1. Reasoning in Psychology Using Statistics
2018

2. Announcements Quiz 2 Don’t forget Exam 1 is coming up (Feb 7) Today
Quiz 2 due Fri. Feb. 2 (11:59 pm)
Don’t forget Exam 1 is coming up (Feb 7)
In class part – multiple choice, closed book
In labs part – open book/notes
Today
Sampling and basic probability
Announcements

3. Ann Landers to readers, “If you had to do it again, would you have children?” (1975-76)
70% said kids not worth it!
Nearly 10,000 responses
Do you believe results?
Does the result accurately reflect population of parents?
Is the sample representative of all parents (the population)?
Landers was syndicated in over 1,200 newspapers, with an estimated readership of 90 million people. So the 10,000 people represent
approximately 0.01% of her readership.
- Source:
Sampling
David Bellhouse’s discussion of the 1976 Landers survey

4. Sampling Those research is about Population
Subset that participates in research (giving us our data)
Sample
Sampling

5. Sampling Population Sampling to make data collection manageable
Inferential statistics to generalize back
Sampling to make data collection manageable
Sample
Sampling

6. Sampling Today’s focus 2nd half of course focus Population
Inferential statistics to generalize back
Sampling to make data collection manageable
Sample
Sampling

7. Sampling Population (N=25) For rate hikes Against rate hikes
Local politician wants to know opinions on proposed rate hikes
For rate hikes
Against rate hikes
Proportion “for hikes” in population
# “for hikes”
Total # = 10 25
0.4
Sampling

8. Sampling Population (N=25) Sample (n=5) SAMPLING ERROR # “for hikes”
Sample is used to represent the population.
But, does the sample always match the population exactly?
SAMPLING ERROR
Proportion “for hikes” in sample
# “for hikes”
Total # = 2 5
0.4
Sampling

9. Sampling Error Population (N=25) Sample (n=5) # “for hikes”
Proportion “
for hikes” in population
# “for hikes”
Total # = 10 25
0.4
Proportion “for hikes” in sample
# “for hikes”
Total # = 2 5
0.4
parameter
statistic
Sampling error = = 0
Sampling Error

10. Sampling Error Population (N=25) Sample (n=5) # “for hikes”
Proportion “
for hikes” in population
# “for hikes”
Total # = 10 25
0.4
Proportion “for hikes” in sample
# “for hikes”
Total # = 3 5
0.6
parameter
statistic
Sampling error = = 0.2
Sampling Error

11. Sampling Goals of sampling: Reduce: Sampling error
Maximize: Representativeness
Minimize: Bias
Sampling

12. Sampling Goals of sampling: Reduce: Sampling error
difference between population parameter and sample statistic
BUT we usually don’t know what the population parameter is!
Maximize: Representativeness
Minimize: Bias
Sampling

13. Sampling Error Population (N=25) Proportion “ for hikes” in population
All possible Samples with (n=5)
Population (N=25)
0.6
0.4
0.6
0.4
0.4
0.6
0.6
0.4
0.4
0.6
0.6
0.4
0.4
0.4
0.6
0.6
1.0
0.2
0.0
1.0
0.2
0.0
0.4
0.4
0.6
0.6
Proportion “
for hikes” in population
# “for hikes”
Total # = 10 25
0.4
We will return to this concept when we begin our inferential statistics discussions
Distribution of Sampling proportions
The most frequent sample proportion will be 0.4, but you’ll get others too
0.04
parameter
Sampling Error

14. Formula we will learn later: SE = SD/√n
Use sample (statistic) to estimate population (parameter)
Problem: Samples vary
different estimates depending on sample
But we know what affects size of sampling error (we will demonstrate this mathematically later in the semester)
Variability in population
As variability increases, sampling error increases
Size of sample
As sample size increases, sampling error decreases
Formula we will learn later: SE = SD/√n
Parameter, Greek for besides the measure (compare paralegal, paramilitary)
Sampling Error

15. Sampling Goals of sampling Reduce: Sampling error
difference between population parameter and sample statistic
to what extent do characteristics of sample reflect those in population
systematic difference between sample and population
Maximize: Representativeness
Minimize: Bias
Sampling

16. Sampling Methods Probability sampling Non-probability sampling
Simple random sampling
Systematic random sampling
Stratified sampling
Convenience sampling
Quota sampling
Sampling Methods
 Click here for detailed Wikipedia entry

