psych 230 – statisticskrupinski.radiology.arizona.edu/documents/notes2.pdf · psych 230 –...

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PSYCH 230 – STATISTICS

1) If you are already registered sit down. 2) If you are on the waiting list or just showed up, stay standing and we will see how many seats are available. 3) We will start adding using the waiting list.

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PSYCHOLOGY 230 - STATS

Elizabeth Krupinski, PhD

Depts. Radiology & Psychology

112 Radiology Research Building

626-4498 [email protected]

http://krupinski.radiology.arizona.edu/psych230.htm

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CAMPUS

• North on Cherry • Left on Drachman • First right = Ring

Road but no signs • Around bend • Lot #1 (blue) on

right • Driveway into

fence on right

Rad Res 112

Speedway

Drachman

Ring Road

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PREREQUISITES

1) Psych 101 or IND 101 2) Math 110 – college algebra

+, x, -, ÷, √, , , | | positive vs negative numbers order of operations rounding: < 5 down, > 5 up decimals: 2 places on quizzes

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QUIZZES 4 quizzes - each 25% of your grade - 100 points each - all of them count (none dropped) ~ 1/3 fill-in-the-blank - comprehension of concepts - ability to apply principles, terms, etc. ~ 2/3 problems - ability identify appropriate equations - ability carry out required math - ability use statistical tables - ability reach proper conclusions formulas & tables provided on quizzes

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EXTRA CREDIT

1) Hand in a MAXIMUM of 5 completed homework assignments - 1 point each - 5 points maximum &

2) Hand in completed worksheet packet at end of semester - 10 points

3) Find a journal article with statistics in it; in 3 pages explain the statistics - why used, what tests, interpret etc. - 10 points - once only

15 POINTS MAXIMUM!!!!!!

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TEXTS Class notes: buy in the bookstore (required) http://www.radiology.arizona.edu/krupinski/index.html Book: Fundamentals of Behavioral Statistics 9th edition Runyon, Coleman & Pittenger (optional)

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CALCULATORS

DO NOT FORGET TO BRING YOUR CALCULATOR TO THE QUIZZES!!!!!!

Required: +, -, x, ,

Helpful:

X (sometimes )- mean

S (SD) - standard deviation (sometimes is )

X - sum X

X2 - sum X squared

N or n – number

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BASIC MATH REVIEW 2 + 2 = 4 2 + (-2) = 0 (-2) + (-2) = (-4) 2 x 2 = 4 2 x (-2) = (-4) (-2) x (-2) = 4 2 – 2 = 0 2 – (-2) = 4 (-2) – (-2) = 0 2/2 = 1 2/(-2) = (-1) (-2)/(-2) = 1 22 = 4 (-2)2 = 4 √4 = 2 √(-4) = error

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+ +

+ -

- +

_ _

GRAPHING QUADRANTS

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true limits = + / - ½ the unit of measurement i = (hi - lo + 1) / # groups midpoint = (hi true + lo true) / 2 PR = cumfll + ((X - Xll) / i)(fi) x 100 N cumf = (PR x N) / 100 X = Xll + [[i (cumf - cumfll)] / fi] cumfll = cum freq at lower true limit of X X = score Xll = score at lower true limit of X i = width fi = # cases in X's group N = total # scores

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- Sam wants to find out if the number of hours people study has any effect on their grade. - Mary wants to find out if gender has any influence on math and verbal SAT scores. - Dr. Jones wants to find out if her current class performs any differently on the final compared to all past students. - A large pharmaceutical company wants to know if their new drug for controlling OCD is effective.

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Chapter 1: What is statistics?

- statistics: the process of collecting data & making decisions based on the analysis of these data descriptive inferential (generalize) Common Terms - constant: # representing a construct that does not change (e.g., ); we will see these in some formulas - variable: measurable characteristic that changes with person, environment, experiment e.g., height, IQ, learning (X or Y) - independent variable (IV): variable examined to determine its effect on outcome of interest (DV); under control of experimenter - manipulated variable; e.g., dose of a drug - dependent variable (DV): outcome of interest measured to assess effects of IV; not under experimenter control; e.g., how a person reacts to the drug - subject or organismic variable: naturally occurring IV; characteristic of people but not controlled e.g., eye color, gender

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- data: numbers, measurements collected - population: complete set of people/objects having some common characteristic - parameter: value summarizing characteristic of population; are constants; use Greek letters to represent - sample: subset of population, share same characteristics - statistic: value summarizing characteristic of a sample; are variable; use Roman letters to represent - simple random sample: subset of population selected so that each population member has = & independent chance of being chosen - random assignment: assign subjects to treatments in = & independent manner to avoid bias - confounding: where DV is affected by variable related to IV so can't assume that IV causes DV effects

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CONFOUNDING Group 1 Group 2 Lecture 3x/week vs lecture 2x/week Lab 1x/week Taught by Dr. Smith Taught by Dr. Jones Results: group #2 performs better on final exam Conclude: lecture + lab > lecture alone WRONG!!!! Confounded by different teachers as well as format differences

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CHAPTER #1 HOMEWORK

1 a-e

3 a-g

4, 6-11

13 a-j

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CHAPTER 1 - HOMEWORK

1. a. statistic b. inference c. data d. data

e. inference

3. a. constant b. variable c. variable d. variable

e. constant f. variable g. variable

4. all vs subset; yes

6. sample

7. variable

8. data

9. statistic

10. populations

11. parameter

13. a. manipulated b. not variable

c. not variable d. subject variable

e. subject variable f. manipulated

g. manipulated h. manipulated

i. subject variable j. subject variable

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- Fred wants to find out what types of pets college students

have.

- Alice wants to find out if birth order has any effect on

GPA.

- Mike wants to look at temperature effects on ice cream

consumption.

- Sally wants to see how fast rats run through a maze as a

function of reward type at the end.

- Rick wants to examine how many kids people have today

compared to 50 years ago.

- Mary wants to examine how tall people are compared to 50

years ago.

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Chapter 2 - Basic Concepts

- X or Y: symbol for a variable

- Xi or Yi: represents individual observation

- N or n: # data points in a set, number

- : indicates summation

EXAMPLES (X = group 1 kids, y = group 2 kids)

X1 = 4 X2 = 6 X3 = 1 X4 = 5 X5 = 2 X6 = 3

Y1 = 3 Y2 = 4 Y3 = 6 Y4 = 1

a) Xi = 1 + 5 + 2 + 3 = 11

b) Yi = 3 + 4 + 6 = 13

* c) Xi2 = 52 + 22 + 32 = 25 + 4 + 9 = 38

* d) ( Xi)2 = (5 + 2 + 3)2 = 102 = 100

e) Xi = 6 + 1 + 5 + 2 + 3 = 17

N = go to the end; use all #s from start point

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i = 3

3

i = 1

6

i = 4

6

i = 4

N

i = 2

NOT THE SAME !!!!

Where you stop Where you start

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types of measurement scales (like inches vs cm)

a) nominal: qualitative (name); mutually exclusive without

logical order (cat, dog, fish)

b) ordinal: mutually exclusive with logical rank ordering

(<,>) (1st grade, 2nd grade; captain, major, colonel)

c) interval: quantitative with = units of measurement and

arbitrary (imaginary) zero point (thermometer, calendar)

d) ratio: quantitative with = units of measurement and

absolute (real) zero point (height, weight, length)

some more terms

- reliability: degree to which repeated measurements in same

conditions give same results

- measurement error: uncontrolled recording error

- validity: accuracy test/measure actually measures thing of

interest

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- discontinuous (discrete) variables: only whole #s allowed

e.g., # kids

- continuous variables: any values allowed

a) true limits: #s that limit where true value lies

+ / - ½ the unit of measurement

- to get unit of measurement

1) no decimals: # by which set increases

e.g., 3,4,5,6 => unit = 1 ½ = 0.5 (limit value)

3 + 0.5 = 3.5 (upper limit) 3 - 0.5 = 2.5 (lower limit)

5,10,15,20 => unit = 5 5/2 = 2.5 (limit value)

10 + 2.5 = 12.5 (upper limit) 10 - 2.5 = 7.5 (lower limit)

2) decimals: a) anything to left = 0

b) last # on right = 1; all others = 0

e.g., 13.63 => 0.01 (unit of measurement)

0.01 / 2 = 0.005 (limit values)

13.63 + 0.005 = 13.635 (upper limit)

13.63 - 0.005 = 13.625 (lower limit)

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some basic descriptive statistics

1) frequency: count

class = 20 13 women; 7 men

2) ratio: 13:7 women to men; DO NOT REDUCE

20: 5 do not reduce to 4:1

3) proportion: fraction 13/20 = 0.65 women

DO OUT THE DIVISION

4) percentage: proportion x 100 7/20 x 100 = 35% men

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CHAPTER #2 HOMEWORK

8 b, d, e

10 b, c

15 a-d

16 a-d

19 a-f

23 a, b, e, f

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8. X1 = 2; X2 = 3; X3 = 5; X4 = 7; X5 = 9; X6 = 10; X7 = 13

b. Xi = 2 + 3 + 5 + 7 + 9 + 10 + 13 = 49

d. Xi = 3 + 5 + 7 + 9 = 24

e. Xi = 2 + 3 + 5 + 7 + 9 + 10 + 13 = 49

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b. X1 + X2 + ...+ Xn = Xi

c. X32 + X42 + X52 + X62 = Xi2

15. a) ratio b) ratio c) nominal d) ordinal

16. a) continuous b) continuous c) discontinuous

d) discontinuous

5

7

i = 1

i = 2 N

i = 1

n

i = 1 6

i = 3

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19. a) 5 1/2 = 0.5 4.5 - 5.5

b) 5.0 0.1/2 = 0.05 4.95 - 5.05

c) 5.00 0.01/2 = 0.005 4.995 - 5.005

d) 0.1 0.1/2 = 0.05 0.05 - 0.15

e) (-10) ½ = 0.5 (-10.5) - (-9.5)

f) 0.8 0.1/2 = 0.05 0.75 - 0.85

23. men women

BA 400 300

E 50 150

H 150 200

S 250 300

SS 200 200

a) 1150/1150 + 1050 = 52.3%

b) BA: 400/1050 x 100 = 38.10%

E: 50/1050 x 100 = 4.76%

H: 150/1050 x 100 = 14.29%

S: 250/1050 x 100 = 23.81%

SS: 200/1050 x 100 = 19.05%

e) 300/700 x 100 = 42.86%

f) 250/550 x 100 = 45.45%

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- I have 23,184 data points from my experiment - what do I

do with all that information?

