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Practice. The Neuroticism Measure = 23.32 S = 6.24 n = 54 How many people likely have a neuroticism score between 29 and 34?. Practice. (29-23.32) /6.24 = .91 area = .3186 ( 34-23.32)/6.26 = 1.71 area =.4564 .4564-.3186 = .1378 .1378*54 = 7.44 or 7 people. Practice.

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Practice
Practice

  • The Neuroticism Measure

    = 23.32

    S = 6.24

    n = 54

    How many people likely have a neuroticism score between 29 and 34?


Practice1
Practice

  • (29-23.32) /6.24 = .91

  • area = .3186

  • ( 34-23.32)/6.26 = 1.71

  • area =.4564

  • .4564-.3186 = .1378

  • .1378*54 = 7.44 or 7 people


Practice2
Practice

  • On the next test I will give an A to the top 5 percent of this class.

  • The average test grade is 56.82 with a SD of 6.98.

  • How many points on the test did you need to get to get an A?



Step 2 look in table z
Step 2: Look in Table Z

Z score = 1.64

.05


Step 3 find the x score that goes with the z score
Step 3: Find the X score that goes with the Z score

  • Must solve for X

  • X =  + (z)()

  • 68.26 = 56.82 + (1.64)(6.98)


Step 3 find the x score that goes with the z score1
Step 3: Find the X score that goes with the Z score

  • Must solve for X

  • X =  + (z)()

  • 68.26 = 56.82 + (1.64)(6.98)

  • Thus, a you need a score of 68.26 to get an A


Practice3
Practice

  • The prestigious Whatsamatta U will only take people scoring in the top 97% on the verbal section SAT (i.e., they reject the bottom 3%).

  • What is the lowest score you can get on the SAT and still get accepted?

  • Mean = 500; SD = 100



Step 2 look in table c
Step 2: Look in Table C

Z score = -1.88

.03


Step 3 find the x score that goes with the z score2
Step 3: Find the X score that goes with the Z score

  • Must solve for X

  • X =  + (z)()

  • 312 = 500 + (-1.88)(100)


Step 3 find the x score that goes with the z score3
Step 3: Find the X score that goes with the Z score

  • Must solve for X

  • X =  + (z)()

  • 312 = 500 + (-1.88)(100)

  • Thus, you need a score of 312 on the verbal SAT to get into this school


Is this quarter fair
Is this quarter fair?

  • How could you determine this?

  • You assume that flipping the coin a large number of times would result in heads half the time (i.e., it has a .50 probability)


Is this quarter fair1
Is this quarter fair?

  • Say you flip it 100 times

  • 52 times it is a head

  • Not exactly 50, but its close

    • probably due to random error


Is this quarter fair2
Is this quarter fair?

  • What if you got 65 heads?

  • 70?

  • 95?

  • At what point is the discrepancy from the expected becoming too great to attribute to chance?









Compare group to population
Compare Group to Population

Population Happiness Score


The theory of hypothesis testing
The Theory of Hypothesis Testing

  • Data are ambiguous

  • Is a difference due to chance?

    • Sampling error


Population
Population

  • You are interested in the average self-esteem in a population of 40 people

  • Self-esteem test scores range from 1 to 10.


Population scores

1,1,1,1

2,2,2,2

3,3,3,3

4,4,4,4

5,5,5,5

6,6,6,6

7,7,7,7

8,8,8,8

9,9,9,9

10,10,10,10

Population Scores



What is the average self esteem score of this population
What is the average self-esteem score of this population?

  • Population mean = 5.5

  • Population SD = 2.87

  • What if you wanted to estimate this population mean from a sample?


What if
What if. . . .

  • Randomly select 5 people and find the average score


Group activity
Group Activity

  • Why isn’t the average score the same as the population score?

  • When you use a sample there is always some degree of uncertainty!

  • We can measure this uncertainty with a sampling distribution of the mean



Internet example
INTERNET EXAMPLE

  • http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/index.html

  • http://onlinestatbook.com/stat_sim/sampling_dist/index.html


Sampling distribution of the mean
Sampling Distribution of the Mean

  • Notice: The sampling distribution is centered around the population mean!

  • Notice: The sampling distribution of the mean looks like a normal curve!

    • This is true even though the distribution of scores was NOT a normal distribution


Central limit theorem
Central Limit Theorem

For any population of scores, regardless of form, the sampling distribution of the mean will approach a normal distribution a N (sample size) get larger. Furthermore, the sampling distribution of the mean will have a mean equal to  and a standard deviation equal to / N


Sampling distribution
Sampling Distribution

  • Tells you the probability of a particular sample mean occurring for a specific population


Sampling distribution1
Sampling Distribution

  • You are interested in if your new Self-esteem training course worked.

  • The 5 people in your course had a mean self-esteem score of 5.5


Sampling distribution2
Sampling Distribution

  • Did it work?

    • How many times would we expect a sample mean to be 5.5 or greater?

      • Theoretical vs. empirical

    • 5,000 random samples yielded 2,501 with means of 5.5 or greater

    • Thus p = .5002 of this happening


Sampling distribution3
Sampling Distribution

5.5

P = .4998 P =.5002

2,499 2,501


Sampling distribution4
Sampling Distribution

  • You are interested in if your new Self-esteem training course worked.

