SPSS Homework

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SPSS Homework
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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?

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Practice

• (29-23.32) /6.24 .91
• area .3186
• ( 34-23.32)/6.26 1.71
• area .4564
• .4564-.3186 .1378
• .137854 7.44 or 7 people

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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?

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Step 1 Sketch out question
.05
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Step 2 Look in Table Z
Z score 1.64
.05
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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)

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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

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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

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Step 1 Sketch out question
.03

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Step 2 Look in Table C
Z score -1.88
.03
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Step 3 Find the X score that goes with the Z
score

• Must solve for X
• X ? (z)(?)
• 312 500 (-1.88)(100)

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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

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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)

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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

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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?

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Basic logic of research
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Start with two equivalent groups of subjects
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Treat them alike except for one thing
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See if both groups are different at the end
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Or Single Group
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Do something
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Measure DV
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Compare Group to Population
Population Happiness Score
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The Theory of Hypothesis Testing

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

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Population

• You are interested in the average self-esteem in
  a population of 40 people
• Self-esteem test scores range from 1 to 10.

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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

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Histogram
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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?

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What if. . . .

• Randomly select 5 people and find the average
  score

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Group Activity

• Why isnt 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

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EXCEL
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INTERNET EXAMPLE

• http//www.ruf.rice.edu/lane/stat_sim/sampling_di
  st/index.html

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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

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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

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Sampling Distribution

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

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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

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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

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Sampling Distribution
5.5
P .4998 P
.5002
2,499 2,501
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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

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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

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Sampling Distribution
5.8
P .59 P .41
2,700 2,300
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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

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Sampling Distribution
9.8
P .9992
P .0008
4,996
4
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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

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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

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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

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Step 1 and 3

• H1 125 gt µ
• Ho 125 lt or µ

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Step 4 Appendix Z shows distribution of Z scores
under null
-3? -2? -1? ? 1? 2 ? 3 ?
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Step 5 Obtain probability
125
-3? -2? -1? ? 1? 2 ? 3 ?
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Step 5 Obtain probability
(125 - 100) / 15 1.66
125
-3? -2? -1? ? 1? 2 ? 3 ?
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Step 5 Obtain probability
Z 1.66
125
.0485
-3? -2? -1? ? 1? 2 ? 3 ?
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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 lt .05
• Thus, your IQ is significantly HIGHER than the
  average IQ

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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?

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One-Tailed
H1 IQ gt µ Ho IQ lt 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
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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?
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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
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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?
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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 lt .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

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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!

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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?

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4.7

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

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4.8

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

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4.9

• Would not be normally distributed
• Positively skewed

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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.

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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.