- How do I present that information to someone else?

- Mitch got a 43 on the quiz – how did he do compared to

everyone else?

- Ann was told she scored at the 75th percentile on the GRE

exam – what does that mean?

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Chapter 3 - Frequency Distributions & Percentiles

- exploratory data analysis: ways to arrange & display #s to

quickly organize & summarize data

- grouping data

1) frequency distribution: high - low

pet type frequency proportion %

dog 20 0.43 (20/46) 43 (0.43 x 100)

cat 15 0.33 33

turtle 11 0.24 24

46 1.0 100

2) grouping in classes

a) aim for 12 - 15 groups

b) mutually exclusive

c) same width

d) don't omit intervals

e) make widths convenient

width = (hi - lo + 1) / # groups = i

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example:

84 85 87 80 81 88 89 90 92 92 93 95 96 96 96 97 97 97 97 98 98 98 98 99 99 99 99 99 99 100 100 100 100 101 101 101 101 102 102 103 103 100 100 100 101 102 103 102 100 101 102 100 100 100 100 100 100 104 104 105 104 106 105 104 105 105 110 110 111 111 111 111 111 111 111 111 111 111 112 112 113 113 114 115 116 117 118 124 124 125 125 126 127 129 134

i = (134 - 80 + 1)/15 = 3.67 ~ 4

START AT BOTTOM WITH LOW #

Interval True Limits f Midpoint 132 - 135 131.5 - 135.5 1 133.5 128 - 131 127.5 - 131.5 1 129.5 124 - 127 123.5 - 127.5 6 125.5 120 - 123 119.5 - 123.5 0 121.5 116 - 119 115.5 - 119.5 3 117.5 112 - 115 111.5 - 115.5 6 113.5 108 - 111 107.5 - 111.5 12 109.5 104 - 107 103.5 - 107.5 9 105.5 100 - 103 99.5 - 103.5 28 101.5 96 - 99 95.5 - 99.5 17 97.5 92 - 95 91.5 - 95.5 4 93.5 88 - 91 87.5 - 91.5 3 89.5 84 - 87 83.5 - 87.5 3 85.5 80 - 83 79.5 - 83.5 2 81.5

midpoint = (hi true + lo true) / 2

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- cumulative data class grades f cum f cum prop cum % 91 - 100 6 32 1.0 100 81 - 90 4 26 0.8125 81.25 71 - 80 9 22 0.6875 68.75 61-70 11 13 0.4062 40.62 51 - 60 2 2 0.0625 6.25 32 Percentiles & Percentile Ranks - score alone means nothing, must compare to standard or base score; can do with percentiles - percentiles: #s that divide distribution into 100 = parts - percentile rank: # that represents the % of cases in a comparison group that achieved scores < the one cited e.g., PR of 95 on SAT means 95% of those taking SAT at the same time did worse than you & 5% did better some symbols cumfll = cum freq at lower true limit of X X = score Xll = score at lower true limit of X i = width fi = # cases in X's group N = total # scores

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1) Getting PR from score (X)

PR = cumfll + ((X - Xll)/i) (fi) x 100 N Class (X) limits f cum f cum %

93 - 95 92.5 - 95.5 4 25 100 90 - 92 89.5 - 92.5 3 21 84 87 - 89 86.5 - 89.5 2 18 72 84 - 86 83.5 - 86.5 7 16 64 81 - 83 80.5 - 83.5 6 9 36 78 - 80 77.5 - 80.5 3 3 12 What is PR of 88? X = 88 cumfll = 16 Xll = 86.5 i = 3 fi = 2 N = 25

NB: PR goes from 0 – 100

PR = 16 + ((88 - 86.5) / 3) (2) x 100 25 PR = 68

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2) Getting score (X) from PR

cumf = (PR x N)/100 X = Xll + [ i (cumf - cumfll) / fi ]

Class (X) limits f cum f cum %

93 - 95 92.5 - 95.5 4 25 100 90 - 92 89.5 - 92.5 3 21 84 87 - 89 86.5 - 89.5 2 18 72 84 - 86 83.5 - 86.5 7 16 64 81 - 83 80.5 - 83.5 6 9 36 78 - 80 77.5 - 80.5 3 3 12

What is score for PR of 75?

cumf = 75 x 25 / 100 = 18.75

Xll = 89.5

i = 3 X = 89.5 + [ 3 (18.75 - 18) / 3 ] = 90.25

cumf = 18.75

cumfll = 18

fi = 3

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CHAPTER 3 HOMEWORK

3 a - use 18 for # groups

3b

c – not in book – What is PR if X = 36 using data from # 3

d – not in book – What is X if PR = 09? Use data from # 3

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3 a) if you want 18 groups: (90 - 5 + 1) / 18 = 4.7 ~ 5

b) group limits mdpt f cumf cum%

90 - 94 89.5 - 94.5 92 1 90 100 85 - 89 84.5 - 89.5 87 0 89 98.89 80 - 84 79.5 - 84.5 82 0 89 98.89 75 - 79 74.5 - 79.5 77 1 89 98.89 70 - 74 69.5 - 74.5 72 0 88 97.78 65 - 69 64.5 - 69.5 67 3 88 97.78 60 - 64 59.5 - 64.5 62 4 85 94.44 55 - 59 54.5 - 59.5 57 7 81 90 50 - 54 49.5 - 54.5 52 5 74 82.22 45 - 49 44.5 - 49.5 47 11 69 76.67 40 - 44 39.5 - 44.5 42 11 58 64.44 35 - 39 34.5 - 39.5 37 10 47 52.22 30 - 34 29.5 - 34.5 32 9 37 41.11 25 - 29 24.5 - 29.5 27 8 28 31.11 20 - 24 19.5 - 24.5 22 5 20 22.22 15 - 19 14.5 - 19.5 17 9 15 16.67 10 - 14 9.5 - 14.5 12 4 6 6.67 5 - 9 4.5 - 9.5 7 2 2 2.22

c) what is PR if X = 36?

PR = 37 + ((36 - 34.5) / 5) (10) x 100 = 44.44 90 d) what is X if PR = 98?

cumf = 98 x 90 / 100 = 88.2

X = 74.5 + [ 5 (88.2 - 88) / 1 ] = 75.5

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- What types of graphs are used most often in psychology?

- Are there rules for which one to use? - Are there rules about how to make them? - Does the shape of the graph mean anything useful?

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Chapter 7 - Graphing

- visual methods to display data a) figure: pictorial; photo, drawing b) table: organized numerical info c) graph: pictorial; axes, #s etc. - basics of graphing a) X-axis (abscissa): horizontal; IV b) Y-axis (ordinate): vertical; DV c) always label axes – note the units d) Y starts at 0; continuous, no breaks X can change start; break; can be discrete e) Y about 0.75 length of X 1) Bar Graph: nominal, sometimes ordinal a) bar = category b) height = frequency c) bars DO NOT touch d) if ordinal must preserve order e) can be vertical or horizontal

02468

101214161820

Fre

qu

ency

DOG CAT FISH BIRD

Women

Men

Pet w m Dog 20 10 Cat 15 15 Fish 8 5 Bird 5 14

TYPE OF PET

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2) Histogram: interval, ratio data, sometimes ordinal a) same rules as bar only bars DO touch b) usually for discrete data

3) Frequency or Line graph: interval, ratio, sometimes ordinal a) usually for continuous data

0

5

10

15

20

25

F D C B A

Grad e

Fre

quen

cy

Grade Freq F 2 D 4 C 20 B 15 A 10

0

1

2

3

4

5

6

7

56 57 58 59 60

Weight

Fre

qu

ency

Weight freq 56 2 57 2 58 4 59 6 60 5

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4) cumulative frequency: can be bar, histogram or line, but uses cumulative freq, proportion or % a) the line graph version is typically s-shaped or ogive b) always increases e.g., 12 people on a drug to cure disease X. Left = # cured each time period. Right = cum % cured over time.

