1. Introduction to Statistics for the Social Sciences SBS200 - Lecture Section 001, Spring 2017 Room 150 Harvill Building 9:00 - 9:50 Mondays, Wednesdays & Fridays.
Welcome

2. A note on doodling

3. By the end of lecture today 2/24/17
Confidence Intervals
Central Limit Theorem

4. Due: Wednesday, March 1st
Homework
Assignment 15 & 16
Please complete the homework modules on the D2L website
HW15-Confidence Intervals
Please complete this homework worksheet 16
Confidence Intervals
Due: Wednesday, March 1st

5. Before next exam (March 3rd)
Schedule of readings
Before next exam (March 3rd)
Please read chapters in OpenStax textbook
Please read Chapters 10, 11, 12 and 14 in Plous
Chapter 10: The Representativeness Heuristic
Chapter 11: The Availability Heuristic
Chapter 12: Probability and Risk
Chapter 14: The Perception of Randomness

6. Lab sessions Everyone will want to be enrolled
in one of the lab sessions
Labs
Exam 2 Prep

8. Used in lots of different contexts. Sometimes with different names
Confidence Intervals
We need three foundational building blocks to create confidence intervals
Used in lots of different contexts. Sometimes with different names
What
is it
for?
How to
Find it
Why
SEM?

9. Now we have all the pieces we need to create confidence intervals

10. 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.
Building towards confidence intervals

11. Confidence Interval of 95% Has and alpha of 5% α = .05
Critical z
-2.58
Critical z
2.58
Confidence Interval of 99%
Has and alpha of 1%
α = .01
99%
Critical z separates rare from common scores
Critical z
-1.96
Critical z
1.96
Confidence Interval of 95% Has and alpha of 5% α = .05
95%
Area associated with most extreme scores is called alpha
Critical z
-1.64
Critical z
1.64
Confidence Interval of 90%
Has and alpha of 10%
α = . 10
90%
Area in the tails is called alpha

12. Writing Assignment: Part 1.
Writing Assignment: Part 1

13. ? ? .9500 .475 .475 . 28 32 30 Writing Assignment: Part 1
Question 1: Please find the (2) raw scores that exactly
border the middle 95% of the curve
Mean of 30 and standard deviation of 2 .
.9500
.475
.475
Building towards confidence intervals ? 28 30 32 ?
Part 1

14. ? ? .9900 .495 .495 . 28 32 30 Writing Assignment: Part 1
Question 2: Please find the (2) raw scores that exactly
border the middle 99% of the curve
Mean of 30 and standard deviation of 2 .
.9900
.495
.495
Building towards confidence intervals ? 28 30 32 ?
Part 1

15. ? ? .9500 .475 .475 . 25 35 30 Writing Assignment: Part 1
Question 3: Please find the (2) raw scores that exactly
border the middle 95% of the curve
Mean of 30 and standard deviation of 5 .
.9500
.475
.475
Building towards confidence intervals ? 25 30 35 ?
Part 1

16. Compare your results from questions 1 and 2. Which interval
is wider and why?
Question 1:
Please find the raw scores that
border the middle 95% of the curve
Question 2:
Building towards confidence intervals
Please find the raw scores that
border the middle 99% of the curve
Part 1

17. Compare your results from questions 1 and 3. Which interval
is wider and why?
Question 1:
Please find the raw scores that
border the middle 95% of the curve
Mean of 30 standard deviation of 2 .
Question 3:
Please find the raw scores that
border the middle 95% of the curve
Mean of 30 standard deviation of 5
Building towards confidence intervals
Part 1

18. Question 6: What are the two ways to make the confidence interval smaller (narrower)?.
95%
95%

20. Writing Assignment: Part 1.
Writing Assignment: Part 1
Solutions

21. .
Question 1: Please find the (2) raw scores that exactly
border 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
Part 1

22. .
Question 2: Please find the (2) raw scores that exactly
border 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
Part 1

23. .
Question 3: Please find the (2) raw scores that exactly
border the middle 95% of the curve
Mean of 30 and standard deviation of 5
Go to
table
.4750
nearest z = 1.96
mean + z σ = 30 + (1.96)(5) = 39.8
Go to
table
.4750
nearest z = -1.96
mean + z σ = 30 + (-1.96)(5) = 20.2
.9500
.475
.475
Building towards confidence intervals
20.2 ?
39.8 28 32 ? 30
Part 1

24. Compare your results from questions 1 and 2. Which interval
is wider and why?
26.08
33.92
Please find the raw scores that
border the middle 95% of the curve
24.84
35.16
Building towards confidence intervals
Please find the raw scores that
border the middle 99% of the curve
Part 1

25. Compare your results from questions 1 and 3. Which interval
is wider and why?
26.08
33.92
Please find the raw scores that
border the middle 95% of the curve
Mean of 30 standard deviation of 2
20.2
39.8
Please find the raw scores that
border the middle 95% of the curve
Mean of 30 standard deviation of 5
Building towards confidence intervals
Part 1

26. Question 6: What are the two ways to make the confidence interval smaller (narrower)?
Decrease level of confidence
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) .
95%
95%

27. 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
99% Confidence Interval:
We can be 99% confident that the
estimated score really does fall
between these two scores
Building towards confidence intervals
Part 1

28. 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 and standard error of the mean
“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

29. 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

30. 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

31. 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

32. ? ? 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.

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

34. Thank you!
See you next time!!