Lecture 6
1Outline
1. Random Variablesa. Discrete Random Variables
b. Continuous Random Variables
2. Symmetric Distributions
3. Normal Distributions
4. The Standard Normal Distribution
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21. Random Variables
Two kinds of random variables:
a. Discrete (DRV) Outcomes have countable values Possible values can be listed E.g., # of people in this room
Possible values can be listed: might be …28 or 29 or 30…
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31. Random Variables
Two kinds of random variables:
b. Continuous (CRV) Not countable Consists of points in an interval E.g., time till coffee break
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41. Random Variables
The form of the probability distribution for a CRV is a smooth curve. Such a distribution may also be called a
Frequency Distribution Probability Density Function
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51. Random Variables
In the graph of a CRV, the X axis is whatever you are measuring (e.g., exam scores, depression scores, # of widgets produced per hour).
The Y axis measures the frequency of scores.
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X
The Y-axis measures frequency. It is usually not shown.
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72. Symmetric Distributions
In a symmetric CRV, 50% of the area under the curve is in each half of the distribution.
P(x ≤ ) = P(x ≥ ) = .5
Note: Because points are infinitely thin, we can only measure the probability of intervals of X values – not of individual X values.
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8
µ
50% of area
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93. Normal Distributions
A particularly important set of CRVs have probability distributions of a particular shape: mound-shaped and symmetric. These are “normal distributions”
Many naturally-occurring variables are normally distributed.
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10Normal Distributions
are perfectly symmetrical around their mean, .
have the standard deviation, , which measures the “spread” of a distribution – an index of variability around the mean.
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11
µ
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12Standard Normal Distribution
The area under the curve between and some value X ≥ has been calculated for the “standard normal distribution” and is given in the Z table (Table IV).
E.g., for Z = 1.62, area = .4474
(Note that for the mean, Z = 0.)
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XZ = 1.62Z = 0
Area gives the probability of finding a score between the mean and X when you make an observation
.4474
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14Using the Standard Normal Distribution
Suppose average height for Canadian women is 160 cm, with = 15 cm.
What is the probability that the next Canadian woman we meet is more than 175 cm tall?
Note that this is a question about a single case and that it specifies an interval.
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15Using the Standard Normal Distribution
160 175
We need this areaTable gives this area
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Remember that area above the mean, , is half (.5) of the distribution.
µ
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17Using the Standard Normal Distribution
160 175
Call this shaded area P. We can get P from Table IV
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18Using the Standard Normal Distribution
Z = X - = 175-160
15
= 1.00
Now, look up Z = 1.00 in the table.
Corresponding area (= probability) is P = .3413.
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19Using the Standard Normal Distribution
160 175
This area is .3413
So this area must be .5 – .3413 = .1587
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20Using the Standard Normal Distribution
Z = 0 Z = 1.0
This area is .3413
So this area must be .5 – .3413 = .1587
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21Using the Standard Normal Distribution
What is the probability that the next Canadian woman we meet is more than 175 cm tall?
Answer: .1587
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22Review
Area under curve gives probability of finding X in a given interval. Area under the curve for Standard Normal Distribution is given in Table IV. For area under the curve for other normally-distributed variables first compute:
Z = X -
Then look up Z in Table IV.