Is this quarter fair?

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Example

• You give 100 random students a questionnaire
  designed to measure attitudes toward living in
  dormitories
• Scores range from 1 to 7
• (1 unfavorable 4 neutral 7 favorable)
• You wonder if the mean score of the population is
  different then 4

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Hypothesis

• Alternative hypothesis
• H1 ?sample 4
• In other words, the population mean will be
  different than 4

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Hypothesis

• Alternative hypothesis
• H1 ?sample 4
• Null hypothesis
• H0 ?sample 4
• In other words, the population mean will not be
  different than 4

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Results

• N 100
• X 4.51
• s 1.94
• Notice, your sample mean is consistent with H1,
  but you must determine if this difference is
  simply due to chance

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Results

• N 100
• X 4.51
• s 1.94
• To determine if this difference is due to chance
  you must calculate an observed t value

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Observed t-value

• tobs (X - ?) / Sx

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Observed t-value

• tobs (X - ?) / Sx

• This will test if the null hypothesis H0 ?
  sample 4 is true
• The bigger the tobs the more likely that H1 ?
  sample 4 is true

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Observed t-value

• tobs (X - ?) / Sx

Sx S / N
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Observed t-value

• tobs (X - ?) / .194

.194 1.94/ 100
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Observed t-value

• tobs (4.51 4.0) / .194

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Observed t-value

• 2.63 (4.51 4.0) / .194

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t distribution
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t distribution
tobs 2.63
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t distribution
tobs 2.63
Next, must determine if this t value happened due
to chance or if represent a real difference in
means. Usually, we want to be 95 certain.
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t critical

• To find out how big the tobs must be to be
  significantly different than 0 you find a tcrit
  value.
• Calculate df N - 1
• Page 747
• First Column are df
• Look at an alpha of .05 with two-tails

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t distribution
tobs 2.63
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t distribution
tcrit -1.98
tcrit 1.98
tobs 2.63
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t distribution
tcrit -1.98
tcrit 1.98
tobs 2.63
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t distribution
tcrit -1.98
tcrit 1.98
tobs 2.63
If tobs fall in critical area reject the null
hypothesis Reject H0 ? sample 4
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t distribution
tcrit -1.98
tcrit 1.98
tobs 2.63
If tobs does not fall in critical area do not
reject the null hypothesis Do not reject H0 ?
sample 4
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Decision

• Since tobs falls in the critical region we reject
  Ho and accept H1
• It is statistically significant, students tend to
  think favorably about living in the dorms.
• p lt .05

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Example

• You wonder if the average IQ score of students at
  Villanova significantly different (at alpha
  .05)than the average IQ of the population (which
  is 100). You sample the students in this room.
• N 54
• X 130
• s 18.4

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The Steps

• Try to always follow these steps!

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Step 1 Write out Hypotheses

• Alternative hypothesis
• H1 ?sample 100
• Null hypothesis
• H0 ?sample 100

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Step 2 Calculate the Critical t

• N 54
• df 53
• ? .05
• tcrit 2.0

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Step 3 Draw Critical Region
tcrit 2.00
tcrit -2.00
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Step 4 Calculate t observed

• tobs (X - ?) / Sx

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Step 4 Calculate t observed

• tobs (X - ?) / Sx

Sx S / N
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Step 4 Calculate t observed

• tobs (X - ?) / Sx

2.5 18.4 / 54
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Step 4 Calculate t observed

• tobs (X - ?) / Sx
• 12 (130 - 100) / 2.5

2.5 18.4 / 54
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Step 5 See if tobs falls in the critical region
tcrit 2.00
tcrit -2.00
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Step 5 See if tobs falls in the critical region
tcrit 2.00
tcrit -2.00
tobs 12
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Step 6 Decision

• If tobs falls in the critical region
• Reject H0, and accept H1
• If tobs does not fall in the critical region
• Fail to reject H0

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Step 7 Put answer into words

• We reject H0 and accept H1.
• The average IQ of students at Villanova is
  statistically different (? .05) than the
  average IQ of the population.