0

0.5

1

1.5

2

2.5

3

3.5

1 3 6 9 12

months on drug

# cu

red

01020304050607080

1 3 6 9 12

months on drug

Cu

m %

cu

red

Forms of Frequency Curves 1) Normal (bell-shaped) curve: symmetric

a) mesokurtic: ideal (middle)

b) leptokurtic: peaked (leaping)

c) platykurtic: flat (prairie) 2) skew: not symmetric a) positive skew: fewer scores at high end;

shifted to left b) negative skew: fewer scores at low end;

shifted to right

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CHAPTER 7 HOMEWORK

Chapter 3: 5 a-e Chapter 7: 7 do 1b 6 7 12 – schizophrenic data only

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CHAPTER 7 - HOMEWORK Chap 3 # 5: a) b) c) d) e) Chap 7: 1b)

6)

02468

10121416

30-39 40-49 50-59 60-69 70-79 80-89 90-99

scores (midpt)

F

0102030405060708090

100

0 3 6 9 12 15 18

interval (sec)

%

43

7)

12) schizophrenic data only

0

10

20

30

40

50

60

70

1 2 3 4 5 6 7 8 9 10

minutes

F

100%

60%

010203040506070

80

cat dis par und

type schizophrenia

f

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z = (X - X) / s = (X - ) / SIR = (Q3 - Q1) / 2 X = X / n s3 = [3(X - median)] / s Range = hi - lo

Xw = fX / ntot

s4 = 3 + [ (Q3 - Q1) / 2 (P90 - P10)] md = Xll + i [ ((N/2) - cumfll) / fi] s2 = (X - X)2 / n s = s2 SS = X2 - (X)2/n s2 = SS/n s = s2

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- Sid wants to know what is the average age of people in the mall before the stores open? - Dr. Smith has 4 classes each with a different number of pupils. He has the average grade on the last quiz for each of the 4 classes but wants to know the overall average. - If we include all the billionaires in the calculation of the average US income will it be inflated because of the few very high values? Is there a better measure than the mean?

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Chapter 4 - Central Tendency A) Arithmetic Mean (average): X = X/n 4 + 2 + 6 + 4 + 5 = 21 21/5 = 4.2 = X 1) from ungrouped frequency distribution: X = fX/n X f fX 10 4 40 9 2 18

8 6 48 X = 155/20 = 7.75 7 2 14 6 5 30 5 1 5 20 155 2) Weighted Mean: mean of a group of means e.g., 4 classes with mean exam scores of 75, 78, 72, 80. What is the overall or grand mean? a) if each class has same # of people: (75 + 78 + 72 + 80)/4 = 76.25 b) if each class has different # people must account for it class X F fX 75 30 2250 Xw = fX/Ntot 78 40 3120 72 25 1800 Xw = 11170/145 = 77.03 80 50 4000 145 11170

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B) Median: midpoint of a distribution of scores so ½ fall above & ½ fall below = 50th percentile 1) for continuous scores md = Xll + i [ ((N/2) - cumfll) / fi] true limits f cumf 68.5 - 71.5 13 101 1) to find box = N/2 65.5 - 68.5 15 88 62.5 - 65.5 20 73 101/2 = 50.5 59.5 - 62.5 28 53 find 50.5 in cumf column 56.5 - 59.5 19 25 53.5 - 56.5 6 6

md = 59.5 + 3 [((101/2) - 25) / 28 ] = 62.2 Good for skewed, truncated & open-ended distributions - truncated: use only part of the distribution - open-ended: top or bottom category has only 1 limit e.g., 68.5 + for top category < 53.5 for bottom category

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2) median for arrays of scores a) if N is odd => put in ascending order, find middle #

56, 6, 13, 31, 28 => 6, 13, 28, 31, 56 b) if N is even => ascending order, take X of 2 middle #s 6, 13, 28, 31, 56, 72 => (28 + 31) / 2 = 29.5 c) N is even but middle 2 #s are the same => use formula 1, 2, 4, 6, 6, 6, 7, 121 x f cumf 121 1 8 8/2 = 4 => box 7 1 7 6 3 6 md = 5.5 + 1 [ ((8/2) - 3) / 3] = 5.83 4 1 3 2 1 2 1 1 1 C) Mode: most common score; crude measure 1) 1, 3, 4, 6, 7, 7, 7, 9, 9 mode = 7 2, 2, 4, 9, 9 mode = 2, 9

2) class f 68.5 - 71.5 10 1) find highest f value 65.5 - 68.5 15 2) report midpoint as mode 62.5 - 65.5 9 59.5 - 62.5 10 mode = (68.5 + 65.5) / 2 = 67

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- Which to use? 1) mode: quick & easy but crude; not unique - can have 2+

2) median: skewed, truncated, open-ended

3) mean: most common, normal distributions

some properties of the mean a) summed deviations = 0 (X - X) = 0 X X - X

4 4 - 5.5 = -1.5 3 3 - 5.5 = -2.5 9 9 - 5.5 = 3.5 6 6 - 5.5 = 0.5 0

b) sensitive to extreme values (skew) 2, 3, 5, 7, 8 X = 5 md = 5 2, 3, 5, 7, 33 X = 10 md = 5 c) can't use with open-ended distribution Mean, Median & Skew relationship

a) mean > median => positive skew b) mean < median => negative skew c) mean = median => no skew

50

CHAPTER 4 HOMEWORK 1 a – c 2 8 a – d 18

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1a) 0, 0, 2, 3, 5, 6, 8, 8, 8, 10

X = 50/10 = 5; mode = 8; md = (5 + 6) / 2 = 5.5

b) 1, 3, 3, 5, 5, 5, 7, 7, 9

X = 45/9 = 5; mode = 5; md = 5

c) 119, 5, 4, 4, 4, 3, 1, 0 X = 140/8 = 17.5; mode = 4 X f cumf 119 1 8 8/2 = 4 5 1 7 4 3 6 md = 3.5 + 1 [ ((8/2) - 3) /3] = 3.83 3 1 3 1 1 2 0 1 1 2) c, it's skewed 8) a) - b) + c) no d) no, bimodal 18) X f fX 1.75 4 7 2.0 5 10 Xw = 50.01/24 = 2.08 2.4 5 12 2.5 4 10 2.0 3 6 1.67 3 5.01 24 50.01

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- Al calculated the average height of people in a random sample to figure out how high he should make the pull-down security bars on a new roller coaster. He says the average height is 5’10” but his boss says not everyone is 5’10”. He wants to know about what height to expect – what is the dispersion or spread of heights? Betty graphs data she collected on frequency of failing grades for grammar school students as a function of tv shows watched and finds a very peaked graph shifted to the left. She knows it’s leptokurtic and skewed but can she attach values to say how leptokutic and how skewed?

53

Chapter 5 - Dispersion - dispersion: spread or variability of scores around central tendency measure 1) range: hi score - lo score 11, 17, 9, 3, 20, 36 36 - 3 = 33 2) semi-interquartile range (SIR) or Q2: use with median; median + SIR cuts off middle 50% of scores SIR = Q2 = (Q3 - Q1) / 2 Q3 = score at 75th PR PR X Q1 = score at 25th PR 90 80 75 70 SIR = Q2 = (70 - 10) / 2 = 30 50 40 35 30 25 10 10 5 3) variance or mean square (s2 or 2) & standard deviation or root mean square (s or ) a) use with mean b) can use to compare distributions c) quite precise d) used in statistical tests later on e) large values = high error, low precision small values = low error, high precision

54

1) Mean Deviation Method: long, but shows how scores vary from the mean s2 = (X - X)2 / n = SS/n s = s2 X X - X (X - X)2

65 -14.375 206.64 n = 8 X = 79.375 90 10.625 112.89 84 4.625 21.39 s2 = 1123.87/8 = 140.48 76 -3.375 11.39 81 1.625 2.64 s = 140.48 = 11.85 98 18.625 346.89 82 2.625 6.89 59 -20.375 415.14 0 1123.87 = SS 2) Raw Score Method: easier; less intuitive about mean SS = X2 - (X)2/n s2 = SS/n s = s2

X X2 65 4225 90 8100 SS = 51527 - (635)2/8 = 1123.875 84 7056 76 5776 s2 = 1123.875/8 = 140.48 81 6561 98 9604 s = 140.48 = 11.85 82 6724 59 3481 635 51527

55

- homogeneous sample: data values similar => low s2 & s - heterogeneous sample: data values dissimilar => high s2 & s - Pearson's Coefficient of Skew: + or - and how much

s3 = [3(X - median)] / s

X = 20 s = 5 md = 24 s3 = [ 3(20 - 24)] / 5 = -2.4

Generally + 0.5 is ~ symmetrical/normal - Kurtosis: peaked or flat s4 = 3 + [ (Q3 - Q1) / 2 (P90 - P10)] P90 = score at 90th PR P10 = score at 10th PR X PR 100 90 90 75 s4 = 3 + [ (90 - 20) / 2 (100 - 5)] = 3.37 70 60 40 50 3 = mesokurtic 20 25 < 3 = platykurtic 5 10 > 3 = leptokurtic

56

CHAPTER 5 HOMEWORK

5 6 a – use mean deviation method b – use raw score method 8 a – d NOT IN BOOK PR X X = 30 s = 5 md = 25 100 90 90 85 1) Find SIR 75 70 60 50 2) Find SKEW 50 40 35 20 3) Find KURTOSIS 25 10 10 5 5 2

57

CHAPTER 5 HOMEWORK 5) all the same # 6a) X X - X (X - X)2 10 5.3 28.09 8 3.3 10.89 X = 4.7 n = 10 6 1.3 1.69 0 -4.7 22.09 s2 = 124.1/10 = 12.41 8 3.3 10.89 3 -1.7 2.89 s = 12.41 = 3.52 2 -2.7 7.29 2 -2.7 7.29 8 3.3 10.89 0 -4.7 22.09 0 124.1 b) X X2 1 1 3 9 SS = 273 - (45)2/9 = 48 3 9 5 25 s2 = 48/9 = 5.33 5 25 5 25 s = 5.33 = 2.31 7 49 7 49 9 81 45 273

58

8 a) 10 - 0 = 10 b) 9 - 1 = 8 c) 20 - 0 = 20 d) 5 - 5 = 0 this one is misleading For this data find: SIR, skew, kurtosis (not in book) PR X X = 30 s = 5 md = 25 100 90 90 85 SIR = (70 - 10)/2 = 30 75 70 60 50 s3 = [3(30 - 25)]/5 = 3 50 40 35 20 25 10 s4 = 3 + [ (70 - 10)/2(85 - 5)] = 3.375 10 5 5 2

59

- Is there a simpler method to examine percentile ranks and compare values other than the PR formula? - Mitch has the mean and standard deviation values for a quiz that a class just took. He also has his grade on the quiz. How can he determine how many people did worse than him and how many did better? - If you know a country club takes people whose income is in the top 5% of the city and you know the average income of the city and standard deviation, can you use your income to figure out if you can get in the club?

