# University of Central Florida Psychology Statistics Questions

9/10/22Normal Distribution

Normal Distribution

•

•

•

•

•

•

Standard Deviation

160

Reading ability

Introversion

Job satisfaction

Intelligence

Empathy

Aggression

• C. F. Gauss

• A. Quételet (l’homme moyen)

• US Army Corps of Engineers

Z scores

161

Z-Scores Example 1

Z-Scores Example 2

• Assuming that CAT scores among U.S. college

students are approximately normally

distributed with a mean of 302 (and a made

up standard deviaBon of 10).

• Assuming the same mean (302) and standard

deviation (10), what is the range of scores for

the middle 95% of CAT scores?

• What is the z-score for someone who scored a

312 on the CAT?

162

163

Extra practice

• In the United States, the average IQ is 100,

with a standard deviation of 15. What

percentage of the population would you

expect to have an IQ lower than 85?

164

Extra practice

• z = (x – xbar)/s

• z = (85-100)/15 = -15/15

• z = -1.00

• Which portion of the normal distribution?

• Smaller or Larger?

• p = 0.159

165

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9/10/22

Extra practice

Extra practice

• z = (x – xbar)/s

• z90 = (90-100)/15 = -10/15

• z90 = -0.66

• p120 = 0.092

• p90 ∞ , sampling distribution -> Normal

186

187

• We use sample mean as an esOmate

of the populaOon mean (µ = 𝑥! ) .

• We use s,n to calculate the SE

• What does LLN predict about SE?

• n -> ∞ , the sampling

distribuOon becomes normal.

• SE = s/√n

• What does CLT predict about SE?

• n -> ∞ , 𝑥 -> μ

• n -> ∞ , 𝑆𝐸 -> 0

188

189

Why should I care about the SE?

• Means of multiple samples taken from the same

population vary (i.e., they differ from each other)

• This variation among means can be considered a

type of error in sampling (i.e., Sampling variability)

• We can use this estimate of error to create

confidence intervals.

• 90% CI = 𝑥 ± 𝑆𝐸 ∗ 𝑍3.45

• 95% CI = 𝑥 ± 𝑆𝐸 ∗ 𝑍3.64

• 99% CI = 𝑥 ± 𝑆𝐸 ∗ 𝑍7.58

190

191

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9/10/22

Example.

Sperm count of

Japanese Quail

• Population mean

of 15 million

• 𝑥 and s vary

among samples

• 95% CI around

each men

• 95% of samples

will contain pop

mean within CIs

• Any sample CI has

a .95 probability

of containing the

true population

value

192

6

9/16/22

Hypothesis Testing

• Karl Popper and Null Hypothesis Testing

Framework

• Psychology was not a science, because the field

made untestable claims à Sigmund Freud

• Falsifiability: Easier to falsify H0 than validate HA

• Confidence Intervals are a way to test H0

• Find p that H0 falls within a Confidence Interval?

• If p -> 0, the it would be absurd to not reject H0 ;

thus validating HA*

• Consider the Claim: Physical activity affects

cognitive performance

197

Claim: CF affected by PA

• Testable?

– Not yet

• DV/Outcome?

– CF

• How operationalize?

– Test scores

– LoM?

• IV/Predictor?

– PA

• How operationalize?

– Cheapest -> one sample compared to known population

• Testable H0

– No difference test scores (𝑥 = μ)

198

CI around μ vs. 𝑥̅

Claim: CF affected by PA

HA is true

∆ = 200; PA affects CF

Population

H0 is true

∆ ≠ 200; Sampling variability

• In one sample of size n, you observe one 𝑥,̅

which leads to one ∆ (𝑥̅ − 𝜇) ≠ 0

• Decision (Null hypothesis significance testing)

– 𝐻!: ∆ ≠ 0; 𝑥 ≠ 𝜇; PA -> CF

– 𝐻”: ∆ = 0; 𝑥 = 𝜇; sampling variability

2 possibilities for the raw

observed effect (∆= 200):