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Practice

• You recently finished giving 5 of your friends
  the MMPI paranoia measure. Is your friends
  average average paranoia score significantly (?
  .10) different than the average paranoia of the
  population (? 56.1)?

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Scores
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Step 1 Write out Hypotheses

• Alternative hypothesis
• H1 ?sample 56.1
• Null hypothesis
• H0 ?sample 56.1

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Step 2 Calculate the Critical t

• N 5
• df 4
• ? .10
• tcrit 2.132

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Step 3 Draw Critical Region
tcrit 2.132
tcrit -2.132
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Step 4 Calculate t observed

• tobs (X - ?) / Sx
• -.48 (55.2 - 56.1) / 1.88

1.88 4.21/ 5
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Step 5 See if tobs falls in the critical region
tcrit 2.132
tcrit -2.132
tobs -.48
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Step 6 Decision

• If tobs falls in the critical region
• Reject H0, and accept H1
• If tobs does not fall in the critical region
• Fail to reject H0

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Step 7 Put answer into words

• We fail to reject H0
• The average paranoia of your friends is not
  statistically different (? .10) than the
  average paranoia of the population.

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SPSS
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One-tailed test

• In the examples given so far we have only
  examined if a sample mean is different than some
  value
• What if we want to see if the sample mean is
  higher or lower than some value
• This is called a one-tailed test

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Remember

• You recently finished giving 5 of your friends
  the MMPI paranoia measure. Is your friends
  average paranoia score significantly (? .10)
  different than the average paranoia of the
  population (? 56.1)?

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Hypotheses

• Alternative hypothesis
• H1 ?sample 56.1
• Null hypothesis
• H0 ?sample 56.1

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What if. . .

• You recently finished giving 5 of your friends
  the MMPI paranoia measure. Is your friends
  average paranoia score significantly (? .10)
  lower than the average paranoia of the population
  (? 56.1)?

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Hypotheses

• Alternative hypothesis
• H1 ?sample lt 56.1
• Null hypothesis
• H0 ?sample or gt 56.1

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Step 2 Calculate the Critical t

• N 5
• df 4
• ? .10
• Since this is a one-tail test use the
  one-tailed column
• Note one-tail directional test
• tcrit -1.533
• If H1 is lt then tcrit negative
• If H1 is gt then tcrit positive

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Step 3 Draw Critical Region
tcrit -1.533
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Step 4 Calculate t observed

• tobs (X - ?) / Sx

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Step 4 Calculate t observed

• tobs (X - ?) / Sx
• -.48 (55.2 - 56.1) / 1.88

1.88 4.21/ 5
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Step 5 See if tobs falls in the critical region
tcrit -1.533
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Step 5 See if tobs falls in the critical region
tcrit -1.533
tobs -.48
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Step 6 Decision

• If tobs falls in the critical region
• Reject H0, and accept H1
• If tobs does not fall in the critical region
• Fail to reject H0

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Step 7 Put answer into words

• We fail to reject H0
• The average paranoia of your friends is not
  statistically less then (? .10) the average
  paranoia of the population.

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Practice

• You just created a Smart Pill and you gave it
  to 150 subjects. Below are the results you
  found. Did your Smart Pill significantly (?
  .05) increase the average IQ scores over the
  average IQ of the population (? 100)?
• X 103
• s 14.4

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Step 1 Write out Hypotheses

• Alternative hypothesis
• H1 ?sample gt 100
• Null hypothesis
• H0 ?sample lt or 100

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Step 2 Calculate the Critical t

• N 150
• df 149
• ? .05
• tcrit 1.645

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Step 3 Draw Critical Region
tcrit 1.645
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Step 4 Calculate t observed

• tobs (X - ?) / Sx
• 2.54 (103 - 100) / 1.18

1.1814.4 / 150
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Step 5 See if tobs falls in the critical region
tcrit 1.645
tobs 2.54
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Step 6 Decision

• If tobs falls in the critical region
• Reject H0, and accept H1
• If tobs does not fall in the critical region
• Fail to reject H0

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Step 7 Put answer into words

• We reject H0 and accept H1.
• The average IQ of the people who took your Smart
  Pill is statistically greater (? .05) than the
  average IQ of the population.

