1. Hand out z tables

2. Introduction to Statistics for the Social Sciences SBS200, COMM200, GEOG200, PA200, POL200, or SOC200 Lecture Section 001, Spring 2016 Room 150 Harvill Building 9:00 - 9:50 Mondays, Wednesdays & Fridays
Welcome

4. Before next exam (March 4th)
Schedule of readings
Before next exam (March 4th)
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

5. By the end of lecture today 2/19/16
Counting ‘standard deviationses’ – z scores
Connecting raw scores, z scores and probability Connecting probability, proportion and area of curve
Percentiles

6. Homework Assignment Assignment 10
Please complete the homework modules on the D2L website
HW10-Normal Curve, z scores and probabilities
Due: Monday, February 22nd
Assignment 11
Please complete the homework worksheet
Finding z scores and areas under the curve.
Also Due: Monday, February 22nd

7. Lab sessions Everyone will want to be enrolled
in one of the lab sessions
Labs continue
With Project 2

10. Scores, standard deviations, and probabilities
The normal curve always has the same shape.
They differ only by having different means and standard deviation

11. z table Formula 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

12. Writing Assignment Let’s do some problems
Mean = 50 Standard deviation = 10
Writing Assignment Let’s do some problems

13. Find the percentile rank for score of 60
Mean = 50 Standard deviation = 10 ?
Let’s do some problems 60
Find the area under the curve that falls below 60
means the same thing as
Find the percentile rank for score of 60
Problem 1

14. Find the percentile rank for score of 60
Mean = 50 Standard deviation = 10 ?
Let’s do some problems
Find the percentile rank for score of 60 60
.5000
.3413
1) Find z score
z score = 10
= 1
2) Go to z table
- find area under correct column (.3413)
3) Look at your picture - add to = .8413
4) Percentile rank or score of 60 = 84.13%
Problem 1
Hint always draw a picture!

15. Find the percentile rank for score of 75
Mean = 50 Standard deviation = 10 ?
Find the percentile rank for score of 75 75
.4938
1) Find z score
z score = 10
z score = 10
= 2.5
2) Go to z table
Problem 2
Hint always draw a picture!

16. Find the percentile rank for score of 75
Mean = 50 Standard deviation = 10 ?
Find the percentile rank for score of 75 75
.4938
.5000
1) Find z score
z score = 10
z score = 10
= 2.5
2) Go to z table
3) Look at your picture - add to = .9938
4) Percentile rank or score of 75 = 99.38%
Problem 2
Hint always draw a picture!

17. Find the percentile rank for score of 45
Mean = 50 Standard deviation = 10 ?
Find the percentile rank for score of 45 45
1) Find z score
z score = 10
z score = 10
= -0.5
2) Go to z table
Problem 3

18. Find the percentile rank for score of 45
Mean = 50 Standard deviation = 10 ?
Find the percentile rank for score of 45
.1915 45 ?
1) Find z score
z score = 10
z score = 10
= -0.5
2) Go to z table
Problem 3

19. Find the percentile rank for score of 45
Mean = 50 Standard deviation = 10 ?
Find the percentile rank for score of 45
.1915 45
.3085
1) Find z score
z score = 10
z score = 10
= -0.5
2) Go to z table
3) Look at your picture - subtract = .3085
Problem 3
4) Percentile rank or score of 45 = 30.85%

20. Find the percentile rank for score of 55
Mean = 50 Standard deviation = 10 ?
Find the percentile rank for score of 55 55
1) Find z score
z score = 10
z score = 10
= 0.5
2) Go to z table
Problem 4

21. Find the percentile rank for score of 55
Mean = 50 Standard deviation = 10 ?
Find the percentile rank for score of 55 55
.1915
1) Find z score
z score = 10
z score = 10
= 0.5
2) Go to z table
Problem 4

22. Find the percentile rank for score of 55
Mean = 50 Standard deviation = 10 ?
Find the percentile rank for score of 55 55
.1915 .5
1) Find z score
z score = 10
z score = 10
= 0.5
2) Go to z table
3) Look at your picture - add = .6915
4) Percentile rank or score of 55 = 69.15%
Problem 4

23. Find the score that is associated
Mean = 50 Standard deviation = 10 ?
Find the score for z = -2 30
Hint always draw a picture!
Find the score that is associated
with a z score of -2
raw score = mean + (z score)(standard deviation)
Raw score = 50 + (-2)(10)
Raw score = (-20) = 30
Please note: When we are looking for the score from proportion we use the z-table ‘backwards’. We find the closest z to match our proportion

24. percentile rank of 77%ile
Mean = 50 Standard deviation = 10 ?
Find the score for
percentile rank of 77%ile
.7700 ?
Please note: When we are looking for the score from proportion we use the z-table ‘backwards’. We find the closest z to match our proportion
Problem 5

25. percentile rank of 77%ile
Mean = 50 Standard deviation = 10
.27 ?
Find the score for
percentile rank of 77%ile .5
= .77 .5
.27
.7700 ?
1) Go to z table - find z score for
for area ( ) = .27
Please note: When we are looking for the score from proportion we use the z-table ‘backwards’. We find the closest z to match our proportion
area = .2704
(closest I could find to .2700)
z = 0.74
Problem 5

