Download October 7th lecture

To the world you may be
one person,
but
to one person
you
may be the world.
Probability
The probability of an event is the
proportion of times we would expect
the event to occur in an infinitely long
series of identical sampling
experiments.
Probability
If all the possible outcomes are equally
likely, the probability of the
occurrence of an event is equal to the
proportion of the possible outcomes
characterized by the event.
Probability
Probability is a very useful notion in
situations involving at least some
degree of uncertainty; it gives us a
way of expressing the degree of
assurance that a particular event
will occur.
Probability
Chance factors inherent in forming
samples always affect sample results.
Sample results must therefore be
interpreted with that in mind.
Probability
The probability of occurrence of either
one event OR another OR… is
obtained by adding their individual
probabilities, provided the events are
mutually exclusive.
Probability
The probability of the joint occurrence
of one event AND another AND
another AND another… is obtained by
multiplying their separate
probabilities, provided the events are
independent.
Probability
Any relative frequency distribution may
be interpreted as a probability
distribution.
Probability
Grade
Relative Frequency
A
.15
B
.30
C
.40
D
.10
F
.05
Pr = .15 + .30 + .40 = .85
(85% chance)
Confidence Statement
Statistic ± Margin of Error
Margin of Error
Confidence Interval
A range of values constructed from
sample data so the parameter occurs
within that range at a specified
probability. The specified level of
probability is called the level of
confidence.
Confidence Interval for a
Sample Mean
Confidence Interval
95% Confidence Interval
95% of those surveyed will fall into a certain range
surrounding the mean
Confidence Interval
The average size of a mortgage applied
for in 1993 was $116,991 as opposed
to $119,999 in 1992. A sample of 64
mortgages showed that the standard
deviation of the amount applied for
was $6019. Find a 95% confidence
interval for the average size of a
mortgage applied for in 1993.
Confidence Interval
n = 64
x = $116,991
s = 6019
Ci = 95%
Z = 1.96
Confidence Interval
X ± Z( s / √n) =
116,991 ± 1.96( 6019 / √64 ) =
116,991 ± 1.96(752.38) =
116,991 ± 1474.66 =
115516.34 to 118465.66
Confidence Interval
According to the Family Economic Research
Group of the US Department of Agriculture,
middle income couples who had babies in
1992 will spend an average of $128,670 by
the time the baby is 18 years old. Assume
the standard deviation of a sample of 100
families was $8473. Find a 90%
confidence interval for the average cost to
raise a child born in 1992.
Confidence Interval
n = 100
s = 8473
X = 128,670
Ci = 90%
Z = 1.645
Confidence Interval
X ± Z( s / √n) =
128,670 ± 1.645( 8473 / √100) =
128,670 ± 1.645(847.3) =
128670 ± 1393.81 =
127276.19 to 130063.81
Confidence Interval for a
Sample Proportion
p
p(1-p)
n
Confidence Interval for a
Sample Proportion
Suppose 1,600 of 2000 union members
sampled said they plan to vote for the
proposal to merge with the UMA. Using
the .95 level of confidence, what is the
interval estimate for the population
proportion? Based on the confidence
interval,what conclusion can be drawn?
Confidence Interval for a
Sample Proportion
p+z
P(1-p)
n
= .80 + 1.96
=.80 + 1.96
.80(1-.80)
2,000
.00008
= .782 and .818
Point Estimate
A value, computed from sample
information that is used to estimate
the population parameter