60

Chapter 6 - z-scores or standard scores - z-score: represents distance between score & mean relative to s 1) can use to compare 2 different variables because z-scores are abstract #s without units

2) if scores are normally distributed can relate directly to PR via the "Standard Normal Distribution" = a theoretically ideal normal distribution where:

= 0 = 1 total area under curve = 1.0 or 100%

+ above the mean

- below the mean

50% => <= 50%

-4 -3 -2 -1 0 1 2 3 4

68.26% 95.44% 99.74%

61

3) when you transform data to z-scores a) mean = 0 b) sum of squared z-scores = n c) s = 1 z = (X - X)/s z = (X - )/ sample population e.g., for IQ = 100 = 15; someone got an IQ of 130 z = (130 - 100)/15 = +2.00

so are 2 standard deviations above the mean e.g., when 2 scores come from different distributions is hard to compare; z-scores let you do it psych = 50 = 10 bio = 48 = 4 Bob got a 60 on psych & 56 on bio; for which course should he expect a better grade? Psych z = (60 - 50)/10 = +1.0 Bio z = (56 - 48)/4 = +2.0 would expect better grade in bio!!!

62

e.g., of properties

ht ht z-score ht z2 wt z2 wt z-score wt 6' 0.27 0.0729 0.0961 0.31 200 lb 5' -1.1 1.21 0.6084 -0.78 150 lb 5' -1.1 1.21 2.0736 -1.44 120 lb 6' 0.27 0.729 0.2704 0.52 210 lb 7' 1.6 2.56 1.9321 1.39 250 lb

X 5.8 0 0 186 S 0.75 1 1 45.9 N 5 5 5 5 5 5 5 5 =======================================================

1) assume X = 650 = 600 = 100. What % did worse than X? z = (650 - 600) / 100 = 0.5 Table A page 548 - 549 Column a = z-score Column b = area between & z Column c = area beyond z Area between = 0.1915 so 0.1915 + 0.5 = 0.6915 = 69.15% did worse or PR = 69.15 2) X = 400 = 600 = 100. What % did worse? z = (400 - 600) / 100 = -2 Area beyond = 0.0228 = 2.28% did worse or PR = 2.28

0 0.5

-2 0

65

3) What % of cases fall between X = 650 and X = 400 if = 600 = 100? z = (650 - 600) / 100 = 0.5 z = (400 - 600) / 100 = -2 0.1915 + 0.4772 = 0.6687 = 66.87% 4) What % fall between X = 700 and X = 800 if = 600 = 100? z = (700 - 600) / 100 = 1 z = (800 - 600) /100 = 2 0.4772 - 0.3413 = 0.1359 = 13.59% RULE: ++ or -- => subtract column b + - => add column b 5) Suppose a golf club takes only top 3% of population in income where = 500k = 25k. You make 520k. Can you get in? column c gives beyond so find 0.03 in c & get z that goes with it z = 1.88 so.... 1.88 = (X - 500) / 25 X = 547K (1.88) (25) = X - 500 so you cannot get in!!! (1.88) (25) + 500 = X

0 ?

0 1 2

-2 0 0.5

0.03 or 3%

66

6) Suppose = 600 = 100, what is the score at the 60th percentile?

7)

0.40 above

0 ?

Column c => 0.4013 => z = 0.25 So … 0.25 = X – 600/100 0.25 (100) = X – 600 0.25 (100) + 600 = X X = 625

7) Suppose = 600 = 100, between what scores do the middle 30% lie? Column b => 0.15 => +/- 0.39 0.39 = X - 600/100 = 639 -0.39 = X – 600/100 = 561

0.150.15

? 0 ?

8) Suppose = 600 = 100, beyond what scores do the most extreme 20% lie? Column c => 0.10 => +/- 1.28 1.28 = X – 600/100 = 728 -1.28 = X – 600/100 = 472

0.10

0.10

? 0 ?

67

CHAPTER 6 - HOMEWORK 1 a, c, e 2 a, c, e, g, I 3 a (60 & 25) b (70 & 45) c (60 & 70, 45 & 70) 7 a, f, g

68

CHAPTER 6 - HOMEWORK 1a) z = (55 - 45.2) / 10.4 = 0.94 c) z = (45.2 - 45.2) / 10.4 = 0 e) z = (68.4 - 45.2) / 10.4 = 2.23 2a) 0.4798 c) 0.0987 e) 0.4505 g) 0.4901 i) 0.4990 3a) (60 - 50) / 10 = 1 0.3413 x1000 = 341.3 cases (25 - 50) / 10 = -2.5 0.4938 x 1000 = 493.8 cases b) (70 - 50) / 10 = 2 0.0228 x 1000 = 22.8 cases (45 - 50) / 10 = -0.5 0.6915 x 1000 = 691.5 cases c) (60 - 50) / 10 = 1 (70 - 50) / 10 = 2 0.4772 - 0.3413 = 0.1359 x 1000 = 135.9 cases (45 - 50) / 10 = -0.5 (70 - 50) / 10 = 2 0.4772 + 0.1915 = 0.6687 x 1000 = 668.7 cases

0 1

-2.5 0

0 2

-0.5 0

0 1 2

-0.5 0 2

69

7a) -0.67 = (X - 72) / 12 X = 63.96 f) 0.68 = (X - 72) / 12 X = 80.16 -0.68 = (X - 72) / 12 X = 63.84 g) 1.64 = (X - 72) / 12 X = 91.68 -1.64 = (X - 72) / 12 X = 52.32

0.25

0.25 0.25

? = -0.67

? = -0.68 0 ? = 0.68

0.05 0.05

? = -1.64 ? = 1.64

70

sesty = sy [N ( 1 - r2)] / (N - 2) r = (zxzy) / N by = (r) (sy/sx) a = Y - byX Y = a + byX rs = 1 - [ (6D2) / [N (N2 - 1)]] 1 = r2 + k2 zy' = (r)(zx) Y' = Y + (zy')(sy) Y' = Y + [ (r)(sy/sx)(X - X)] r = XY - [(X)(Y) / N] [X2 - [(X)2 /N]] [Y2 - [(Y)2 / N]]

71

- Sue wants to know if there is a relationship between how well students do on a quiz and how much test anxiety they report prior to taking it. - Bill has teachers rank their students by how popular they think they are and then wants to know if there is a relationship between the popularity ranks and the students’ GPA. - Sandy wants to know if there is a relationship between number of depressed people and SES.

72

Chapter 8 - Correlation - correlation: relationship between 2 variables - correlation coefficient: measure used to express extent or strength of relationship 1) positive correlation: 0 < r < 1; score high on 1 variable & score high on the other; score low on 1 variable score & score low on the other; positive slope; 1.0 = perfect correlation 2) negative correlation: -1 < r < 0; score high on 1 variable & score low on the other; negative slope; -1.0 = perfect correlation positive negative 3) 0 = no correlation, no linear relationship 4) looking for a linear relationship - others exist (e.g., u-shaped), but correlation only measures linear 5) correlation = causation 6) |r| < 0.29 small correlation, weak relationship |r| 0.3 - 0.49 medium correlation / relationship |r| 0.5 - 1.0 large correlation, strong relationship

73

- scatter diagram: graphic means to show data points & correlation & (later) regression - centroid: X, Y point ( )

1) Pearson r: for interval & ratio data a) z-score method r = (zxzy) / N N = # pairs X Zx Y Zy ZxZy 1 -1.5 4 -1.5 2.25 3 -1 7 -1.0 1 5 -0.5 10 -0.5 0.25 r = 7/7 = 1.0 7 0 13 0 0 9 0.5 16 0.5 0.25 11 1 19 1.0 1 13 1.5 22 1.5 2.25 = 7 Good if already have z-scores, otherwise is a pain! If already have info: (zxzy) = 4.9 N = 7, then 4.9/7 = 0.7 then it's easy.