1000

Sampling Distribution

(Theoretical)

Sample

m = 1200

s = 150

n = 25

1200

199

200

Confidence Interval around

population mean

Confidence Interval: As N -> ∞

• Confidence interval

SE = s/√n

• 95% Confidence interval

– UB/LB: 𝜇 ± (1.96 x s/√n)

– Upper Bound: Mean + (z x SE)

– Lower Bound: Mean – (z x SE)

• Our small sample of students

– s = 150 and n=25

– UB/LB: 𝜇 ± (1.96 x 150/√25)

– UB/LB: 𝜇 ± (1.96 x 30)

– UB/LB: 𝜇 ± 58.8

– UB/LB: 1000 ± 58.8

– [941.2 1058.8]

– Does 𝑥 (1200) fall outside 95%CI around population mean?

Z = 1.64 for 90% CI

Z = 1.96 for 95% CI

Z = 2.58 for 99% CI

• 95% Confidence interval

– UB/LB: Mean ± (1.96 x s/√n)

201

202

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9/16/22

CI around μ vs. 𝑥̅

2 possibilities for the raw

observed effect (∆= 200):

2 ways of saying the same thing

HA is true

∆ = 200; PA affects CF

Population

H0 is true

∆ ≠ 200; Sampling variability

If H0 were true: 95% of

sample means from

941-1059 just from

sampling variability.

1000

Sampling Distribution

(Theoretical)

[941.2 — 1000 — 1058.8]

Sample

(Physical Activity

Intervention)

1200

– 95% p a sample mean will fall within this range

just from sampling variability

• CI around sample mean:

– 95% p the population mean falls within this range

just from sampling variability

p of 𝑥 b/t 941-1059 is

0.95

p of 𝑥 of 1200 (i.e.,

𝑥 < 941 or 𝑥 > 1059)

just from sampling

variability is less than

5% (p < 0.05)
m = 1200
s = 150
n = 25
• CI around population mean:
• Same decision each time with respect to H0
p(∆=200|H0) < 0.05
203
204
Confidence Interval around
sample mean
• 95% Confidence interval
– UB/LB: 𝑥 ± (1.96 x s/√n)
CI around 𝑥̅ vs. μ
2 possibilities for the raw effect (∆=200)
H A is true
(i.e., ∆ ≠ 0; PA affects CF)
H 0 is true
(i.e., ∆ = 0; Sampling variability)
𝜇 = 1000
• Our small sample of students
– UB/LB: 𝑥 ± 58.8
– UB/LB: 1200 ± 58.8
– [1141.2 1258.8]
– Does 𝜇 (1000) fall outside 95% CI around sample
mean?
95% Confidence Interval
In other words, the
probability of
observing a difference
this large (∆ = 200)
just from sampling
variability is so low
(p 𝛼 = 0.32
• Y axis is n (%) of sample
• 90% CI (1.64 SE) -> 𝛼 = 0.10

• % -> p

• 𝛼 = (100% – CI%) / 100

• 68% CI (1.00 SE) -> p = 0.68

• 90% CI (1.64 SE) -> p = 0.90

Confidence Interval for any z

• 95% CI?

– 100% – 95% = 5%

– 5% = 𝛼 of 0.05

– But 0.05 in both tails of Normal Distribution!

– 0.05 ÷ 2 = 0.025

– z table: 𝛼 -> z

– z = 1.96

34

DISCOVERING STATISTICS USING SPSS

FIGURE 1.15

0.40

The probability

density function

of a normal

distribution

0.35

Probability = .95

0.30

Density

0.25

0.20

0.15

Probability = .025

Probability = .025

0.10

0.05

0.00

−4.00

212

213

−3.00

−1.96

−1.00

0.00

z

1.00

1.651.96

3.00

4.00

data. That is to go where eagles dare, and no one should fly where eagles dare; but to

become scientists we have to, so the rest of this book attempts to guide you through the

various models that you can fit to the data.

