Today

1
Todays Agenda

• Review Homework 1 not posted
• Probability ? Application to Normal Curve
• Inferential Statistics
• Sampling

2
Probability Basics

• What is the probability of picking a red marble
  out of a bowl with 2 red and 8 green?

p(red) 2 divided by 10 p(red) .20
3
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

4
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

5
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

6
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

7
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 gt 89) .0985 or 9.85
• ALL WE ARE DOING IS THINKING ABOUT AREA UNDER
  CURVE A BIT DIFFERENTLY (SAME MATH)

8
Probability the Normal Distribution

• Bottom line
• Normal distribution can also be thought of as
  probability distribution
• Probabilities 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

9
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

10
WHY SAMPLE?

• Why sample?
• Its 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)

11
WHY USE PROBABILITY SAMPLING?

• Representative sample
• One that, in the aggregate, closely approximates
  the population from which it is drawn

12
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

13
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)

14
INFERENTIAL STATISTICS

• Samples are only estimates of the population
• Sample statistics will be slightly off from the
  true values of its populations parameters
• Sampling error
• The difference between a sample statistic and a
  population parameter

15
EXAMPLE OF HOW SAMPLE STATISTICS VARY FROM A
POPULATION PARAMETER
x7 x0 x3 x1 x5 x8
x5 x3 x8 x7 x4 x6
X4.0
X5.5
µ 4.5 (N50)
x1 x7 x3 x4 x5 x6
CHILDRENS AGE IN YEARS
X4.3
x2 x8 x4 x5 x9 x4
x5 x9 x3 x0 x6 x5
X5.3
X4.7
16
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 doesnt exist

17
Nonprobability Sampling 2 Types

• CONVENIENCE SAMPLING
• When the researcher simply selects a requisite
  number of cases that are conveniently available
• SNOWBALL SAMPLING
• Researcher asks interviewed subjects to suggest
  additional people for interviewing

18
Probability vs. Nonprobability SamplingResearch
Situations

• For the following research situations, decide
  whether a probability or nonprobability sample
  would be more appropriate
• You plan to conduct research delving into the
  motivations of serial killers.
• You want to estimate the level of support among
  adult Duluthians for an increase in city taxes to
  fund more snow plows.
• You want to learn the prevalence of alcoholism
  among the homeless in Duluth.

19
(Back to Probability Sampling)The Catch-22 of
Inferential Stats

• When we collect a sample, we know nothing about
  the populations 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

20
PROBABILITY SAMPLING

• 2 Advantages of probability sampling
• Probability samples are typically more
  representative than other types of samples
• Allow us to apply probability theory
• This permits us to estimate the accuracy or
  representativeness of the sample

21
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

22
EXAMPLE OF HOW SAMPLE STATISTICS VARY FROM A
POPULATION PARAMETER
x7 x0 x3 x1 x5 x8
x5 x3 x8 x7 x4 x6
X4.0
X5.5
µ 4.5 (N50)
x1 x7 x3 x4 x5 x6
CHILDRENS AGE IN YEARS
X4.3
x2 x8 x4 x5 x9 x4
x5 x9 x3 x0 x6 x5
X5.3
X4.7
23
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

24
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)

25
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