0

2

4

6

8

10

12

0 2 4 6 8 10

ht

wt

Ht Wt 2 3 4 7 5 10 9 11 5 7.75 mean

74

2) Raw Score Method r = XY - [(X)(Y) / N] [X2 - [(X)2 /N]] [Y2 - [(Y)2 / N]] numerator = covariance: degree to which 2 variables share common variance; high covariance = more linear, closer to +1 low covariance = less linear, closer to 0 X X2 Y Y2 XY X = 49 1 1 7 49 7 X2 = 455 3 9 4 16 12 Y = 91 5 25 13 169 65 Y2 = 1435 7 49 16 256 112 XY = 775 9 81 10 100 90 N = 7 11 121 22 484 242 (X)2 = 2401 13 169 19 361 247 (Y)2 = 8281 49 455 91 1435 775 r = 775 - [(49)(91) / 7] [455 - [2401/7]] [1435 - [8281 / 7]] r = + 0.82 N.B. can get negative on top but not on bottom

75

- If r = + 1 all data fall in a line; if |r| < 1 data are scattered. There are 3 types of variation: total = explained (r2) + unexplained (k2) if r = + 1 all is explained; if r = 0 all is unexplained a) r2 = coefficient of determination: proportion of 1 variable explained by the other b) k2 = coefficient of non-determination: proportion of 1 variable not explained by the other total = 1 or 100% so.... 1 = r2 + k2 => k2 = 1 - r2 e.g., r = 0.84 r2 = 0.71 k2 = 1 - 0.71 = 0.29 - cautions with Pearson r 1) measures linearity so low r means not linear; could still have a non-linear relationship 2) distribution need not be normal but must be unimodal 3) of truncated will get spuriously low r

76

2) Spearman r: with ordinal data; rs a) both variables must be rank ordered b) non-parametric test: looks at ranks only (parametric uses actual #s) rs = 1 - [ (6D2) / [N (N2 - 1)]] D = rank X - rank Y D = 0 N = # pairs X rank X Y rank Y D D2 140 1 63 6 -5 25 120 5 70 3 2 4 136 2 72 1 1 1 100 6 69 4 2 4 129 3 65 5 -2 4 125 4 71 2 2 4 0 42 rs = 1 - [ (6 42) / [6 ( 36 - 1)]] = - 0.2 - Tied Scores: if tied must take this into account to be fair X rank X adjusted rank X 140 1 1 120 4 4.5 (4 + 5) / 2 = 4.5 136 2 2 100 6 6 take mean of tied ranks 120 5 4.5 assign mean rank 125 3 3

77

- Correlation matrix: table to visualize many correlations kindergarten grammar high college kinder ------ 0.93 0.74 0.61 grammar ------ ----- -0.63 -0.54 high ------ ----- ------ 0.36 college ------ ----- ------ ------ e.g., what 2 groups correlate the most? Grammar & kindergarten e.g., which 2 groups correlate the least? High school & college e.g., what is the correlation between grammar & high? -0.63

78

CHAPTER 8 HOMEWORK 2 a – d for c also find k2 7 a, c 8 a 9 b, c 15 NOT IN BOOK – RANK ORDER THESE 1) X 2) X 3) X 7 76 -41 4 79 -38 6 81 -42 7 76 -41 9 63 -26 4 28 -26 2 -41

79


2a)

b) X X2 Y Y2 XY 90 8100 94 8836 8460 X = 710 85 7225 92 8464 7820 X2 = 51750 80 6400 81 6561 6480 Y = 738 75 5225 78 6084 5850 Y2 = 56244 70 4900 74 5476 5180 XY = 53890 70 4900 73 5329 5110 (X)2 = 504100 70 4900 75 5625 5250 (Y)2 = 544644 60 3600 66 4356 3960 N = 10 60 3600 53 2809 3180 50 2500 52 2704 2600 710 51750 738 56244 53890 r = 53890 - [(710 738) / 10] [ 51750 - (504100 /10)] [ 56244 - (544644 / 10)]

r = 0.97 c) r2 = 0.972 = 0.9409 k2 = 1 - 0.9409 = 0.0591 d) yes

40

50

60

70

80

90

100

40 60 80 100

test

gra

de

80

7a) - .410 c) they test many of the same things 8a) 0.633 9b) Spearman rank c) only use % recall & % recognition % recall rank recall % recog. rank recog. D2 86 1 91 3 4 81 2 95 1 1 75 4 86 4 0 78 3 93 2 1 58 6 80 6 0 62 5 70 7 4 38 7 84 5 4 14 rs = 1 - [ ( 6 14) / [7 (49 - 1)]] = 0.75 15) 41.3 / 50 = 0.826 Not in book: rank order these data 1) X rank 2) X rank 3) X rank 7 2.5 76 3.5 -41 5 4 5.5 79 2 -38 3 6 4 81 1 -42 7 7 2.5 76 3.5 -41 5 9 1 63 5 -26 1.5 4 5.5 28 6 -26 1.5 2 7 -41 5

81

- Joe has a set of data correlating number of books read per month with age. He wants to plot these data on a graph and draw a line to show the general linear trend of the data. - Carol has a set of data on height as a function of how many grams of protein children had on average per day. She then wants to predict the height of an individual assuming they had 10 grams of protein on average per day.

82

Chapter 9 - Regression - regression: allows you to predict relationships - remember Y = mX + b as the equation for a line? We re-write it in regression analysis as Y = a + byX X, Y = variables by = slope (m) (tilt) a = y-intercept (b) (where it hits y-axis) a) if r = + 1 it's easy to predict & draw the line if r < + 1 you must draw a "best fit" line b) some properties of the regression line 1) squared deviations around line are minimal 2) sum deviations = 0 3) new symbols X' & Y' for predictions - To find regression line equation: by = (r) (sy/sx) a = Y - byX Y = a + byX X Y 1 5 r = -1.0 by = (-1)(1.41/1.41) = -1 2 4 3 3 a = 3 - (-1)(3) = 6 4 2 5 1 Y = 6 + (-1) X leave X & Y as letters 3 3 mean 1.41 1.41 s

83

- To Draw the regression line for Y = 6 + (-1) X 1) pick 2 reasonable values for X 2) put in equation & solve for Y 3) plot the 2 pairs of X,Y points 4) connect the dots with a line

- In regression analysis you can also find X = a + bxY and get 2 regression lines that have certain relationship r = 1 r = 0.75 r = 0.25 r = 0 r = + 1 => superimposed r = 0 => perpendicular intersection point = X,Y the centroid

0

1

2

3

4

5

6

0 2 4 6

X

Y

If X = 5 Y = 6 + (-1)(5) = 1 If X = 1 Y = 6 + (-1)(1) = 5

centroid

84

- To predict Y if know X Y' = Y + [ (r)(sy/sx)(X - X)] Given: X = 70 sx = 4 Y = 75 sy = 8 r = 0.6 If Sue got a 62 on X what did she get on Y?

Y' = 75 + [ (0.6) (8/4) (62 - 70) ] = 65.40 - If you have z-scores zy' = (r)(zx) Y' = Y + (zy')(sy) Given: X = 62 X = 70 sx = 4 zx = -2 Y = 75 sy = 8 r = 0.6 a) zy' = (0.6) (-2) = -1.2 b) Y' = 75 + (-1.2)(8) = 65.40

85

- Standard Error of the Estimate (sesty): estimate of the standard deviation of data around the regression line; k2 was a version of this but not really in terms of standard deviation sesty = sy [N ( 1 - r2)] / (N - 2) r = + 1 => sesty = 0 no errors / deviation r = 0 => sesty is maximal Given: X = 70 sx = 4 Y = 75 sy = 8 N = 20 r = 0.6 sesty = 8 [ 20 (1 - 0.62)] / (20-2) = 6.75 Larger sesty => less accurate predictions - recall: Y' was a prediction not a fact. Using sesty we can find an interval where are 68% sure that the true Y will be Ytrue = Y' + sesty 1 + (1/N) + [(X - X)2 / SSx] Sesty & Ytrue are influenced by magnitude of X & Y variance: low variance => better / lower sesty => better Ytrue

86

- Homoscedasticity: where variance of 1 variable is constant at all levels of the other variable - Heteroscedasticity: where variance of 1 variable is not constant at all levels of the other variable Homoscedasticity Heteroscedasticity - Post-Hoc Fallacy: assuming a cause & effect relationship from correlation data

87

CHAPTER 9 HOMEWORK 3 a – e 11 14 15 NOT IN BOOK X = 20 Sx = 5 X = 24 Zx = 0.8 Y = 50 Sy = 7 r = 0.7 a) Zy’ = ? b) Y’ = ?

88


3a) by = (0.36)(0.5 / 12) = 0.015 a = 2.85 - (0.015)(49) = 2.115 Y = 2.115 + 0.015 X b) Y = 2.115 + (0.015)(1) = 2.13 Y = 2.115 + (0.015) (3) = 2.16

c) Y' = 2.85 + (0.36)(0.5 / 12)(65 - 49) = 3.09 d) sesty = 0.5 [60(1 - 0.362)] / (60 - 2) = 0.474 e) r2 = 0.362 = 0.1296 k2 = 1 – 0.1296 = 0.8704 11) no = post hoc fallacy 14) no, could be curvilinear or some other relationship 15) 0.20 => yes, will probably do different 0.90 => no, will do about the same Given: X = 20 sx = 5 X = 24 zx = 0.8 Y = 50 sy = 7 r = 0.7 a) zy' = (0.7)(0.8) = 0.56 b) Y' = 50 + (0.56)(7) = 53.92

0

1

2

3

0 1 2 3 4

x

y

89

2 = [(Oi - Ei)2 / Ei]

est 2 = (t2 - 1) / (t2 + N1 + N2 - 1) df = (r - 1)( c - 1) est 2 = [SSbet - (k - 1)(s2

w)] / (SStot + s2w)

OR est 2 = [dfbet(F - 1)] / [dfbet (F - 1) + Ntot] HSD = q s2

w / n x = / N z = (X - ) / x upper limit = X + (t 0.05)(sx) lower limit = X - (t 0.05)(sx) sx = s / N - 1 t = (X - ) / sx df = N - 1 SS1 = X1

2 - [(X1)2 / N1]

SS2 = X2

2 - [(X2)2 / N2]

Sx1x2 = [(SS1 + SS2) / (N1 + N2 - 2)][(1/N1) + (1/N2)] t = [(X1 - X2) - (1 - 2)] / sx1x2

df = N1 + N2 - 2

SStot = Xtot2 - [(Xtot)

2 / Ntot] dfw = Ntot - k

SSbet = [(Xi)

2 /Ni] - [(Xtot)2 / Ntot] s2

bet = SSbet / dfbet SSw = SStot - SSbet s2

w = SSw / dfw

dfbet = k - 1 F = s2bet / s

2w

90

- Are there any underlying concepts that guide our choice of statistical tests? - Are there standards that we can compare our results to in order to see if there are statistically significant differences? - Are we always right or are there errors we should be aware of?

