332
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Looking up probabilities in the standard normal
table
What is the area to the left of Z1.50 in a
standard normal curve?
Area is 93.32
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Looking up probabilities in the standard normal
table
What is the area to the left of Z1.51 in a
standard normal curve?
Area is 93.45
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Probit function the inverse

•   ?(area) Z gives the Z-value that goes with
  the probability you want
•  For example, recall SAT math scores example.
  Whats the score that corresponds to the 90th
  percentile?
• In the Table, find the Z-value that corresponds
  to an area of 90...

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90 area corresponds to a Z score of about 1.28.
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Probit function the inverse

• Z1.28 convert back to raw SAT score ?
• 1.28 X 500 1.28 (50)
• X1.28(50) 500 564 (1.28 standard
  deviations above the mean!)

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Practice problem

• If birth weights in a population are normally
  distributed with a mean of 109 oz and a standard
  deviation of 13 oz,
• What is the chance of obtaining a birth weight of
  141 oz or heavier when sampling birth records at
  random?
• What is the chance of obtaining a birth weight of
  120 or lighter?

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Answer

1. What is the chance of obtaining a birth weight of
   141 oz or heavier when sampling birth records at
   random?

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Area to the left of Z2.46 is .9931
Area to the right of 2.46 is 1-.9931 .0069 or
.69
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Answer

• b. What is the chance of obtaining a birth
  weight of 120 or lighter?

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Area to the left of Z0.85 is .8023 or 80.23.
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Practice problem 2 DSST (a measure of cognitive
function) is a normally distributed trait
Normally distributed Mean 28 points Standard
deviation 10 points
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Practice problem 2

• a. What percent of people have values of DSST
  above 38?
• b. What percent of people have values of DSST
  below 8?

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Answers

• a. What percent of people have values of DSST
  above 38?

Thus, 16 of people have DSSTs above 38.
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Answers

• b. What percent of people have values of DSST
  below 8?

Thus, 2.5 of people have DSSTs below 8.
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Review question 1

• The probability that a standardized normal
  variable Z is positive is ____.
• 100
• 50
• 10
• 0

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Review question 2

• The probability that Z is between -2 and -1 is
  _____.
• 50
• 34
• 25.5
• 13.5

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Review question 3

• The probability that Z values are larger than
  _____ is 0.6985.
• Z1
• Z0
• Z-.5
• Z.5

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Review question 4

• 27 of Z values are smaller than ____.
• Z0
• Z1
• Z-.6
• Z.6

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Are my data normal?

• Not all continuous random variables are normally
  distributed!!
• It is important to evaluate how well the data are
  approximated by a normal distribution

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Are my data normally distributed?

1. Look at the histogram! Does it appear bell
   shaped?
2. Compute descriptive summary measuresare mean,
   median, and mode similar?
3. Do 2/3 of observations lie within 1 std dev of
   the mean? Do 95 of observations lie within 2 std
   dev of the mean?
4. Look at a normal probability plotis it
   approximately linear?
5. Run tests of normality (such as
   Kolmogorov-Smirnov). But, be cautious, highly
   influenced by sample size!

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Data from our class
Median 8 Mean 8.8 Mode 0
SD 8.3 Range 0 to 32 ( 4 s)
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Data from our class
Median 45 Mean 41 Mode 6
SD 23 Range 0 to 83 ( 3.5 s)
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Data from our class
Median 4 Mean 3.7 Mode 4
SD 1.8 Range 0.5 to 7 ( 3.5 s)
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Data from our class
Median 18 Mean 20 Mode 20
SD 16 Range 2 to 70 (4 s)
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Data from our class
8.8 /- 8.3 0.5 17.1
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Data from our class
8.8 /- 28.3 0 25.4
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Data from our class
8.8 /- 38.3 0 33.7
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Data from our class
41 /- 23 18 64
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Data from our class
41 /- 223 0 87
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Data from our class
41 /- 323 0 100
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Data from our class
3.7 /- 1.8 1.9 5.5
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Data from our class
3.7 /- 21.8 0.1 7.3
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Data from our class
3.7 /- 31.8 0 9.1
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Data from our class
20 /- 16 4 36
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Data from our class
20 /- 216 0 52
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Data from our class
20 /- 316 0 68
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The Normal Probability Plot