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So far. . .

• We have been doing hypothesis testing with a
  single sample
• We find the mean of a sample and determine if it
  is statistically different than the mean of a
  population

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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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Notice

• This means that we need to see if two samples are
  statistically different from each other
• We can use the same logic we learned earlier with
  single sample hypothesis testing

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Example

• You just invented a magic math pill that will
  increase test scores.
• You give the pill to 4 subjects and another 4
  subjects get no pill
• You then examine their final exam grades

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HypothesisTwo-tailed

• Alternative hypothesis
• H1 ?pill ?nopill
• In other words, the means of the two groups will
  be significantly different
• Null hypothesis
• H0 ?pill ?nopill
• In other words, the means of the two groups will
  not be significantly different

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HypothesisOne-tailed

• Alternative hypothesis
• H1 ?pill gt ?nopill
• In other words, the pill group will score higher
  than the no pill group
• Null hypothesis
• H0 ?pill lt or ?nopill
• In other words, the pill group will be lower or
  equal to the no pill group

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For current example, lets just see if there is a
difference

• Alternative hypothesis
• H1 ?pill ?nopill
• In other words, the means of the two groups will
  be significantly different
• Null hypothesis
• H0 ?pill ?nopill
• In other words, the means of the two groups will
  not be significantly different

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Results

• Pill Group
• 5
• 3
• 4
• 3

• No Pill Group
• 1
• 2
• 4
• 3

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Remember before. . . Step 2 Calculate the
Critical t

• df N -1

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NowStep 2 Calculate the Critical t

• df N1 N2 - 2
• df 4 4 - 2 6
• ? .05
• t critical 2.447

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Step 3 Draw Critical Region
tcrit 2.447
tcrit -2.447
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Remember before. . .Step 4 Calculate t observed

• tobs (X - ?) / Sx

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NowStep 4 Calculate t observed

• tobs (X1 - X2) / Sx1 - x2

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NowStep 4 Calculate t observed

• tobs (X1 - X2) / Sx1 - x2

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NowStep 4 Calculate t observed

• tobs (X1 - X2) / Sx1 - x2
• X1 3.75
• X2 2.50

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NowStep 4 Calculate t observed

• tobs (X1 - X2) / Sx1 - x2

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Standard Error of a Difference

• Sx1 - x2
• When the N of both samples are equal
• If N1 N2
• Sx1 - x2 Sx12 Sx22

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Results

• Pill Group
• 5
• 3
• 4
• 3

• No Pill Group
• 1
• 2
• 4
• 3

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Standard Deviation
S
-1
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Standard Deviation

• Pill Group
• 5
• 3
• 4
• 3

• No Pill Group
• 1
• 2
• 4
• 3

??X2 10 ??X22 30
??X1 15 ??X12 59
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Standard Deviation

• Pill Group
• 5
• 3
• 4
• 3

• No Pill Group
• 1
• 2
• 4
• 3

??X2 10 ??X22 30
??X1 15 ??X12 59
S .96
S 1.29
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Standard Deviation

• Pill Group
• 5
• 3
• 4
• 3

• No Pill Group
• 1
• 2
• 4
• 3

??X2 10 ??X22 30
??X1 15 ??X12 59
S .96
S 1.29
Sx .48
Sx . 645
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Standard Error of a Difference

• Sx1 - x2
• When the N of both samples are equal
• If N1 N2
• Sx1 - x2 Sx12 Sx22

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Standard Error of a Difference

