1. Introduction to Statistics for the Social Sciences SBS200 - Lecture Section 001, Fall 2018 Room 150 Harvill Building 10: :50 Mondays, Wednesdays & Fridays.
Welcome
10/15/18

2. Alyson Lecturer’s desk Chris Flo Jun Trey Projection Booth Screen
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Harvill 150
renumbered

3. ..
The
Green
Sheets

5. Before next exam (November 16th)
Schedule of readings
Before next exam (November 16th)
Please read chapters in OpenStax textbook
Please read Chapters 2, 3, and 4 in Plous
Chapter 2: Cognitive Dissonance
Chapter 3: Memory and Hindsight Bias
Chapter 4: Context Dependence

6. Lab sessions
Presentations
Project 2

9. What are they for? Confidence Intervals:
Combining these three skills to build confidence intervals
What are they for? Confidence Intervals:
We are estimating a value but providing two scores between which we believe the true value lies. We can be 95% confident that our mean falls between these two scores.
Central Limit Theorem
Central Limit Theorem highlights importance of standard error of the mean (which is built on standard deviation)
Using standardized scores (z scores) to find raw score values that border the middle part of the curve.
Combining these three skills to build confidence intervals
What confidence intervals are most useful for
Finding scores that border the middle 95% of curve
Using what we know about Central Limit Theorem to use the standard error of the mean rather than the standard deviation
Building towards confidence intervals

10. Central Limit Theorem

11. x will approach µ Central Limit Theorem
Proposition 1: If sample size (n) is large enough (e.g. 100)
The mean of the sampling distribution will
approach the mean of the population
As n ↑
x will approach µ
Proposition 2: If sample size (n) is large enough (e.g. 100)
The sampling distribution of means will
be approximately normal, regardless of the
shape of the population
As n ↑ curve
will approach normal shape
Proposition 3: The standard deviation of the sampling
distribution equals the standard deviation of
the population divided by the square root of the
sample size. As n increases SEM decreases.
As n ↑ curve
variability gets smaller X

12. Sampling distribution: is a theoretical probability distribution of
the possible values of some sample statistic that would
occur if we were to draw an infinite number of same-sized
samples from a population
Eugene
Frequency distributions of individual scores
derived empirically
we are plotting raw data
this is a single sample X X
Melvin X X X X X X X X X X X
23rd sample
Sampling distributions sample means
theoretical distribution
we are plotting means of samples
2nd sample

13. Distribution of raw scores: is an empirical probability distribution
Central Limit Theorem
Distribution of raw scores: is an empirical probability distribution
of the values from a sample of raw scores from a population
Frequency distributions of individual scores
derived empirically
we are plotting raw data
this is a single sample
Eugene X X X X X X X
Melvin X X X X X X
Take a single score
Repeat
over and over x x x
Population x x x x

14. same-sized sample = “fixed n”
Central Limit Theorem
Sampling distribution: is a theoretical probability distribution of
the possible values of some sample statistic that would
occur if we were to draw an infinite number of same-sized
samples from a population
important note:
same-sized sample = “fixed n”
Sampling distributions of sample means
theoretical distribution
we are plotting means of samples
Take sample – get mean
Repeat
over and over
Population

15. Sampling distribution: is a theoretical probability distribution of
Central Limit Theorem
Sampling distribution: is a theoretical probability distribution of
the possible values of some sample statistic that would
occur if we were to draw an infinite number of same-sized
samples from a population
important note:
same-sized sample = “fixed n”
Sampling distributions of sample means
theoretical distribution
we are plotting means of samples
Take sample – get mean
Repeat
over and over
Population
Distribution of means of samples

16. Central Limit Theorem: If random samples of a fixed N are drawn
from any population (regardless of the shape of the
population distribution), as N becomes larger, the
distribution of sample means approaches normality, with
the overall mean approaching the theoretical population
mean.
Distribution of Raw Scores
Sampling Distribution of Sample means
Melvin X
Eugene
23rd sample X
2nd sample

17. Central Limit Theorem µ = 100 Population Take sample – get mean Repeat
Question:
What if we made our sample absurdly large, as large as possible??
What if our sample size was HUGE - as large as the whole population?
Take sample – get mean
Repeat
over and over
Distribution of means of samples
100
Population
µ = 100

