SPSS Homework

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?

Step 1: Sketch out question

.05

Step 2: Look in Table Z

.05

Z score = 1.64

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)



• 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

Step 1: Sketch out question

.03

Step 2: Look in Table CZ score = -1.88

.03



• X = + (z)()

• 312 = 500 + (-1.88)(100)



• 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)


• 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?

Basic logic of research

Start with two equivalent groups of subjects

D ep en d en t V ariab leIf p e rson lives

E xp erim en ta l G rou pG ive m ed ica tion

S u b jec ts


C on tro l G rou pD o n o t g ive m ed ica tion

S u b jec ts

Treat them alike except for one thing



S u b jec ts



S u b jec ts

See if both groups are different at the end



S u b jec ts



S u b jec ts

Or – Single Group

Subjects

Give Treatment -- Prozac

Dependent VariableHappiness

Do something

Subjects



Measure DV

Subjects



Compare Group to Population

Subjects



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.

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

Histogram

012345

6789

10

1 2 3 4 5 6 7 8 9 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

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

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

http://onlinestatbook.com/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


• 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


5.5

2,499 2,501

P = .4998 P =.5002


• 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



5.8

2,700 2,300

P = .59 P =.41


• 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



9.8

4,996 4

P = .9992 P =.0008

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

• Your IQ = 125

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

-3 -2 -1 1 2 3

125


-3 -2 -1 1 2 3

125

(125 - 100) / 15 = 1.66


-3 -2 -1 1 2 3

125

Z = 1.66

.0485

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

-3 -2 -1 1 2 3

p = .05

Did you score HIGHER than population mean?

Want to see if score falls in top .05

μ

H1: IQ > μ

Ho: IQ < or = μ

Two-Tailed

-3 -2 -1 1 2 3

p = .05

Did you score DIFFERNTLY than population mean?

μ

p = .05

H1: IQ = μ

Ho: IQ = μ

Two-Tailed

-3 -2 -1 1 2 3

p = .05


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

μ

p = .05

H1: IQ = μ

Ho: IQ = μ

Two-Tailed

-3 -2 -1 1 2 3

p = .025


μ

p = .025

H1: IQ = μ

Ho: IQ = μ

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.

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