• Sx1 - x2
• When the N of both samples are equal
• If N1 N2
• Sx1 - x2 (.48)2 (.645)2

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Standard Error of a Difference

• Sx1 - x2
• When the N of both samples are equal
• If N1 N2
• Sx1 - x2 (.48)2 (.645)2

.80
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Standard Error of a Difference Raw Score Formula

• When the N of both samples are equal
• If N1 N2
• Sx1 - x2

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??X1 15 ??X12 59 N1 4
??X2 10 ??X22 30 N2 4

• Sx1 - x2

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??X1 15 ??X12 59 N1 4
??X2 10 ??X22 30 N2 4

• Sx1 - x2

10
15
100
??X1 15 ??X12 59 N1 4
??X2 10 ??X22 30 N2 4

• Sx1 - x2

10
15
59
30
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??X1 15 ??X12 59 N1 4
??X2 10 ??X22 30 N2 4

• Sx1 - x2

10
15
59
30
4
4
4 (4 - 1)
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??X1 15 ??X12 59 N1 4
??X2 10 ??X22 30 N2 4

• Sx1 - x2

10
15
59
30
56.25
25
4
4
12
103
??X1 15 ??X12 59 N1 4
??X2 10 ??X22 30 N2 4

10
15
.80
59
30
56.25
25
7.75
4
4
12
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NowStep 4 Calculate t observed

• tobs (X1 - X2) / Sx1 - x2

Sx1 - x2 .80 X1 3.75 X2 2.50
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NowStep 4 Calculate t observed

• tobs (3.75 - 2.50) / .80

Sx1 - x2 .80 X1 3.75 X2 2.50
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NowStep 4 Calculate t observed

• 1.56 (3.75 - 2.50) / .80

Sx1 - x2 .80 X1 3.75 X2 2.50
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Step 5 See if tobs falls in the critical region
tcrit 2.447
tcrit -2.447
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Step 5 See if tobs falls in the critical region
tcrit 2.447
tcrit -2.447
tobs 1.56
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Step 6 Decision

• If tobs falls in the critical region
• Reject H0, and accept H1
• If tobs does not fall in the critical region
• Fail to reject H0

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Step 7 Put answer into words

• We fail to reject H0.
• The final exam grades of the pill group were
  not statistically different (? .05) than the
  final exam grades of the no pill group.

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SPSS
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Practice

• You wonder if psychology majors have higher IQs
  than sociology majors (? .05)
• You give an IQ test to 4 psychology majors and 4
  sociology majors

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Results

• Psychology
• 110
• 150
• 140
• 135

• Sociology
• 90
• 95
• 80
• 98

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Step 1 Hypotheses

• Alternative hypothesis
• H1 ?psychology gt ?sociology
• Null hypothesis
• H0 ?psychology or lt ?sociology

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Step 2 Calculate the Critical t

• df N1 N2 - 2
• df 4 4 - 2 6
• ? .05
• One-tailed
• t critical 1.943

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Step 3 Draw Critical Region
tcrit 1.943
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NowStep 4 Calculate t observed

• tobs (X1 - X2) / Sx1 - x2

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??X1 535 ??X12 72425 N1 4 X1 133.75
??X2 363 ??X22 33129 N2 4 X2 90.75

• 9.38

363
535
72425
33129
4
4
4 (4 - 1)
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Step 4 Calculate t observed

• 4.58 (133.75 - 90.75) / 9.38

Sx1 - x2 9.38 X1 133.75 X2 90.75
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Step 5 See if tobs falls in the critical region
tcrit 1.943
tobs 4.58
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Step 6 Decision

• If tobs falls in the critical region
• Reject H0, and accept H1
• If tobs does not fall in the critical region
• Fail to reject H0

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Step 7 Put answer into words

• We Reject H0, and accept H1
• Psychology majors have significantly (? .05)
  higher IQs than sociology majors.

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Practice
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SPSS Problem 2

• 7.37
• 7.11