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Today’s Agenda
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 8 green marbles, 2 red marbles, 10 total p(Green) = .8 p(Red) = .2
Frequencies & Probability 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
32 2.3 2.4 2.4
171 12.3 13.0 15.4
186 13.4 14.1 29.5
486 35.0 36.8 66.3
205 14.8 15.5 81.9
198 14.3 15.0 96.9
41 3.0 3.1 100.0
1319 95.1 100.0
62 4.5
6 .4
68 4.9
1387 100.0
1 EXTREMELY LIBERAL
2 LIBERAL
3 SLIGHTLY LIBERAL
4 MODERATE
5 SLGHTLYCONSERVATIVE
6 CONSERVATIVE
7 EXTRMLYCONSERVATIVE
Total
Valid
8 DK
9 NA
Total
Missing
Total
Frequency Percent Valid PercentCumulative
Percent
PROBABILITY & THE NORMAL DISTRIBUTION
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
PROBABILITY & THE NORMAL DISTRIBUTION
The probability of a particular outcome is the proportion of times that outcome would occur in a long run of repeated observations.
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%
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?
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%
ALL 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 distributionProbabilities always range from 0 – 1
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
μ = 4.5 (N=50)
x=7x=0 x=3x=1 x=5x=8 x=5 x=3
x=8 x=7x=4 x=6
x=2 x=8 x=4 x=5 x=9 x=4
x=5 x=9x=3 x=0x=6 x=5
x=1 x=7 x=3x=4 x=5x=6
EXAMPLE OF HOW SAMPLE STATISTICSVARY FROM A POPULATION PARAMETER
X=4.0
X=5.5
X=4.3
X=5.3 X=4.7
CHILDREN’S AGE IN YEARS
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
2. Allow us to apply probability theory– 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
μ = 4.5 (N=50)
x=7x=0 x=3x=1 x=5x=8 x=5 x=3
x=8 x=7x=4 x=6
x=2 x=8 x=4 x=5 x=9 x=4
x=5 x=9x=3 x=0x=6 x=5
x=1 x=7 x=3x=4 x=5x=6
EXAMPLE OF HOW SAMPLE STATISTICSVARY FROM A POPULATION PARAMETER
X=4.0
X=5.5
X=4.3
X=5.3 X=4.7
CHILDREN’S AGE IN YEARS
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
POPULATION
SAMPLING DISTRIBUTION
(Distribution of sample outcomes)
SAMPLE
•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)
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