17. Sampling Methods Probability sampling Non-probability sampling
Simple random sampling
Systematic random sampling
Stratified sampling
Convenience sampling
Quota sampling
Every individual has equal & independent chance of being selected from population 3 2
Sampling Methods

18. Sampling Methods Probability sampling Non-probability sampling
Simple random sampling
Systematic random sampling
Stratified sampling
Convenience sampling
Quota sampling
Step 1: compute K = population size/sample size
Step 2: randomly select Kth person
23/6 = 3.8
K = 4 4 1
Sampling Methods

19. Sampling Methods Probability sampling Non-probability sampling
Simple random sampling
Systematic random sampling
Stratified sampling
Convenience sampling
Quota sampling
Step 2: randomly select from each group (proportional to size of group: 8/23= /23=.48 4/23=.17)
Step 1: Identify groups (strata)
blue
green
red
If n =5, 2 1
Sampling Methods

20. Sampling Methods Probability sampling Non-probability sampling
Simple random sampling
Systematic random sampling
Stratified sampling
Convenience sampling
Quota sampling
Step 1: Identify groups
blue
green
red
Step 2: pick first # from each group (not proportional)
If n =6, 2
Sampling Methods

21. Sampling Methods Probability sampling Non-probability sampling
Simple random sampling
Systematic random sampling
Stratified sampling
Convenience sampling
Quota sampling
70% of parents say kids not worth it!
Convenience sampling: voluntary response method of sampling
Using easily available participants
Results typically biased
Typical respondents with very strong opinions (NOT representative of population)
Newsday random sample (n = 1373) found 91% said “yes”
Gallop Poll (2013)
For more discussion: David Bellhouse
Sampling Methods

22. Sampling Methods Probability sampling Non-probability sampling
Simple random sampling
Systematic random sampling
Stratified sampling
Convenience sampling
Quota sampling
Good
Poor
Representativeness
Stacked Deck
Bias
Sampling Methods

23. Inferential statistics
Where does “probability” fit in?
Population
Randomness in sampling leads to variability in sampling error
“Randomness” in short run is unpredictable but in long run is predictable!
Odds in games of chance
Allows predictions about likelihood of getting particular samples
Possible Samples
Inferential statistics

24. Inferential statistics
Where does “probability” fit in?
Probability of 4 of a kind =
Probability of a sample with particular characteristics
If we know the proportions in the population
And we know how we sampled:
Deal 5 cards
Allows predictions about likelihood of getting particular samples
Inferential statistics
Tools that use our estimates of sampling error to generalize from observations from samples to statements about the populations
Flipping the logic

25. Basics of probability: Derived from games with all outcomes known
Draw lettered tiles from bag
Bag contains:
A’s B’s and C’s.
Both upper and lower case letters A a b B c C
What is the probability of getting an A (upper or lower case)?
Total number of outcomes classified as A
Prob. of A = p(A) =
Total number of possible outcomes
Sample space
Basics of probability: Derived from games with all outcomes known

26. Flipping a coin example: 1 flip
What are odds of getting heads?
One outcome classified as heads = 1 2
Total of two outcomes =
0.5
This simplest case is known as the binomial 2n
= 21 = 2 total outcomes
pn=(0.5)1= the prob of a single outcome
Flipping a coin example: 1 flip

27. Flipping a coin example: 2 flips
What are the odds of getting all heads?
Number of heads 2
Four total outcomes
One 2 heads outcome 1 =
0.25 1 2n
= 22 = 4 total outcomes
pn = (0.5)2 = 0.25 for 1 outcome twice in a row
Flipping a coin example: 2 flips
All heads on 3 flips? 23 = 8 outcomes
p3 = (0.5)3 = or ⅛

28. Flipping a coin example: 2 flips
What are the odds of getting only one heads?
Number of heads 2
Four total outcomes 1
Two 1 heads outcome =
0.50 1
Flipping a coin example: 2 flips

29. Flipping a coin example: 2 flips
What are the odds of getting at least one heads?
Number of heads 2
Four total outcomes
Three at least one heads outcome 1 =
0.75 1
Flipping a coin example: 2 flips

30. Flipping a coin example: 2 flips
What are the odds of getting no heads?
Number of heads 2
Four total outcomes 1
One no heads outcome =
0.25 1
Flipping a coin example: 2 flips
Skip poker odds