• Normal probability plot
• Order the data.
• Find corresponding standardized normal quantile
  values
• Plot the observed data values against normal
  quantile values.
• Evaluate the plot for evidence of linearity.

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Normal probability plot coffee

Right-Skewed! (concave up)
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Normal probability plot love of writing

A wiggly line!
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Norm prob. plot Exercise

Mostly a straight line!
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Norm prob. plot Wake up time

Right-Skewed! (concave up)
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Formal tests for normality

• Results
• Coffee Moderate evidence of non-normality
  (p.008 to p.11)
• Writing love No evidence of non-normality (all
  pgt.15)
• Exercise No evidence of non-normality (all
  pgt.15)
• Homework Strong evidence of non-normality (all
  plt.01)

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Review question 5

• Which of the following does NOT support the
  conclusion that your data are normally
  distributed
• The histogram is bell-shaped.
• The normal probability plot is approximately a
  straight line.
• The mean and the median are far apart.
• Formal tests of normality (with fancy Russian
  names) yield high p-values.

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Normal approximation to the binomial

• When you have a binomial distribution where the
  expected value is greater than 5 (npgt5), then
  the binomial starts to look like a normal
  distribution?
•  
• Recall What is the probability of being a smoker
  among a group of cases with lung cancer is .6,
  whats the probability that in a group of 8 cases
  you have less than 2 smokers?

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Normal approximation to the binomial

• When you have a binomial distribution where n is
  large and p isnt too small (rule of thumb
  meangt5), then the binomial starts to look like a
  normal distribution?  
• Recall smoking example

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Normal approximation to binomial
What is the probability of fewer than 2 smokers?
Exact binomial probability (from before) .00065
.008 .00865
  Normal approximation probability ?4.8 ?1.39
P(Zlt2).022
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• A little off, but in the right ballpark we
  could also use the value to the left of 1.5 (as
  we really wanted to know less than but not
  including 2 called the continuity correction)

A fairly good approximation of the exact
probability, .00865.
P(Z-2.37) .0069
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Practice problem

• 1. You are performing a cohort study. If the
  probability of developing disease in the exposed
  group is .25 for the study duration, then if you
  sample (randomly) 500 exposed people, Whats the
  probability that at most 120 people develop the
  disease?

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Answer
OR use, normal approximation ?np500(.25)125
and ?2np(1-p)93.75 ?9.68

•  P(Zlt-.52) .3015

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Review question 6

• If you flip a coin 1600 times, what is the
  approximate probability that you will get fewer
  than 860 heads?
• 25
• 2.5
• 0.5
• 0.005

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Review Problem 7

• Which of the following about the normal
  distribution is NOT true?
• Theoretically, the mean, median, and mode are the
  same.
• About 2/3 of the observations fall within 1
  standard deviation from the mean.
• It is a discrete probability distribution.
• Its parameters are the mean, ? , and standard
  deviation, ?.

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Proportions

• The binomial distribution forms the basis of
  statistics for proportions.
• A proportion is just a binomial count divided by
  n.
• For example, if we sample 200 cases and find 60
  smokers, X60 but the observed proportion.30.
• Statistics for proportions are similar to
  binomial counts, but differ by a factor of n.

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Stats for proportions

• For binomial

For proportion
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It all comes back to Z

• Statistics for proportions are based on a normal
  distribution, because the binomial can be
  approximated as normal if npgt5

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Homework

• Problem Set 3
• Reading Vickers 10-15
• Journal article/article review sheet