18. Central Limit Theorem µ = 100 Population Take sample – get mean Repeat
Question:
What if we made our sample absurdly large, as large as possible??
What if our sample size was HUGE - as large as the whole population?
What is the variability??
It is zero!
Take sample – get mean
Repeat
over and over
Distribution of means of samples
100
Population
µ = 100

19. x will approach µ Central Limit Theorem
Proposition 1: If sample size (n) is large enough (e.g. 100)
The mean of the sampling distribution will
approach the mean of the population
As n ↑
x will approach µ
Proposition 2: If sample size (n) is large enough (e.g. 100)
The sampling distribution of means will
be approximately normal, regardless of the
shape of the population
As n ↑ curve
will approach normal shape
Proposition 3: The standard deviation of the sampling
distribution equals the standard deviation of
the population divided by the square root of the
sample size. As n increases SEM decreases.
As n ↑ curve
variability gets smaller X

20. Calculating Two Scores that Border Middle 95% of curve
Step 1: Find mean (expected value)
x is mean of sample
Step 2: Find standard deviation  “σ” if population
Step 3: Decide on area of interest
- Middle 95% α = .05  z = 1.96
- Middle 99% α = .01  z = 2.58
Step 4: Calculations for confidence intervals
Step 5: Conclusion
- tie findings with statement about what you are estimating

21. 95% ? ? Find the scores that border the middle 95%
Mean = 50 Standard deviation = 10
Find the scores that border
the middle 95% ? ?
95%
x = mean ± (z)(standard deviation)
30.4
69.6
.9500
.4750
.4750
Please note:
We will be using this same logic for “confidence intervals” ? ?
1) Go to z table - find z score for
for area .4750
z = 1.96
2) x = mean + (z)(standard deviation)
x = 50 + (-1.96)(10)
x = 30.4
30.4
3) x = mean + (z)(standard deviation)
x = 50 + (1.96)(10)
x = 69.6
69.6
Scores capture the
middle 95% of the curve

22. Calculating Confidence Intervals
Step 1: Find mean (expected value)
x is mean of sample
Step 2: Find standard deviation and standard error of the mean
“σ” if population
Step 3: Decide on level of confidence
- 95% Confident α = .05  z = 1.96
- 99% Confident α = .01  z = 2.58
Step 4: Calculations for confidence intervals
Step 5: Conclusion
- tie findings with statement about what you are estimating

23. σ n 95% ? ? 10 = = √ 100 √ Construct a 95% confidence interval
Mean = 50 Standard deviation = 10
n = 100
s.e.m. = 1 ? ?
95%
48.04
51.96 ?
.9500
.4750
For “confidence intervals”
same logic – same z-score
But - we’ll replace standard deviation with the standard error of the mean
standard error
of the mean σ √ n = 10 =
x = mean ± (z)(s.e.m.) √
100
x = 50 + (1.96)(1)
x =
x = 50 + (-1.96)(1)
x =
95% Confidence Interval
is captured by the scores – 51.96

24. ? ? Construct a 95 percent confidence interval around the mean
95% ?
We know this
raw score = mean ± (z score)(s.d.)
Some Mean Some Variability
We used this one when finding raw scores
associated with an area under the curve.
We had all population info. Not really a
“confidence interval” because we know the mean of the population,
so there is nothing to estimate or be “confident about”.
Hint always draw a picture!
We used this one when finding raw scores associated with an
area under the curve. We used this to provide an interval within
which we believe the mean falls. We have some level of confidence about our guess. We know the population standard deviation.
raw score = mean ± (z score)(s.e.m.)
Similar, but uses
standard error the mean
based on population s.d.

25. Its own mean  x Its own variability  σx
Construct a 95 percent confidence
interval around the mean ?
95% ?
Its own mean  x Its own variability  σx
Choose confidence level  z
raw score = mean ± (z score)(s.e.m.)
We used this one when finding an interval within which we believe the mean falls. We have some level of confidence about our guess. We know the population standard deviation.

26. Thank you!
See you next time!!