1. Introduction to Statistics for the Social Sciences SBS200 - Lecture Section 001, Fall 2016 Room 150 Harvill Building 10: :50 Mondays, Wednesdays & Fridays.
Welcome

3. By the end of lecture today 10/21/16
Law of Large Numbers
Central Limit Theorem

4. Before next exam (November 18th)
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

5. Lab sessions Everyone will want to be enrolled
in one of the lab sessions
Labs continue
Next week
With Project 3

6. Homework Assignments 15 & 16
On class website:
Homework Assignments 15 & 16
Please complete the homework modules on the D2L website
HW15-Confidence Intervals
Please complete this homework worksheet 16 - Confidence Intervals
Both are due: Monday, October 24th

7. Review Central Limit Theorem
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
Review

8. x will approach µ Review 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
Review

9. Animation for creating sampling distribution of sample means
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
Animation for creating
sampling distribution
of sample means
Distribution of single sample
Eugene
Sampling Distribution of Sample means
Mean for sample 12
Sampling Distribution of Sample means
Mean for sample 7

10. What are the three propositions of the Central Limit Theorem?
As n goes up …
1. Sample mean approaches
true population mean
2. Curve becomes more “normal”
3. Variability goes down
(includes standard deviation, variance, width of the curve, and random error)

11. What is the formula for the standard error of the mean?

12. What are confidence intervals for…
Estimating a value (could be a single score, a mean of a sample or a mean of a population) by providing a range within we believe (with a certain level of confidence) it falls

13. Confidence Intervals: What are they used for?
We are estimating a value by providing two scores between which we believe the true value lies. We can be 95% confident that our mean falls between these two scores.
95% Confidence Interval: We can be 95% confident that our population mean falls between these two scores
99% Confidence Interval: We can be 99% confident that our population mean falls between these two scores

14. Confidence Intervals: What are they used for?
We are using this to estimate a value such as a mean,
with a known degree of certainty with a range of values
The interval refers to possible values of the population mean.
We can be reasonably confident that the population mean
falls in this range (90%, 95%, or 99% confident)
Subjective vs
Empirical
In the long run, series of intervals, like the one we
figured out will describe the population mean about 95%
of the time.
Greater confidence implies loss of precision. (95% confidence is most often used)
Can actually generate CI for any confidence level you want – these are just the most common

15. How to find the two scores that border the middle 95% of the curve up …

16. Building towards confidence intervals
Normal distribution
Raw scores z-scores probabilities
Have z
Find raw score Z
Scores
Have z
Find area
z table
Formula
Have area
Find z
Area &
Probability
Have raw score
Find z
Raw Scores
Building towards confidence intervals

17. .
Try this one:
Please find the (2) raw scores that border exactly the middle 95% of the curve
Mean of 30 and standard deviation of 2
Go to
table
.4750
nearest z = 1.96
mean + z σ = 30 + (1.96)(2) = 33.92
Go to
table
.4750
nearest z = -1.96
mean + z σ = 30 + (-1.96)(2) = 26.08
.9500
.475
.475
Building towards confidence intervals
26.08 ?
33.92 28 32 ? 30

18. .
Try this one:
Please find the (2) raw scores that border exactly the middle 99% of the curve
Mean of 30 and standard deviation of 2
Go to
table
.4950
nearest z = 2.58
mean + z σ = 30 + (2.58)(2) = 35.16
Go to
table
.4950
nearest z = -2.58
mean + z σ = 30 + (-2.58)(2) = 24.84
.9900
.495
.495
Building towards confidence intervals
24.84 ?
35.16 28 32 ? 30

19. Finding the interval that borders the middle of the curve.
Finding the interval that borders the middle of the curve
Which is wider?
Please find the raw scores that
border the middle 95% of the curve
Building towards confidence intervals
Please find the raw scores that
border the middle 99% of the curve

20. Remember Confidence Intervals.
Remember
Confidence Intervals
95% Confidence Interval:
We can be 95% confident that the
estimated score really does fall
between these two scores
Please find the raw scores that
border the middle 95% of the curve
99% Confidence Interval:
We can be 99% confident that the
estimated score really does fall
between these two scores
Building towards confidence intervals
Please find the raw scores that
border the middle 99% of the curve
Part 1

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. 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

23. Confidence interval uses SEM
Homework Worksheet:
Confidence interval uses SEM

24. .99 2.58 sd 2.58 sd ? 55 ? Homework Worksheet: Problem 1
29.2
Upper boundary raw score
x = mean + (z)(standard deviation)
x = 55 + (+ 2.58)(10)
x = 80.8
80.8
Lower boundary raw score
x = mean + (z)(standard deviation)
x = 55 + (- 2.58)(10)
x = 29.2
Standard deviation = 10
Mean = 55
2.58 sd
2.58 sd
.99
29.2 ? 55
80.8 ?

25. .99 49 2.58 sem 2.58 sem 1.42 ? 55 ? Homework Worksheet: Problem 1
29.2
Upper boundary raw score
x = mean + (z)(standard error mean)
x = 55 + (+ 2.58)(1.42)
x = 58.7
80.8
51.3
58.7
Lower boundary raw score
x = mean + (z)(standard error mean)
x = 55 + (- 2.58)(1.42)
x = 51.3
Standard deviation = 10
Mean = 55 10 49
2.58 sem
2.58
sem
1.42
.99
51.3 ? 55
58.7 ?

26. Confidence Intervals: A range of values that, with a known degree of
certainty, includes an estimated value (like a mean)
How can we make our confidence interval smaller?
Decrease variability (make standard deviation smaller)
1. Increase sample size (This will decrease variability)
2. Very careful assessment and measurement practices (improve reliability will minimize noise)
Decrease level of confidence .
95%
95%

27. Thank you!
See you next time!!