31. Flipping a coin example: 2 flips
What are the odds of getting all heads?
One 2 heads outcome
Four total outcomes =
0.25
# of heads 2
What are the odds of getting only one heads?
Four total outcomes =
0.50
Two 1 heads outcome 1
Four total outcomes =
0.75
Three at least one heads outcome
What are the odds of getting at least one heads? 1
What are the odds of getting no heads?
Four total outcomes =
0.25
One no heads outcome
Flipping a coin example: 2 flips
Summary Slide

32. Sampling Error: Odds in Poker
Population (N=52)
Lots of Samples (hands n=5)
Sampling Error: Odds in Poker
Lucky numbers: Marcus du Sautoy (~14 mins)

33. Odds in Poker What are the odds of being dealt a “Royal Flush”?
Total number of possible outcomes
Total number of outcomes classified as A
Prob. of A = p(A) = 4
p(Royal Flush) = =
2,598,960
~1.5 hands out of every million hands
Odds in Poker

34. Odds in Poker What are the odds of being dealt a “Straight Flush”?
Total number of possible outcomes
Total number of outcomes classified as A
Prob. of A = p(A) = 40
p(straightflush) = =
2,598,960
~15 hands out of every million hands
Odds in Poker

35. Odds in Poker What are the odds of being dealt a …?
Total number of possible outcomes
Total number of outcomes classified as A
Prob. of A = p(A) =
Odds in Poker

36. Inferential statistics
Where does “probability” fit into statistics?
Most research uses samples rather than populations.
The predictability in the long run, allows us to know quantify the probable size of the sampling error.
Inferential statistics use our estimates of sampling error to generalize from observations from samples to statements about the populations.
Inferential statistics

37. Wrap up Today’s lab: Try out sampling and probability Questions?
Breaking down probability sampling (~4 mins)
Sampling: Simple Random, Convenience, systematic, cluster, stratified (~4 mins)
Non-Probability Sampling (~4 mins)
Basics Probability and Statistics | Khan Academy (~8 mins)
Example 2 | Probability and Statistics | Khan Academy (~10 mins)
Probability with playing cards | Khan Academy (~10 mins)
Wrap up

38. What’s the probability of getting an A (upper or lower case) on the first pick and another on a second pick? A a b B c C
1/6
2/6 + a
First Pick:
Prob. of A = p(A) =
2/6 = 0.33 A
Second Pick: ? – it depends on how you sample
Sampling with replacement
Sampling without replacement
The probabilities of selecting the titles change from 1st to 2nd pick A a b B c C a b B c C A b B c C
2/6
1/5
1/5
Basics of probability

39. What’s the probability of getting an A (upper or lower case) on the first pick and another on a second pick? A a b B c C
Sampling with replacement
Sampling without replacement
(2/6)*(1/5) =
1st pick
2nd pick a A b B c C
30 total outcomes
2 outcomes of 2 A’s
2/30 =
1st picks
2nd picks
1st picks
2nd picks a a A b B c C A a A b B c C b a A b B c C B a A b B c C c a A b B c C C a A b B c C
36 total outcomes
4 outcomes of 2 A’s
4/36 = 0.11
(2/6)*(2/6) = 0.11
1st pick
2nd pick
Basics of probability

40. Most statistical procedures assume sampling with replacement
For large populations it turns out not to matter much
e.g., suppose your population is N=1,000,000. Starting probability of selecting a particular item 1 in 1,000,000.
Sampling with replacement, odds stay at 1 in 1,000,000
Sampling without replacement, odds change to 1 in 999,999 the change is so small that it may not matter
In experiments, you typically don’t want to use sampling with replacement because of the potential for lasting effects of your independent variable
Basics of probability

41. Sampling Error: Odds in Poker
Population (N=52)
Sample (n=5) 13
Proportion of spades = 52
in deck = 0.25 1
Proportion of spades = 5
in a draw = 0.20
parameter
statistic
Sampling error = 0.25 – 0.20 = 0.05
Sampling Error: Odds in Poker

42. Sampling Error: Odds in Poker
Population (N=52)
Sample (n=5) 13
Proportion of any suit = 52
in deck = 0.25 5
Proportion of suit =
in a draw = 1.0
parameter
statistic
Sampling error = 0.25 – 1.0 = 0.75
Sampling Error: Odds in Poker