1.7. Reporting data !

1.7.1. Dissemination of research !

Confidence Interval for any z

• 90% CI?

Confidence Interval for any z

• 60% CI?

– 100% – 90% = 10%

– 10% = 𝛼 of 0.10

– But 0.10 in both tails of Normal Distribution!

– 0.10 ÷ 2 = 0.05

– Check table for corresponding z

– z = 1.64

214

– 100% – 60% = 40%

– 40% = 𝛼 of 0.40

– But 0.40 in both tails of Normal Distribution!

– 0.40 ÷ 2 = 0.20

– Check table for corresponding z

– z = 0.84

01-Field 4e-SPSS-Ch-01.indd 34

07/11/2012 6:14:56 PM

215

We could create any size (0-100)* confidence interval

using all the z’s in the z table

Sample

(Physical Activity

Intervention)

m = 1200

s = 150

n = 25

1200

[ 1150.8 —- 1249.2 ]

[ 1141.2 ————- 1258.8 ]

[ 1122.6 ———————- 1277.4 ]

216

Having established a theory and collected and started to summarize data, you might

want to tell other people what you have found. This sharing of information is a fundamental part of being a scientist. As discoverers of knowledge, we have a duty of

care to the world to present what we find in a clear and unambiguous way, and with

enough information that others can challenge our conclusions. Tempting as it may be

to cover up the more unsavoury aspects of our results, science should be about ‘the

truth’. We tell the world about our findings by presenting them at conferences and in

articles published in scientific journals. A scientific journal is a collection of articles

written by scientists on a vaguely similar topic. A bit like a magazine, but more tedious. These articles can describe new research, review existing research, or might put

forward a new theory. Just like you have magazines such as Modern Drummer, which is

about drumming, or Vogue, which is about fashion (or Madonna, I can never remember which), you get journals such as Journal of Anxiety Disorders, which publishes

articles about anxiety disorders, and British Medical Journal, which publishes articles

about medicine (not specifically British medicine, I hasten to add). As a scientist, you

submit your work to one of these journals and they will consider publishing it. Not

everything a scientist writes will be published. Typically, your manuscript will be given

to an ‘editor’ who will be a fairly eminent scientist working in that research area who

has agreed, in return for their soul, to make decisions about whether or not to publish

articles. This editor will send your manuscript out to review, which means they send it

to other experts in your research area and ask those experts to assess the quality of the

work. The reviewers’ role is to provide a constructive and even-handed overview of

the strengths and weaknesses of your article and the research contained within it. Once

these reviews are complete the editor reads them all, and assimilates the comments

z = 1.64 for 90% CI

z = 1.96 for 95% CI

z = 2.58 for 99% CI

LLN

• Sampling distribution

becomes normal when

sample sizes gets larger

• SE = s/√n

• In order for sampling

distribution to be

perfectly normal, n = ∞

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9/16/22

Experimental design

z vs. t

• In practice, z is often not used for inferential

statistics

– data are rarely normal enough

– often related to sample size

• t is for degenerate cases of normal distribution

– i.e., when sample is not large enough to be

perfectly normal

218

219

t for CIs

t for CIs

• Confidence Intervals

• (𝑥̅ ± t * SE)

• (𝑥̅ ± t * s/√n)

• t indexed by 3 criteria

• df (i.e., n-1)

• α (total area under curve outside ± t)

• Always 2 sided for CIs

• Confidence Intervals

• (𝑥̅ ± tα, df, 2-sided * SE)

• (𝑥̅ ± tα, df, 2-sided * s/√n)

