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Practice

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- The Neuroticism Measure
= 23.32

S = 6.24

n = 54

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

- (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

- 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

- Must solve for X
- X = + (z)()
- 68.26 = 56.82 + (1.64)(6.98)

- 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

- 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

- Must solve for X
- X = + (z)()
- 312 = 500 + (-1.88)(100)

- 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

- 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)

- Say you flip it 100 times
- 52 times it is a head
- Not exactly 50, but its close
- probably due to random error

- What if you got 65 heads?
- 70?
- 95?
- At what point is the discrepancy from the expected becoming too great to attribute to chance?

Population Happiness Score

- Data are ambiguous
- Is a difference due to chance?
- Sampling error

- 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

- Population mean = 5.5
- Population SD = 2.87
- What if you wanted to estimate this population mean from a sample?

- Randomly select 5 people and find the average score

- 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

- http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/index.html
- http://onlinestatbook.com/stat_sim/sampling_dist/index.html

- 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

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

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

- 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

- 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

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

5.5

P = .4998 P =.5002

2,499 2,501

- 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

- 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

- 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

9.8

P = .9992 P =.0008

4,996 4

- 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

- 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

- 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

- H1: 125 > μ
- Ho: 125 < or = μ

-3 -2 -112 3

125

-3 -2 -112 3

(125 - 100) / 15 = 1.66

125

-3 -2 -112 3

Z = 1.66

125

.0485

-3 -2 -112 3

- 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

- 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?

H1: IQ > μ

Ho: IQ < or = μ

p = .05

μ

-3 -2 -112 3

Did you score HIGHER than population mean?

Want to see if score falls in top .05

H1: IQ = μ

Ho: IQ = μ

p = .05

p = .05

μ

-3 -2 -112 3

Did you score DIFFERNTLY than population mean?

H1: IQ = μ

Ho: IQ = μ

p = .05

p = .05

μ

-3 -2 -112 3

Did you score DIFFERNTLY than population mean?

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

H1: IQ = μ

Ho: IQ = μ

p = .025

p = .025

μ

-3 -2 -112 3

Did you score DIFFERNTLY than population mean?

- 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 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!

- 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?

- Z = (490-650) / 50 = -3.2
- p = .0007 (490 or lower)

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

- Would not be normally distributed
- Negatively skewed

- 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.

- 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.