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# Practice PowerPoint PPT Presentation

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.

Practice

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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

• 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?

.05

Z score = 1.64

.05

### 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 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

### 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

.03

Z score = -1.88

.03

### 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 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?

• 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 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 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

Population Happiness Score

### The Theory of Hypothesis Testing

• Data are ambiguous

• Is a difference due to chance?

• Sampling error

### Population

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

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

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

### 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. . . .

• Randomly select 5 people and find the average score

### 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

• 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

• 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

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

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

### 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 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 Distribution

5.5

P = .4998 P =.5002

2,499 2,501

### 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 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

5.8

P = .59 P =.41

2,700 2,300

### 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 Distribution

9.8

P = .9992 P =.0008

4,996 4

### 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

### 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

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

• Population IQ = 100 ; SD = 15

### Step 1 and 3

• H1: 125 > μ

• Ho: 125 < or = μ

### Step 4: Appendix Z shows distribution of Z scores under null

-3 -2 -112  3 

### Step 5: Obtain probability

125

-3 -2 -112  3 

### Step 5: Obtain probability

(125 - 100) / 15 = 1.66

125

-3 -2 -112  3 

### Step 5: Obtain probability

Z = 1.66

125

.0485

-3 -2 -112  3 

### 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

• 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

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

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-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-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: 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 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!

### 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?

### 4.7

• Z = (490-650) / 50 = -3.2

• p = .0007 (490 or lower)

### 4.8

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

### 4.9

• Would not be normally distributed

• Negatively skewed

### 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.

### 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.