Download Math 314: Statistics - Chapter 17: The Expected Value and the

Math 314: Statistics
Chapter 17: The Expected Value and the Standard Error
Dr. Ralph Wojtowicz
CME Department
Introduction
SD Shortcut
1
Introduction
2
SD Shortcut
3
Using the Normal Curve
Math 314: Statistics
Using the Normal Curve
Chapter 17: The Expected Value and the Standard Error
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Introduction
SD Shortcut
Using the Normal Curve
Expected Value and Standard Error
The sum of draws made at random with replacement from a box is likely to
be around its expected value, but to be off by a chance error similar in size
to the standard error.
The expected value for the sum of draws made at random with replacement
from a box equals
EV = (number of draws) × (average of box)
The standard error for the sum of draws made at random with replacement
from a box equals
√
SE = number of draws × (SD of box)
The average and SD are properties of the box.
The expected value (EV) and standard error (SE) are properties of the
random sample.
Math 314: Statistics
Chapter 17: The Expected Value and the Standard Error
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Introduction
SD Shortcut
Using the Normal Curve
SD Shortcut
If a box contains only two different numbers, a big one and a small one
(possibly many different times each), the SD of the box equals
s
fraction with
fraction with
big
small
×
×
−
big number
small number
number
number
Example: For a box containing only H = 1 and T = 0, the SD is
r
r
1
1 1
1
× =
=
(1 − 0) ×
2 2
4
2
Example: For a box containing 2 red beads (R = 1) and 8 green beads
(G = 0), the SD is
r
r
2
16
8
4
2
×
=
=
=
(1 − 0) ×
10 10
100
10
5
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Chapter 17: The Expected Value and the Standard Error
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Introduction
SD Shortcut
Using the Normal Curve
Using the Normal Curve
We can use (areas under) the normal curve to estimate the probability that
the sum of random draws from a box will fall within a given range.
To estimate the probability that the sum will be in the interval [a, b]:
Let EV be the expected value of the sum and let SE be the standard
error of the sum.
E
.
Let z1 = a−SEEV and let z2 = b−
SE
Calculate the area under the normal curve between z1 and z2 .
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Chapter 17: The Expected Value and the Standard Error
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