  • The 5 people in your course had a mean self-esteem score of 5.8


Sampling distribution5
Sampling Distribution

  • Did it work?

    • How many times would we expect a sample mean to be 5.8 or greater?

    • 5,000 random samples yielded 2,050 with means of 5.8 or greater

    • Thus p = .41 of this happening


Sampling distribution6
Sampling Distribution

5.8

P = .59 P =.41

2,700 2,300


Sampling distribution7
Sampling Distribution

  • The 5 people in your course had a mean self-esteem score of 9.8.

  • Did it work?

    • 5,000 random samples yielded 4 with means of 9.8 or greater

    • Thus p = .0008 of this happening


Sampling distribution8
Sampling Distribution

9.8

P = .9992 P =.0008

4,996 4


Logic
Logic

  • 1) Research hypothesis

    • H1

    • Training increased self-esteem

    • The sample mean is greater than general population mean

  • 2) Collect data

  • 3) Set up the null hypothesis

    • H0

    • Training did not increase self-esteem

    • The sample is no different than general population mean


Logic1
Logic

  • 4) Obtain a sampling distribution of the mean under the assumption that H0 is true

  • 5) Given the distribution obtain a probability of a mean at least as large as our actual sample mean

  • 6) Make a decision

    • Either reject H0 or fail to reject H0


Hypothesis test single subject
Hypothesis Test – Single Subject

  • You think your IQ is “freakishly” high that you do not come from the population of normal IQ adults.

  • Population IQ = 100 ; SD = 15

  • Your IQ = 125


Step 1 and 3
Step 1 and 3

  • H1: 125 > μ

  • Ho: 125 < or = μ



Step 5 obtain probability
Step 5: Obtain probability

125

-3 -2 -1 1 2  3 


Step 5 obtain probability1
Step 5: Obtain probability

(125 - 100) / 15 = 1.66

125

-3 -2 -1 1 2  3 


Step 5 obtain probability2
Step 5: Obtain probability

Z = 1.66

125

.0485

-3 -2 -1 1 2  3 


Step 6 decision
Step 6: Decision

  • Probability that this score is from the same population as normal IQ adults is .0485

  • In psychology

    • Most common cut-off point is p < .05

    • Thus, your IQ is significantly HIGHER than the average IQ


One vs two tailed tests
One vs. Two Tailed Tests

  • Previously wanted to see if your IQ was HIGHER than population mean

    • Called a “one-tailed” test

    • Only looking at one side of the distribution

  • What if we want to simply determine if it is different?


One tailed
One-Tailed

H1: IQ > μ

Ho: IQ < or = μ

p = .05

μ

-3 -2 -1 1 2  3 

Did you score HIGHER than population mean?

Want to see if score falls in top .05


Two tailed
Two-Tailed

H1: IQ = μ

Ho: IQ = μ

p = .05

p = .05

μ

-3 -2 -1 1 2  3 

Did you score DIFFERNTLY than population mean?


Two tailed1
Two-Tailed

H1: IQ = μ

Ho: IQ = μ

p = .05

p = .05

μ

-3 -2 -1 1 2  3 

Did you score DIFFERNTLY than population mean?

PROBLEM: Above you have a p = .10, but you want to test at a p = .05


Two tailed2
Two-Tailed

H1: IQ = μ

Ho: IQ = μ

p = .025

p = .025

μ

-3 -2 -1 1 2  3 

Did you score DIFFERNTLY than population mean?


Step 6 decision1
Step 6: Decision

  • Probability that this score is from the same population as normal IQ adults is .0485

  • In psychology

    • Most common cut-off point is p < .05

    • Note that on the 2-tailed test the point of significance is .025 (not .05)

    • Thus, your IQ is not significantly DIFFERENT than the average IQ


Problems
Problems

  • Problems with Null hypothesis testing

  • Logic is backwards:

  • Most think we are testing the probability of the hypothesis given the data

  • Really testing the probability of the data given the null hypothesis!


Practice4
Practice

  • A recently admitted class of graduate students at a large university has a mean GRE verbal score of 650 with a SD of 50. One student, whose mom is on the board of trustees, has a GRE score of 490. Do you think the school was showing favoritism?

  • Why is there such a small SD?

  • Why might (or might not) the GRE scores in this sample be normally distributed?


Practice
4.7

  • Z = (490-650) / 50 = -3.2

  • p = .0007 (490 or lower)


Practice
4.8

  • Because students are being selected with high GREs (restricted range)


Practice
4.9

  • Would not be normally distributed

    • Negatively skewed


Practice5
Practice

  • Last nights NHL game resulted in a score of 26 – 13. You would probably guess that I misread the paper. In effect you have just tested and rejected a null hypothesis.

  • 1) What is the null hypothesis

  • 2) Outline the hypothesis testing precede you just applied.


Practice
4.1

  • a) Null = last nights game was an NHL game (i.e., the scores come from the population of all NHL scores)

  • B) Would expect that a team would score between 0 – 6 points (null hypothesis).

  • Because the actual scores are a lot different we would reject the null.