1. Stat 111 - Lecture 7 - Normal Distribution
Probability
Normal Distribution
and Standardization
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2. Stat 111 - Lecture 7 - Normal Distribution
Administrative Notes
Homework 2 due on Monday
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3. Stat 111 - Lecture 7 - Normal Distribution
Outline
Law of Large Numbers
Normal Distribution
Standardization and Normal Table
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4. Data versus Random Variables
Data variables are variables for which we actually observe values
Eg. height of students in the Stat 111 class
For these data variables, we can directly calculate the statistics s2 and x
Random variables are things that we don't directly observe, but we still have a probability distribution of all possible values
Eg. heights of entire Penn student population
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5. Stat 111 - Lecture 7 - Normal Distribution
Law of Large Numbers
Rest of course will be about using data statistics (x and s2) to estimate parameters of random variables ( and 2)
Law of Large Numbers: as the size of our data sample increases, the mean x of the observed data variable approaches the mean  of the population
If our sample is large enough, we can be confident that our sample mean is a good estimate of the population mean!
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6. The Normal Distribution
The Normal distribution has the shape of a “bell curve” with parameters  and 2 that determine the center and spread:  
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7. Different Normal Distributions
Each different value of  and 2 gives a different Normal distribution, denoted N(,2)
We can adjust values of  and 2 to provide the best approximation to observed data
If  = 0 and 2 = 1, we have the Standard Normal distribution
N(0,1)
N(2,1)
N(-1,2)
N(0,2)
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8. Property of Normal Distributions
Normal distribution follows the rule:
68% of observations are between  -  and  + 
95% of observations are between  - 2 and  + 2
99.7% of observations are between  - 3 and  + 3  2
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9. Calculating Probabilities
For more general probability calculations, we have to do integration
For the standard normal distribution, we have tables of probabilities already made for us!
If Z follows N(0,1):
P(Z < -1.00) =
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10. Stat 111 - Lecture 7 - Normal Distribution
Standard Normal Table
If Z has N(0,1):
P(Z > 1.46)
= 1 - P(Z < 1.46) = =
What if we need to do a probability calculation for a non-standard Normal distribution?
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11. Stat 111 - Lecture 7 - Normal Distribution
Standardization
If we only have a standard normal table, then we need to transform our non-standard normal distribution into a standard one
This process is called standardization  1 
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12. Standardization Formula
We convert a non-standard normal distribution into a standard normal distribution using a linear transformation
If X has a N(,2) distribution, then we can convert to Z which follows a N(0,1) distribution
Z = (X-)/
First, subtract the mean  from X
Then, divide by the standard deviation  of X
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13. Linear Transformations of Variables Fahrenheit = 9/5 x Celsius + 32
Sometimes need to do simple mathematical operations on our variables, such as adding and/or multiplying with constants
Y = a·X + b
Example: changing temperature scales
Fahrenheit = 9/5 x Celsius + 32
How are means and variances affected?
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14. Mean/Variances of Linear Transforms
For transformed variable Y = a·X + b
mean(Y) = a·mean(X) + b
Var(Y) = a2·Var(X)
SD(Y) = |a|·SD(X)
Note that adding a constant b does not affect measures of spread (variance and sd)
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15. More complicated linear functions
We can also do linear transformations involving with more than one variable:
Z = a·X + b·Y + c
The mean formula is similar:
mean(Z) = a·mean(X) + b·mean(Y) + c
If X and Y are also independent then
var(Z) = a2·var(X) + b2·var(Y)
Need more complicated variance formula (in book) if the variables are not independent
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16. Standardization Example
Dear Abby,
You wrote in your column that a woman is pregnant for 266 days. Who said so? I carried my baby for 10 months and 5 days. My husband is in the Navy and it could not have been conceived any other time because I only saw him once for an hour, and I didn’t see him again until the day after the baby was born. I don’t drink or run around, and there is no way the baby isn’t his, so please print a retraction about the 266-day carrying time because I am in a lot of trouble!
-San Diego Reader
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17. Standardization Example
According to well-documented data, gestation time follows a normal distribution with mean  of 266 days and SD  of 16
Let X = gestation time. What percent of babies have gestation time greater than 310 days (10 months & 5 days) ?
Need to convert X = 310 into standard Z
Z = (X-)/ = ( )/16 = 44/16 = 2.75
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18. Standardization Example
P(X > 310)
= P(Z > 2.75)
= 1 - P(Z < 2.75) = =
So, only a 0.3%
chance of a
pregnancy lasting
as long as 310 days!
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19. Reverse Standardization
Sometimes, we need to convert a standard normal Z into a non-standard normal X
Example: what is the length of pregnancy below which we have 10% of the population?
From table, we see P(Z <-1.28) = 0.10
Reverse Standardization formula:
X = σ⋅Z +μ
For Z = -1.28, we calculate
X = -1.28· = 246 days (8.2 months)
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20. Stat 111 - Lecture 7 - Normal Distribution
Another Example
NCAA Division 1 SAT Requirements: athletes are required to score at least 820 on combined math and verbal SAT
In 2000, SAT scores were normally distributed with mean  of 1019 and SD  of 209
What percentage of students have scores greater than 820 ?
Z = (X-)/ = ( )/209 = -199/209 = -.95
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21. Stat 111 - Lecture 7 - Normal Distribution
Another Example
P(X > 820) = P(Z > -0.95) = 1- P(Z < -0.95)
P(Z < -0.95) = 0.17 so P(X > 820) = 0.83
83% of students meet NCAA requirements
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22. Stat 111 - Lecture 7 - Normal Distribution
SAT Verbal Scores
Now, just look at X = Verbal SAT score, which is normally distributed with mean  of 505 and SD  of 110
What Verbal SAT score will place a student in the top 10% of the population?
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23. Stat 111 - Lecture 7 - Normal Distribution
SAT Verbal Scores
From the table, P(Z >1.28) = 0.10
Need to reverse standardize to get X:
X = σ⋅Z + μ = 110⋅ = 646
So, a student needs a Verbal SAT score of 646 in order to be in the top 10% of all students
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24. Stat 111 - Lecture 7 - Normal Distribution
Next Class - Lecture 8
Chapter 5: Sampling Distributions
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