91

Chapter 11 - Inferential Statistics & Errors

- goal: estimate parameters of pop. from descriptive stats; compare 2+ groups of data 1) hypothesis testing: compare samples for differences - Step #1 = formulate all hypotheses 1) typically have experimental & control groups: manipulated vs comparison groups respectively 2) hypotheses a) null hypothesis (H0): expect no difference b) alternative hypothesis (H1): expect a difference 1) 1-tailed / directional: states how they differ (<, >) 2) 2-tailed / non-directional: just states they differ - Step #2 = conduct the study, collect the data, generate summary statistics (e.g., mean, SD, etc.) - Step #3 = choose appropriate statistical test (i.e., formulas) that will assess the evidence (data) against the null hypothesis by generating a test statistic = a single number that assesses the compatibility of the data with H0

- Step #4 = generate the p-value = the likelihood/probability that the result observed is due to random occurrence if H0 is correct or if H0 is true what is the probability of observing a test statistic as extreme as the one obtained in #3? p-values typically generated by statistical software packages

92

- Step #5a (using software) = compare p-value to a fixed significance level () that the scientific community agrees that there is statistical significance (most common = 0.05 & 0.01): Rule: p < => reject H0 p > => accept H0

= 0.05 p = 0.03 reject H0, are different = 0.01 p = 0.06 accept H0, no different

- Step #5b (by hand) = a) each statistical test is associated with a theoretical

distribution of values (sampling distribution) of what would happen (theoretically) if every sample of a particular size were studied (i.e., what test statistic would you expect for a given sample size)

b) when you generate a test statistic (using a formula) you can then go to a table with the sampling distribution and for a given -level & sample size find what test statistic value would expect if H0 is true – if your test statistic > table value reject H0 = there is a statistically significant difference

- Central Limit Theorum (CLT): method to construct a sampling distribution of the population mean, providing a way to test H0; assumes that if random samples of fixed N from any pop. are drawn & X calculated then: 1) distribution of means becomes normal 2) grand mean approaches mean of pop. 3) standard deviation decreases

93

- standard error of the means: the overall standard deviation of the sample means Since all of this is based on probabilities there is always the risk that you can make an error in your decisions. - decision errors a) Type I (): reject H0 when it's true b) Type II (): accept H0 when it's false true status of null Ho true H0 false your accept correct II / decision Ho (1 - ) reject I / correct Ho (1 - )

- = 0.05 2-tail p = 0.03 1-tail H0: false 0.03 x 2 = 0.06 p > => accept H0 => Type II - = 0.05 1-tail p = 0.06 2-tail H0: true 0.06 / 2 = 0.03 p < => reject H0 => Type I - = 0.05 1-tail p = 0.03 1-tail H0: false p < => reject H0 => correct

Rule: always fix the p-value

94

CHAPTER 11 - HOMEWORK 7 14 15 21 22-26

95

CHAPTER 11 - HOMEWORK 7) approaches normal, mean approaches mean, s decreases 14) no 15) yes; 1-tail => <,> 2-tail => just differ 21) 0.05 p H0 22) p < => reject => Type I 0.01 0.008 T 23) p > => accept => correct 0.05 0.08 T 24) p > => accept => Type II 0.05 0.06 F 25) p < => reject => correct 0.05 0.03 F 26) p < => reject => correct 0.01 0.005 F

96

- John has access to all the records for inductees into the US Army since it began and knows the average IQ and standard deviation for this population. He has a group of new inductees and wants to know if their average IQ differs significantly from past years. - Kelly knows that sampling errors always exist so the sample mean will not exactly match the true population mean. Can she determine a range of values that will cover the true mean with some degree of confidence?

97

Chapter 12 - Single Sample Tests 1) z-test: know & X x = / N z = (X - ) / x

x = standard error of the mean e.g., = 250 = 50 X = 263 N = 100 do the means differ? Use = 0.01 2-tailed x = 50 / 100 = 5 z = (263 - 250) / 5 = 2.60 from z-table: at 0.05 reject if |z| > 1.96 at 0.01 reject if |z| > 2.58 so....2.60 > 2.58 => reject null - they differ

Rule: test statistic > table value => reject null Note: you are now getting the actual test statistic not p-value! Alpha guides you to a place in the table to decide if test statistic is < or > that criterion. Computers provide p-value along with answers.

98

2) Student's t-test: , X & s known sx = s / N - 1 t = (X - ) / sx df = N - 1

e.g., X = 85.1 s = 9.61 N = 10 = 72 do the means differ? Use = 0.01 1-tailed sx = 9.61 / 10 - 1 = 3.2 t = (85.1 - 72) / 3.2 = 4.09

df = 10 - 1 = 9 go to t-table page 551 1) choose 1-tail or 2-tail row 2) get for that row 3) find df = degrees of freedom = # of values free to vary after certain restrictions placed on data (reflection of sample size) so...... 4.09 > 2.821 => reject null, they differ df: # independent scores; e.g., if X = 4.5 & n = 4 and you know 3 of scores are 3, 4 & 5. Total scores must = 18 since 18/4 = 4.5. so last number must be 6.

100

a) confidence limits for X: range of values representing probability that more samples drawn from pop. will fall within it 95% limits 99% limits upper limit = X + (t 0.05)(sx) upper limit = X + (t 0.01)(sx) lower limit = X - (t 0.05)(sx) lower limit = X - (t 0.01)(sx) e.g., X = 108 s = 15 N = 26 sx = 3 df = 25 upper = 108 + (2.06)(3) = 114.18 95% limits lower = 108 - (2.06)(3) = 101.82 t-table at 0.05 ALWAYS 2-TAILED upper = 108 + (2.787)(3) = 116.36 99% limits lower = 108 - (2.787)(3) = 99.64 t-table at 0.01 ALWAYS 2-TAILED NB: 95% limits are "tighter" than 99% 99 95 99 101 108 114 116

101

CHAPTER 12 – HOMEWORK

14 15 29 30

102


14) = 78 = 7 n = 22 X = 82 = 0.01 2-tailed x = 7 / 22 = 1.5 z = (82 - 78) / 1.5 = 2.67 > 2.58 => reject H0 15) = 78 n = 22 X = 82 s = 7 = 0.01 2-tailed sx = 7 / 21 = 1.53 t = (82 - 78) / 1.53 = 2.61 < 2.831 => accept H0

29) X = 45 sx = 2.2 df = 15 upper = 45 + (2.131)(2.2) = 49.69 lower = 45 - (2.131)(2.2) = 40.31 30) upper = 45 + (2.947) (2.2) = 51.48 lower = 45 - (2.947)(2.2) = 38.52

103

- Andy has two groups of rats and wants to see if what he feeds them affects how fast they run through a maze. One group gets mashed protein bars to eat and the other gets mashed bananas. He runs them through the maze and times them. The protein group runs it in 6.5 seconds on average and the banana group runs it in 10.3 seconds. Is there a significant difference? - Is there a way to estimate the degree to which the IV really contributes to the effect seen on the DV?

104

Chapter 13 - 2-Sample Tests - Student's t-test for unknown population SS1 = X1

2 - [(X1)2 / N1]

SS2 = X2

2 - [(X2)2 / N2]

Sx1x2 = [(SS1 + SS2) / (N1 + N2 - 2)][(1/N1) + (1/N2)] t = [(X1 - X2) - (1 - 2)] / sx1x2 ** 1 - 2 = 0 **

df = N1 + N2 - 2 e.g., X1 = 477 X2 = 11 = 0.05 X1

2 = 29845 X22 = 101 1-tail

X1 = 59.625 X2 = 5.5 N1 = 8 N2 = 2 SS1 = 29845 - [(4772)/8] = 1403.875 SS2 = 101 - [(112)/2] = 40.5 Sx1x2 = [(1403 + 40) / (8 + 2 - 2)] [ (1/8) + (1/2)] = 10.62 t = (59.652 - 5.5) / 10.62 = 5.097 > 1.86 => reject H0 df = 8 + 2 - 2 = 8

105

- est 2 (omega-squared): many things contribute to p-level and whether you accept of reject the null; one is 2 or degree to which IV accounts for variance in DV - how much are the 2 variables related? est 2 = (t2 - 1) / (t2 + N1 + N2 - 1) - interpret like r2 - higher 2 means have significant findings e.g., t = 5.097 in previous problem est 2 = (5.0972 - 1) / (5.0972 + 8 + 2 - 1) = 0.714 IV accounts for 71.4% of variance in DV - fairly significant Can follow this with the confidence limits

106

CHAPTER 13 HOMEWORK 4 also find 2

107

CHAPTER 13 - HOMEWORK 4) X1 = 324 X2 = 256 X1

2 = 6516 X22 = 4352 = 0.05

X1 = 18 X2 = 16 2-tailed N1 = 18 N2 = 16 SS1 = 6516 - (3242) / 18 = 684 SS2 = 4352 - (2562) / 16 = 256 Sx1x2 = [(684 + 256) / (18 + 16 - 2)] [(1/18) + 1/16)] = 1.86 t = (18 - 16) / 1.86 = 1.08 < 2.042 => accept H0 df = 32 2 = (1.082 - 1) / (1.082 + 18 + 16 - 1) = 0.005

108

- June has a new drug to control the number of manic episodes patients experience each month, but she is not sure of the most effective dose. She gets 30 manic patients and divides them randomly into 3 groups. She gives one group a low dose, one group a medium dose and one group a high dose of the drug. She then monitors them for one month, recording the number of manic episodes they experience. Group 1 has an average of 6 episodes, group 2 has 3, and group 3 has 5. Do they differ significantly in their effect on the number of manic episodes? - Exactly which doses differ from each other?