26. percentile rank of 77%ile
Mean = 50 Standard deviation = 10
.27 ?
Find the score for
percentile rank of 77%ile .5
x = 57.4 .5
.27
.7700 ?
2) x = mean + (z)(standard deviation)
x = 50 + (0.74)(10)
x = 57.4
x = 57.4
Please note: When we are looking for the score from proportion we use the z-table ‘backwards’. We find the closest z to match our proportion
Problem 5

27. percentile rank of 55%ile
Mean = 50 Standard deviation = 10 ?
Find the score for
percentile rank of 55%ile
.5500 ?
Please note: When we are looking for the score from proportion we use the z-table ‘backwards’. We find the closest z to match our proportion
Problem 6

28. percentile rank of 55%ile
Mean = 50 Standard deviation = 10
.05 ?
Find the score for
percentile rank of 55%ile .5
= .55 .5
.05
.5500 ?
1) Go to z table - find z score for
for area ( ) = .05
Please note: When we are looking for the score from proportion we use the z-table ‘backwards’. We find the closest z to match our proportion
area = .0517
(closest I could find to .0500)
z = 0.13
Problem 7

29. percentile rank of 55%ile
Mean = 50 Standard deviation = 10
.05 ?
Find the score for
percentile rank of 55%ile .5 .5
.05
.5500 ?
1) Go to z table - find z score for
for area ( ) = .05
Please note: When we are looking for the score from proportion we use the z-table ‘backwards’. We find the closest z to match our proportion
area = .0517
(closest I could find to .0500)
z = 0.13
Problem 7

30. percentile rank of 55%ile
Mean = 50 Standard deviation = 10
.05 ?
Find the score for
percentile rank of 55%ile .5
x = 51.3 .5
.05
.5500 ?
1) Go to z table - find z score for
for area ( ) = .0500
area = .0517
(closest I could find to .0500)
z = 0.13
2) x = mean + (z)(standard deviation)
x = 50 + (0.13)(10)
x = 51.3
Please note: When we are looking for the score from proportion we use the z-table ‘backwards’. We find the closest z to match our proportion
x = 51.3
Problem 7

31. Normal Distribution has a mean of 50 and standard deviation of 4.
Determine value below which 95% of observations will occur Note: sounds like a percentile rank problem
Go to
table
.4500
nearest z = 1.64
x = mean + z σ = 50 + (1.64)(4) = 56.56
.9500
.4500
.5000
Additional practice
Problem 8 38 42 46 50 54
56.60 ? 58 62

32. Normal Distribution has a mean of $2,100 and s.d. of $250.
What is the operating cost for the lowest 3% of airplanes Note: sounds like a percentile rank problem = find score for 3rd percentile
Go to
table
.4700
nearest z =
x = mean + z σ = (-1.88)(250) = 1,630
.0300
.4700
Additional practice
Problem 9
1,630 ?
2100

33. Normal Distribution has a mean of 195 and standard deviation of 8.5.
Determine value for top 1% of hours listened.
Go to
table
.4900
nearest z = 2.33
x = mean + z σ = (2.33)(8.5) =
.4900
.5000
.0100
Additional practice
Problem 10
195
214.8 ?

34. ? .7500 .25 .5000 Find score associated with the 75th percentile .
Go to
table
nearest z = .67
.2500
x = mean + z σ = 30 + (.67)(2) = 31.34
.7500
.25
.5000 24 26 28 30 ? 34 36
31.34
Additional practice
Problem 11
z = .67

35. ? .2500 .25 .25 Find the score associated with the 25th percentile .
Go to
table
nearest z = -.67
.2500
x = mean + z σ = 30 + (-.67)(2) = 28.66
.2500
.25
.25 24 26
28.66 28 ? 30 34 36
Additional practice
Problem 12
z = -.67

36. Mean of 30 and standard deviation of 2.
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
Additional practice
Problem 13
26.08
33.92 24 ? 28 30 32 ? 36

37. Mean of 100 and standard deviation of 5.
Try this one:
Please find the (2) raw scores that border exactly the middle 95% of the curve
Mean of 100 and standard deviation of 5
Go to
table
.4750
nearest z = 1.96
mean + z σ = (1.96)(5) =
Go to
table
.4750
nearest z = -1.96
mean + z σ = (-1.96)(5) = 90.20
.9500
.475
.475
Additional practice
Problem 14
90.2
109.8 85 ? 95
100
105 ?
115

38. Mean of 30 and standard deviation of 2.
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
.4750
nearest z = 1.96
mean + z σ = 30 + (2.58)(2) = 35.16
Go to
table
.4750
nearest z = -1.96
mean + z σ = 30 + (-2.58)(2) = 24.84
.9900
.495
.495
Additional practice
Problem 15
24.84 ?
35.16 28 32 ? 30

39. Thank you!
See you next time!!