• t indexed by 3 criteria

• df

•α

• # sides to test

220

34

FIGURE 1.15

221

z for 95% confidence interval

DISCOVERING STATISTICS USING SPSS

How to find the correct t for any CI

0.40

The probability

density function

of a normal

distribution

• All CIs are 2 sided by definition

• Find 𝛂 corresponding with __% CI

0.35

Probability = .95

0.30

Density

0.25

• Calculate df (n-1)

0.20

• Find corresponding t in t table

0.15

Probability = .025

Probability = .025

80%

0.10

90%

95%

98%

99%

0.05

0.00

−4.00

−3.00

−1.96

−1.00

0.00

z

1.00

1.651.96

3.00

4.00

data. That is to go where eagles dare, and no one should fly where eagles dare; but to

become scientists we have to, so the rest of this book attempts to guide you through the

various models that you can fit to the data.

For Z = 1.96 (95% CI)

• 2 side test = 0.050

1.7. Reporting data

• 1!side test = 0.025

1.7.1. Dissemination of research !

222

Having established a theory and collected and started to summarize data, you might

want to tell other people what you have found. This sharing of information is a fundamental part of being a scientist. As discoverers of knowledge, we have a duty of

care to the world to present what we find in a clear and unambiguous way, and with

enough information that others can challenge our conclusions. Tempting as it may be

to cover up the more unsavoury aspects of our results, science should be about ‘the

truth’. We tell the world about our findings by presenting them at conferences and in

articles published in scientific journals. A scientific journal is a collection of articles

written by scientists on a vaguely similar topic. A bit like a magazine, but more tedious. These articles can describe new research, review existing research, or might put

forward a new theory. Just like you have magazines such as Modern Drummer, which is

about drumming, or Vogue, which is about fashion (or Madonna, I can never remember which), you get journals such as Journal of Anxiety Disorders, which publishes

articles about anxiety disorders, and British Medical Journal, which publishes articles

about medicine (not specifically British medicine, I hasten to add). As a scientist, you

submit your work to one of these journals and they will consider publishing it. Not

everything a scientist writes will be published. Typically, your manuscript will be given

223

4

9/16/22

𝜇

224

225

Confidence intervals

to test H0 involving 𝛍

• When 𝛍 falls outside 95% CI around sample mean

• if H0 were true, then the p of observing a ∆ this

large between 𝑥̅ and 𝛍 just from sampling

variability is ≤ 𝛂 (.05)

• When 𝛍 falls inside 95% CI around sample mean

• if H0 were true, then the p of observing a ∆ this

large between 𝑥̅ and 𝛍 just from sampling

variability is > 𝛂 (.05)

226

5

Assignment 2

i

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

Class A

grade

xbar

56 67.813

79 67.813

62 67.813

70 67.813

72 67.813

77 67.813

61 67.813

62 67.813

74 67.813

79 67.813

80 67.813

71 67.813

68 67.813

62 67.813

42 67.813

70 67.813

2

xi – xbar (xi – xbar)