109

Chapter 14 - Analysis of Variance (ANOVA) - omnibus test: permits analysis of several variables or variable levels at the same time - one-way ANOVA: analysis of various levels or categories of single treatment variables - why not do lots of t-tests? Will give experimentwise errors = drive up probability of making Type I errors ANOVA: divide total variance into between & within subjects variance Rat test 1 test 2 test 3 X s2 1 6.3 1.3 14.6 7.4 30.1 2 8.2 2.4 18.2 9.6 42.6 3 7.1 1.9 17.3 8.8 40.9 X 7.2 1.9 16.7 within subject S2 0.61 0.20 2.34 variances Between subject variances - ANOVA is based on the General Linear Model: a conceptual mathematical model Xij = + i + ij <= random error or error variance

110

e.g., blood pressure study: do the 3 means differ? = 0.05 active (X1) passive (X2) relaxed (X3) totals X 1407 1303 1308 4018 X2 99723 85479 86254 271456 X 70.35 65.15 65.40 -------- N 20 20 20 60 Step 1: add across all rows to get totals; then do equations 1) SStot = Xtot

2 - [(Xtot)2 / Ntot]

271456 - [(40182) / 60] = 2383.94 2) SSbet = [(Xi)

2 /Ni] - [(Xtot)2 / Ntot] i = individual

14072 + 13032 + 13082 - 40182 20 20 20 60 = 344.04 3) SSw = SStot - SSbet

2383.94 - 344.04 = 2039.9 4) dfbet = k - 1 k = # conditions 3 - 1 = 2 5) dfw = Ntot - k 60 - 3 = 57

111

6) s2

bet = SSbet / dfbet s2

bet = MSbet

344.04 / 2 = 172.02 7) s2

w = SSw / dfw s2w = MS w

2039.9 / 57 = 35.79 8) F = s2

bet / s2w

172.02 / 35.79 = 4.81 9) F-table on page 558 - 560 - across top = dfbet - down left = dfw - light # = at 0.05 - bold # = at 0.01 df = 2,57 2,60 at 0.05 = 3.15 so...... 4.81 > 3.15 => reject H0 the 3 means do differ

115

- F was an omnibus test - it just says the 3 means differ but not which ones; need follow-up tests to determine this a) a priori: decide prior to study what tests or comparisons will do; planned b) a posteriori or post hoc: do all possible pair-wise

comparisons; not planned - Tukey HSD (Honestly Significant Difference) Test (post hoc)

HSD = q s2w / n

1) prepare a means table 70.35 65.15 65.40 70.35 ------ 5.20* 4.95* 65.15 ------ ----- -0.25 65.40 ------ ----- ------ 2) do HSD test HSD = 3.40 35.79 / 20 = 4.54

q comes from table L on page 562 using dfw & k Any of the difference values (| |) in the table > to HSD value get an * meaning they differ significantly.

117

- est 2: degree of association IV & DV est 2 = [SSbet - (k - 1)(s2

w)] / (SStot + s2w)

est 2 = [344.04 - (3 - 1)(35.79)] / (2383.94 + 35.79) = 0.11 OR est 2 = [dfbet(F - 1)] / [dfbet (F - 1) + Ntot] est 2 = [2(4.81 - 1)] / [2 (4.81 - 1) + 60] = 0.11

118

CHAPTER 14 HOMEWORK 7 d, e, f, h

119

CHAPTER 14 - HOMEWORK 7 d) X1 X2 X3 totals X 15 20 30 65 X2 81 116 216 413 X 3 4 6 ---- n 5 5 5 15 SStot = 413 - (652) / 15 = 131.33 SSbet = 152 + 202 + 302 - 652 5 5 5 15 = 23.33 SSw = 131.33 – 23.33 = 108 dfbet = 3 - 1 = 2 dfw = 15 - 3 = 12 s2

bet = 23/2 = 11.5 s2

w = 108/12 = 9 F = 11.5/9 = 1.28 e) 1.28 < 3.88 => accept H0

f) est 2 = [23.33 - (3 - 1)(9)] / (131.33 + 9) = 0.038 h) 3 4 6 HSD = 3.77 9/5 = 5.06 3 --- -1 -3 none are different 4 --- --- -2 6 --- --- ---

120

- Ed polls a random sample of people by phone to see how much they agree with the statement that the president is doing a good job: very good, good, neutral, poor, very poor. Is there a difference in the frequency with which people give responses for the different categories? - Kathy wants to know if people will help someone more or less as a function of gender of the person needing help. She has Bob & Ann pretend to drop a bag of groceries on a busy street and records how many times people stop to help either one of them. Was there a significant difference in helping versus non-helping for Bob vs Ann?

121

Chapter 17 - Chi-Squared Test (2) - nonparametric: does not require normality - 2: typically with frequencies or proportions from nominal data 1) one-variable X2 or "goodness of fit" 2 = [(Oi - Ei)

2 / Ei] O = observed data E = expected data i = individual strong strong agree agree undecided disagree disagree 7 12 13 13 10 expected = total answers / # categories = 55/5 = 11 X2 = (7 - 11)2 + (12 - 11)2 + (13 - 11)2 + (13 - 11)2 + (10 - 11)2 11 11 11 11 11 = 2.3 df = N - 1 (n = # categories) df = 5 - 1 = 4 X2 table on page 572 at 0.05 => 9.488 2.3 < 9.488 => accept H0 no difference

123

2) multi-variable X2: same formula but different way to get expected get better get worse drug a 1 b 17 18 placebo c 9 d 12 21 10 29 39 1) label boxes a - d 2) find expected values a) (18/39) (10) = 4.6 b) (18/39) (29) = 13.4 x c) (21/39) (10) = 5.4 d) (21/39) (29) = 15.6 3) use X2 formula a b c d (1 - 4.6)2 + (17 - 13.4)2 + (9 - 5.4)2 + (12 - 15.6)2 4.6 13.4 5.4 15.6 = 7.09 df = (r - 1)(c - 1) r = # rows c = # columns df = (2 - 1)(2 - 1) = 1 7.09 > 6.635 => reject H0 they differ

fe = fcfr/n

124

CHAPTER 17 – HOMEWORK

5 8

125

CHAPTER 17 - HOMEWORK 5) 4 5 6 7 use = 0.05 11 15 13 29 68/4 = 17 (11 - 17)2 + (15 - 17)2 + (13 - 17)2 + (29 - 17)2 17 17 17 17 = 11.78 df = 4 - 1 = 3 11.78 > 7.815 => reject H0

8) A B H a 75 b 45 120 use = 0.05 NH c 40 d 80 120 115 125 240 a) (120/240)(115) = 57.5 b) (120/240)(125) = 62.5 c) (120/240)(115) = 57.5 d) (120/240)(125) = 62.5 (75 - 57.5)2 + (45 - 62.5)2 + (40 - 57.5)2 + (80 - 62.5)2 57.5 62.5 57.5 62.5 = 20.4 df = (2 - 1)(2 - 1) = 1 20.4 > 3.841 => reject H0 they differ

126

true limits = + / - ½ the unit of measurement i = (hi - lo + 1) / # groups midpoint = (hi true + lo true) / 2 PR = cumfll + ((X - Xll) / i)(fi) x 100 N cumf = (PR x N) / 100 X = Xll + [[i (cumf - cumfll)] / fi] cumfll = cum freq at lower true limit of X X = score Xll = score at lower true limit of X i = width fi = # cases in X's group N = total # scores

127

z = (X - X) / s = (X - ) / SIR = (Q3 - Q1) / 2 X = X / n s3 = [3(X - median)] / s Range = hi - lo

Xw = fX / ntot

s4 = 3 + [ (Q3 - Q1) / 2 (P90 - P10)] md = Xll + i [ ((N/2) - cumfll) / fi] s2 = (X - X)2 / n s = s2 SS = X2 - (X)2/n s2 = SS/n s = s2

128

sesty = sy [N ( 1 - r2)] / (N - 2) r = (zxzy) / N by = (r) (sy/sx) a = Y - byX Y = a + byX rs = 1 - [ (6D2) / [N (N2 - 1)]] 1 = r2 + k2 zy' = (r)(zx) Y' = Y + (zy')(sy) Y' = Y + [ (r)(sy/sx)(X - X)] r = XY - [(X)(Y) / N] [X2 - [(X)2 /N]] [Y2 - [(Y)2 / N]]

129

2 = [(Oi - Ei)2 / Ei]

est 2 = (t2 - 1) / (t2 + N1 + N2 - 1) df = (r - 1)( c - 1) est 2 = [SSbet - (k - 1)(s2

w)] / (SStot + s2w)

