Transcript Sampling/probability/inferential statistics

Today’s Agenda
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Review Homework #1 [not posted]
Probability  Application to Normal Curve
Inferential Statistics
Sampling
Probability Basics

What is the probability of picking a red marble out
of a bowl with 2 red and 8 green?
There are 2
outcomes that
are red
THERE ARE 10
POSSIBLE
OUTCOMES
p(red) = 2 divided by 10
p(red) = .20
Frequencies and Probability

The probability of picking a color relates to the
frequency of each color in the bowl
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
8 green marbles, 2 red marbles, 10 total
p(Green) = .8 p(Red) = .2
Frequencies & Probability
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What is the probability of randomly selecting an
individual who is extremely liberal from this sample?
p(extremely liberal) =
32 = .024 (or 2.4%)
1,319
THINK OF SELF AS LIBERAL OR CONSERVATIVE
Vali d
Mis sing
Total
1 EXTREMELY LIBERAL
2 LIBERAL
3 SLIGHTLY LIBERAL
4 MODERATE
5 SLGHTLY
CONSERVATIVE
6 CONSERVATIVE
7 EXTRMLY
CONSERVATIVE
Total
8 DK
9 NA
Total
Frequency
32
171
186
486
Percent
2.3
12.3
13.4
35.0
Vali d Percent
2.4
13.0
14.1
36.8
Cum ulative
Percent
2.4
15.4
29.5
66.3
205
14.8
15.5
81.9
198
14.3
15.0
96.9
41
3.0
3.1
100.0
1319
62
6
68
1387
95.1
4.5
.4
4.9
100.0
100.0
PROBABILITY & THE NORMAL
DISTRIBUTION
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We can use the normal
curve to estimate the
probability of randomly
selecting a case
between 2 scores
Probability distribution:
 Theoretical distribution
of all events in a
population of events,
with the relative
frequency of each
event
1.2
1.0
.8
.6
.4
.2
0.0
-2.07
-1.21
-.36
.50
1.36
Normal Curve, Mean = .5, SD = .7
2.21
3.07
PROBABILITY & THE NORMAL
DISTRIBUTION
1.2

The probability of a
particular outcome is the
proportion of times that
outcome would occur in a
long run of repeated
observations.
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
68% of cases fall within +/- 1
standard deviation of the
mean in the normal curve
The odds (probability) over
the long run of obtaining an
outcome within a standard
deviation of the mean is 68%
1.0
.8
.6
.4
.2
0.0
-2.07
-1.21
-.36
.50
1.36
Normal Curve, Mean = .5, SD = .7
2.21
3.07
Probability & the Normal Distribution
 Suppose
the mean score on a test is 80, with a
standard deviation of 7. If we randomly sample
one score from the population, what is the
probability that it will be as high or higher than
89?



 ALL
Z for 89 = 89-80/7 = 9/7 or 1.29
Area in tail for z of 1.29 = 0.0985
P(X > 89) = .0985 or 9.85%
WE ARE DOING IS THINKING ABOUT
“AREA UNDER CURVE” A BIT DIFFERENTLY
(SAME MATH)
Probability & the Normal Distribution

Bottom line:

Normal distribution can also be thought of as
probability distribution
 Probabilities always range from 0 – 1
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0 = never happens
1 = always happens
In between = happens some percent of the time
 This is where our interest lies
Inferential Statistics (intro)
– Inferential statistics are used to generalize
from a sample to a population
•
•
•
We seek knowledge about a whole class of
similar individuals, objects or events (called a
POPULATION)
We observe some of these (called a SAMPLE)
We extend (generalize) our findings to the entire
class
WHY SAMPLE?
– Why sample?
•
•
It’s often not possible to collect info. on all
individuals you wish to study
Even if possible, it might not be feasible (e.g.,
because of time, $, size of group)
WHY USE PROBABILITY
SAMPLING?
– Representative sample
• One that, in the aggregate, closely approximates
the population from which it is drawn
PROBABILITY SAMPLING
• Samples selected in accord with probability theory,
typically involving some random selection
mechanism
– If everyone in the population has an equal chance of
being selected, it is likely that those who are selected will
be representative of the whole group
» EPSEM – Equal Probability of SElection Method
PARAMETER & STATISTIC
• Population
– the total membership of a defined class of people, objects,
or events
• Parameter
– the summary description of a given variable in a
population
• Statistic
– the summary description of a variable in a sample (used
to estimate a population parameter)
INFERENTIAL STATISTICS
– Samples are only estimates of the population
– Sample statistics will be slightly off from the
true values of its population’s parameters
• Sampling error:
– The difference between a sample statistic and a
population parameter
EXAMPLE OF HOW SAMPLE STATISTICS
VARY FROM A POPULATION PARAMETER
x=7
x=0 x=3
x=1 x=5
x=8
X=4.0
x=5 x=3
x=8 x=7
x=4 x=6
X=5.5
μ = 4.5 (N=50)
x=1
x=7 x=3
x=4 x=5
x=6
X=4.3
CHILDREN’S
AGE IN YEARS
x=2 x=8
x=4 x=5
x=9 x=4
X=5.3
x=5 x=9
x=3 x=0
x=6 x=5
X=4.7
By Contrast:
Nonprobability Sampling
• Nonprobability sampling may be more appropriate
and practical than probability sampling:
– When it is not feasible to include many cases in the
sample (e.g., because of cost)
– In the early stages of investigating a problem (i.e.,
when conducting an exploratory study)
• It is the only viable means of case selection:
– If the population itself contains few cases
– If an adequate sampling frame doesn’t exist
Nonprobability Sampling: 2 Types
1. CONVENIENCE SAMPLING
–
When the researcher simply selects a requisite
number of cases that are conveniently available
2. SNOWBALL SAMPLING
–
Researcher asks interviewed subjects to suggest
additional people for interviewing
Probability vs. Nonprobability Sampling:
Research Situations
•
For the following research situations, decide whether a
probability or nonprobability sample would be more
appropriate:
1. You plan to conduct research delving into the
motivations of serial killers.
2. You want to estimate the level of support among
adult Duluthians for an increase in city taxes to fund
more snow plows.
3. You want to learn the prevalence of alcoholism
among the homeless in Duluth.
(Back to Probability Sampling…)
The “Catch-22” of Inferential Stats:
– When we collect a sample, we know nothing
about the population’s distribution of scores
• We can calculate the mean (X) & standard
deviation (s) of our sample, but  and  are
unknown
• The shape of the population distribution (normal?)
is also unknown
– Exceptions: IQ, height
PROBABILITY SAMPLING
–
2 Advantages of probability sampling:
1.
Probability samples are typically more
representative than other types of samples
Allow us to apply probability theory
2.
–
This permits us to estimate the accuracy or
representativeness of the sample
SAMPLING DISTRIBUTION
• Sampling Distribution
– From repeated random sampling, a
mathematical description of all possible
sampling event outcomes (and the probability
of each one)
– Permits us to make the link between sample
and population…
• & answer the question: “What is the probability that
sample statistic is due to chance?”
– Based on probability theory
EXAMPLE OF HOW SAMPLE STATISTICS
VARY FROM A POPULATION PARAMETER
x=7
x=0 x=3
x=1 x=5
x=8
X=4.0
x=5 x=3
x=8 x=7
x=4 x=6
X=5.5
μ = 4.5 (N=50)
x=1
x=7 x=3
x=4 x=5
x=6
X=4.3
CHILDREN’S
AGE IN YEARS
x=2 x=8
x=4 x=5
x=9 x=4
X=5.3
x=5 x=9
x=3 x=0
x=6 x=5
X=4.7
What would happen…
(Probability Theory)
• If we kept repeating the samples from the
previous slide millions of times?
– What would be our most common sample
mean?
• The population mean
– What would the distribution shape be?
• Normal
• This is the idea of a sampling distribution
– Sampling distribution of means
Relationship between Sample,
Sampling Distribution & Population
•Empirical (exists in reality)
but unknown
•Nonempirical (theoretical or
hypothetical)
Laws of probability allow us
to describe its characteristics
(shape, central tendency,
dispersion)
•Empirical & known (e.g.,
distribution shape, mean,
standard deviation)
POPULATION
SAMPLING
DISTRIBUTION
(Distribution of sample
outcomes)
SAMPLE
THE TERMINOLOGY OF INFERENTIAL
STATS
• Population
– the universe of students at the local college
• Sample
– 200 students (a subset of the student body)
• Parameter
– 25% of students (p=.25) reported being Catholic;
unknown, but inferred from sample statistic
• Statistic
– Empirical & known: proportion of sample that is Catholic
is 50/200 = p=.25
• Random Sampling (a.k.a. “Probability”)
– Ensures EPSEM & allows for use of sampling distribution
to estimate pop. parameter (infer from sample to pop.)
• Representative
– EPSEM gives best chance that the sample statistic will
accurately estimate the pop. parameter