-11.813 139.547

11.187 125.149

-5.813

33.791

2.187

4.783

4.187

17.531

9.187

84.401

-6.813

46.417

-5.813

33.791

6.187

38.279

11.187 125.149

12.187 148.523

3.187

10.157

0.187

0.035

-5.813

33.791

-25.813 666.311

2.187

4.783

z p below p above

-1.18

0.12

0.88

1.11

0.87

0.13

-0.58

0.28

0.72

0.22

0.59

0.41

0.42

0.66

0.34

0.91

0.82

0.18

-0.68

0.25

0.75

-0.58

0.28

0.72

0.62

0.73

0.27

1.11

0.87

0.13

1.21

0.89

0.11

0.32

0.63

0.37

0.02

0.51

0.49

-0.58

0.28

0.72

-2.57

0.01

0.99

0.22

0.59

0.41

STUDENT ID: _____________________

Class B

grade

xbar

79 75.938

80 75.938

80 75.938

69 75.938

73 75.938

82 75.938

77 75.938

81 75.938

74 75.938

79 75.938

61 75.938

90 75.938

90 75.938

57 75.938

73 75.938

70 75.938

xi – xbar (xi – xbar)2

z p below p above

3.062

9.376 0.340

0.63

0.37

4.062

16.500 0.450

0.67

0.33

4.062

16.500 0.450

0.67

0.33

-6.938

48.136 -0.780

0.22

0.78

-2.938

8.632 -0.330

0.37

0.63

6.062

36.748 0.680

0.75

0.25

1.062

1.128 0.120

0.55

0.45

5.062

25.624 0.570

0.72

0.28

-1.938

3.756 -0.220

0.41

0.59

3.062

9.376 0.340

0.63

0.37

-14.938 223.144 -1.670

0.05

0.95

14.062 197.740 1.570

0.94

0.06

14.062 197.740 1.570

0.94

0.06

-18.938 358.648 -2.120

0.02

0.98

-2.938

8.632 -0.330

0.37

0.63

-5.938

35.260 -0.660

0.25

0.75

xbar =

67.813

xbar =

75.938

n =

16.000

n =

16.000

SS = 1512.438

SS = 1196.940

df =

15.000

df =

15.000

s2 =

100.829

s2 =

79.796

s =

10.041

10.04

s =

8.933

8.93

Assignment 2

i

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

Class C

grade

xbar xi – xbar (xi – xbar)2

71 74.125 -3.125

9.766

70 74.125 -4.125 17.016

78 74.125

3.875 15.016

74 74.125 -0.125

0.016

77 74.125

2.875

8.266

76 74.125

1.875

3.516

62 74.125 -12.125 147.016

78 74.125

3.875 15.016

66 74.125 -8.125 66.016

86 74.125 11.875 141.016

88 74.125 13.875 192.516

77 74.125

2.875

8.266

65 74.125 -9.125 83.266

71 74.125 -3.125

9.766

76 74.125

1.875

3.516

71 74.125 -3.125

9.766

z p below p above

-0.45

0.33

0.67

-0.59

0.28

0.72

0.56

0.71

0.29

-0.02

0.49

0.51

0.41

0.66

0.34

0.27

0.61

0.39

-1.74

0.04

0.96

0.56

0.71

0.29

-1.16

0.12

0.88

1.70

0.96

0.04

1.99

0.98

0.02

0.41

0.66

0.34

-1.31

0.10

0.90

-0.45

0.33

0.67

0.27

0.61

0.39

-0.45

0.33

0.67

STUDENT ID: _____________________

Class D

grade

xbar xi – xbar (xi – xbar)2

68 78.750 -10.750 115.563

76 78.750 -2.750

7.563

86 78.750

7.250 52.563

72 78.750 -6.750 45.563

79 78.750

0.250

0.063

77 78.750 -1.750

3.063

69 78.750 -9.750 95.063

80 78.750

1.250

1.563

91 78.750 12.250 150.063

91 78.750 12.250 150.063

84 78.750

5.250 27.563

70 78.750 -8.750 76.563

82 78.750

3.250 10.563

87 78.750

8.250 68.063

77 78.750 -1.750

3.063

71 78.750 -7.750 60.063

xbar =

74.125

xbar =

78.750

n =

16.000

n =

16.000

SS = 729.756

SS = 867.008

df =

15.000

df =

15.000

s2 =

48.650

s2 =

57.801

s =

6.975

6.97

s =

7.603

7.60

z p below p above

-1.41

0.08

0.92

-0.36

0.36

0.64

0.95

0.83

0.17

-0.89

0.19

0.81

0.03

0.51

0.49

-0.23

0.41

0.59

-1.28

0.10

0.90

0.16

0.56

0.44

1.61

0.95

0.05

1.61

0.95

0.05

0.69

0.75

0.25

-1.15

0.13

0.87

0.43

0.67

0.33

1.09

0.86

0.14

-0.23

0.41

0.59

-1.02

0.15

0.85