OR est 2 = [dfbet(F - 1)] / [dfbet (F - 1) + Ntot] HSD = q s2

w / n x = / N z = (X - ) / x upper limit = X + (t 0.05)(sx) lower limit = X - (t 0.05)(sx) sx = s / N - 1 t = (X - ) / sx df = N – 1 upper limit = X + (t 0.01)(sx) lower limit = X - (t 0.01)(sx) SS1 = X1

2 - [(X1)2 / N1]

SS2 = X2

2 - [(X2)2 / N2]

Sx1x2 = [(SS1 + SS2) / (N1 + N2 - 2)][(1/N1) + (1/N2)] t = [(X1 - X2) - (1 - 2)] / sx1x2

df = N1 + N2 - 2

SStot = Xtot2 - [(Xtot)

2 / Ntot] dfw = Ntot - k

SSbet = [(Xi)

2 /Ni] - [(Xtot)2 / Ntot] s2

bet = SSbet / dfbet SSw = SStot - SSbet s2

w = SSw / dfw

dfbet = k - 1 F = s2bet / s

2w

130

EXTRA CREDIT PACKET ANSWERS

Chapter 1

1.a. height

1.b. gender

1.c. yes

2. population

3. sample

4. statistic

5. parameter

131

Chapter 2

1. ratio

2. ordinal

3. nominal

4. interval

5. if your weight = 150 then unit of measurement = 1 and ½

of 1 = 0.5 so 150 + 0.5 = 150.5 & 150 – 0.5 = 149.5

6.

Males Females

Scuba 23 36

Read 41 21

TV 38 40

Visit 8 33

a. 110:130

b. 38

c. 36/(23+36) = 36/59 = 0.61

d. 8/110 = 0.07 x 100 = 7.00

132

Chapter 3

1.a. (81 – (-60) + 1)/6 = 23.67 ~ 24

1.b.

Class true limits midpoint f cum freq cum%

60-83 59.5/83.5 71.5 11 20 100.0

36-59 35.5/59.5 47.5 3 9 45.0

12-35 11.5/35.5 23.5 0 6 30.0

(-12)-11 -12.5/11.5 -0.5 0 6 30.0

(-36)-(-13) -36.5/-12.5 -24.5 0 6 30.0

(-60)-(-37) -60.5/-36.5 -48.5 6 6 30.0

c. if score = (-36)

PR = [[6 + (((-36)-(-36.5))/24)(0)]/20]x100 = 30

d. (75x20)/100 = 15

X = 59.5 + [[24(15 – 9)]/11] = 72.59

133

Chapter 7

Males Females

Scuba 38 41

Read 3 2

TV 27 52

Visit 42 35

1.

0

10

20

30

40

50

60

f

scuba read tv visit

activity

males

females

2.

0

20

40

60

80

100

120

‐48.5 ‐24.5 ‐0.5 23.5 47.5 71.5

mdpt change

cum %

134

3.

Males Females

Senior 51 59

Junior 23 40

Soph 30 27

frosh 6 3

0

10

20

30

40

50

60

f

fr so ju sr

class

males

females

135

Chapter 4

1. a. 46/14 = 3.29

1.b. 220, 181, 155, 155, 155, 114

x f cumf

220 1 6

181 1 5 6/2 = 3

155 3 4

114 1 1

md = 154.5 + 1[((6/2) – 1)/3] = 155.17

1.c. 114, 126, 138

1.d.

Mean f fX

Senior -11.23 85 -954.55

Junior 12.91 62 800.42

Soph -14.28 48 -685.44

frosh 16.78 35 587.30

230 -252.27

Xw = (-252.27)/230 = (-1.10)

136

Chapter 5

1a. 9-(-5) = 14

1.b.

X X-X (X-X)2

5 4.125 17.02 X = 0.875

-4 -4.875 23.77

1 0.125 0.02

-6 -6.875 47.27

8 7.125 50.77

0 -0.875 0.77

5 4.125 17.02

-2 -2.875 8.27

0 164.91

s2 = 164.91/8 = 20.61 s = √20.61 = 4.54

137

1.c.

X X2

4 16

-2 4

6 36

-5 25

0 0

9 81

1 1

-4 16

9 179

SS = 179 – (92)/8 = 168.88

s2 = 168.88/8 = 21.11

s = √21.11 = 4.59

138

Chapter 6

1.a. (4 – -3.2)/1.8 = 4

0.5 + 0.4997 = 0.9997 x 100 = 99.97

1.b. (1 – -3.2)/1.8 = 2.33

0.0099 x 100 = 0.99

1.c. (4 - -3.2)/1.8 = 4

(-4 - -3.2)/1.8 = -0.44

0.4997 + 0.1700 = 0.6697 x 100 = 66.97

1.d. (1 – -3.2)/1.8 = 2.33

(2 – -3.2)/1.8 = 2.89

0.4981 – 0.4901 = 0.008 x 100 = 0.80

1.e. 0.25 = (X – -3.2)/1.8 0.4

X = -2.75

1.f. 0.15 = (X – -3.2)/1.8 0.06 0.06

X = -2.93

-0.15 – (X – -3.2)/1.8

X = -3.47

1.g. 1.96 = (X- -3.2)/1.8 =0.328 0.025 0.025

-1.96 = (X- -3.2)/1.8 = -6.728

139

Chapter 8

1. X X2 Y Y2 XY

-3 9 13 169 -39

1 1 -14 196 -14

7 49 21 441 147

6 36 19 361 114

-5 25 -19 361 95

2 4 -15 225 -30

8 124 5 1753 273

R = 273 – [(8)(5)/6]

√[124 - [(82)/6)][1753 - [(52)/6]

= 273 – 6.67/√(113.33)(1748.83) = 266.33/445.19 = 0.60

2.

‐30

‐20

‐10

0

10

20

30

‐10 ‐5 0 5 10

hrs tv

wt

140

3.

Alone With D D2

Scuba 3 2 1 1

Read 1 4 -3 9

TV 4 3 1 1

Visit 2 1 1 1

0 12

rs = 1 – [(6*12)/[4(16-1)]] = -0.2

141

Chapter 9

1. by = (0.83)(4.19/12.63) = 0.28

a = -16.42 – (0.28)(67.94) = -35.44

Y = -35.44 + 0.28X

2.

‐40

‐30

‐20

‐10

0

0 20 40 60

ht

wt

3. Y’ = -16.42 + [(0.83)(4.19/12.63)(68 – 67.94)] =

-16.40

142

Chapter 12

1. x = 2.6/√230 = 0.17 z = (67.94 – 68.9)/0.17 = -5.65

Table = 1.96 < |-5.65| => reject null

2. sx = 1.63/√229 = 0.11 t = (-2.2 – 6.8)/0.11 = -81.82

df = 229 table = 2.576 < |-81.82| => reject null

3. upper = -2.2 + (1.96)(0.11) = -1.98

lower = -2.2 – (1.96)(0.11) = -2.42

143

Chapter 13

1. SS1 = 93568 – [14322/22] = 357.82

SS2 = 111626 – [17662/28] = 241.86

Sx1x2 = √[357.82 + 241.86)/(22+28-2)][(1/22)+(1/28)]

= √[599.68/48][0.08] = 1.00

t = [65.09 – 63.07]/1.00 = 2.02 df = 22 + 28 – 2 = 48

table = 1.684 < 2.02 => reject null

2. est 2 = (2.022 – 1)/(2.022 + 22 + 28 – 1) = 3.08/53.08 =

0.06

144

Chapter 14

Sr Jr So Fr TOT

X 59 40 92 101 292

X2 285 132 604 707 1728

Mean 3.93 2.67 6.13 6.73 ------

N 15 15 15 15 60

1. SStot = 1728 – [2922/60]= 306.93

SSbet = [(592/15)+(402/15)+(922/15)+(1012/15)]-[ 2922/60]

= [1583.07]-[1421.07] = 162

SSw = 306.93 – 162 = 144.93

dfbet = 4-1 = 3 dfw = 60-4 = 56

s2bet = 162/3 = 54 s2

w = 144.93/56 = 2.59

F = 54/2.59 = 20.85

Table = 2.78 < 20.85 => reject null

145

2.

3.93 2.67 6.13 6.73

3.93 0 1.26 -2.2* -2.8*

2.67 ------ 0 -3.46* -4.06*

6.13 ------ ------- 0 -0.6

6.73 ------ ------- ------- 0

HSD = 3.74√ 2.59/15 = 1.55

3. est2 = [162 – (4-1)(2.59)]/(306.93 + 2.59) = 0.50

146

Chapter 17

1. 24+59+87+41+19/5 = 46 expected

X2 = (24-46)2/46 + (59-46)2/46 + (87-46)2/46 + (41-46)2/46 +

(19-46)2/46 = 10.52 + 3.67 + 36.54 + 0.54 + 15.85 = 67.12

df = 5-1 = 4 table = 13.277 => reject null

2.

Cinnabon salad total

Males 69 (a) 41 (b) 110

Females 70 (c) 50 (d) 120

total 139 91 230

a) (110/230)*139 = 66

b) (110/230)*91 = 44

c) (120/230)*139 = 73

d) (120/230)*91 = 47

X2 = (69-66)2/66 + (41-44)2/44 + (70-73)2/73 +(50-47)2/47 =

0.14 + 0.20 + 0.12 + 0.19 = 0.65

df = (2-1)(2-1) = 1

147

table = 3.841 > 0.65 => accept null

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