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Transcript of Example You give 100 random students a questionnaire designed to measure attitudes toward living in...
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
Hypothesis
• Alternative hypothesis– H1: sample = 4
– In other words, the population mean will be different than 4
Hypothesis
• Alternative hypothesis– H1: sample = 4
• Null hypothesis– H0: sample = 4
– In other words, the population mean will not be different than 4
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
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
Observed t-value
tobs = (X - ) / Sx
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
Observed t-value
tobs = (X - ) / Sx
Sx = S / N
Observed t-value
tobs = (X - ) / .194
.194 = 1.94/ 100
Observed t-value
tobs = (4.51 – 4.0) / .194
Observed t-value
2.63 = (4.51 – 4.0) / .194
t distribution
t distribution
tobs = 2.63
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.
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
t distribution
tobs = 2.63
t distribution
tobs = 2.63
tcrit = 1.98tcrit = -1.98
t distribution
tobs = 2.63
tcrit = 1.98tcrit = -1.98
t distribution
tobs = 2.63
tcrit = 1.98tcrit = -1.98
If tobs fall in critical area reject the null hypothesis
Reject H0: sample = 4
t distribution
tobs = 2.63
tcrit = 1.98tcrit = -1.98
If tobs does not fall in critical area do not reject the null hypothesis
Do not reject H0: sample = 4
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 < .05
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
The Steps
• Try to always follow these steps!
Step 1: Write out Hypotheses
• Alternative hypothesis– H1: sample = 100
• Null hypothesis– H0: sample = 100
Step 2: Calculate the Critical t
• N = 54
• df = 53 = .05
• tcrit = 2.0
Step 3: Draw Critical Region
tcrit = 2.00tcrit = -2.00
Step 4: Calculate t observed
tobs = (X - ) / Sx
Step 4: Calculate t observed
tobs = (X - ) / Sx
Sx = S / N
Step 4: Calculate t observed
tobs = (X - ) / Sx
2.5 = 18.4 / 54
Step 4: Calculate t observed
tobs = (X - ) / Sx
12 = (130 - 100) / 2.52.5 = 18.4 / 54
Step 5: See if tobs falls in the critical region
tcrit = 2.00tcrit = -2.00
Step 5: See if tobs falls in the critical region
tcrit = 2.00tcrit = -2.00
tobs = 12
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
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.
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)?
Scores
Person Score
Charlie 55
Lucy 49
Sally 58
Schroeder 60
Franklin 54
Step 1: Write out Hypotheses
• Alternative hypothesis– H1: sample = 56.1
• Null hypothesis– H0: sample = 56.1
Step 2: Calculate the Critical t
• N = 5
• df =4 = .10
• tcrit = 2.132
Step 3: Draw Critical Region
tcrit = 2.132tcrit = -2.132
Step 4: Calculate t observed
tobs = (X - ) / Sx
-.48 = (55.2 - 56.1) / 1.88 1.88 = 4.21/ 5
Step 5: See if tobs falls in the critical region
tcrit = 2.132tcrit = -2.132
tobs = -.48
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
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.
SPSS
5 55.2000 4.2071 1.8815MMPIN Mean
Std.Deviation
Std. ErrorMean
One-Sample Statistics
-.478 4 .657 -.9000 -6.1239 4.3239MMPIt df
Sig.(2-tailed)
MeanDifference Lower Upper
95% ConfidenceInterval of the Difference
Test Value = 56.1
One-Sample Test
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
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)?
Hypotheses
• Alternative hypothesis– H1: sample = 56.1
• Null hypothesis– H0: sample = 56.1
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)?
Hypotheses
• Alternative hypothesis– H1: sample < 56.1
• Null hypothesis– H0: sample = or > 56.1
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 < then tcrit = negative– If H1 is > then tcrit = positive
Step 3: Draw Critical Region
tcrit = -1.533
Step 4: Calculate t observed
tobs = (X - ) / Sx
Step 4: Calculate t observed
tobs = (X - ) / Sx
-.48 = (55.2 - 56.1) / 1.88 1.88 = 4.21/ 5
Step 5: See if tobs falls in the critical region
tcrit = -1.533
Step 5: See if tobs falls in the critical region
tcrit = -1.533
tobs = -.48
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
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.
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
Step 1: Write out Hypotheses
• Alternative hypothesis– H1: sample > 100
• Null hypothesis– H0: sample < or = 100
Step 2: Calculate the Critical t
• N = 150
• df = 149 = .05
• tcrit = 1.645
Step 3: Draw Critical Region
tcrit = 1.645
Step 4: Calculate t observed
tobs = (X - ) / Sx
2.54 = (103 - 100) / 1.181.18=14.4 / 150
Step 5: See if tobs falls in the critical region
tcrit = 1.645
tobs = 2.54
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
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.
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
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
D ep en d en t V ariab leIf p e rson lives
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
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
D ep en d en t V ariab leIf p e rson lives
C on tro l G rou pD o n o t g ive m ed ica tion
S u b jec ts
See if both groups are different at the end
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
D ep en d en t V ariab leIf p e rson lives
C on tro l G rou pD o n o t g ive m ed ica tion
S u b jec ts
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
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
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
HypothesisOne-tailed
• Alternative hypothesis– H1: pill > nopill
– In other words, the pill group will score higher than the no pill group
• Null hypothesis– H0: pill < or = nopill
– In other words, the pill group will be lower or equal to the no pill group
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
Results
Pill Group
5
3
4
3
No Pill Group
1
2
4
3
Remember before. . . Step 2: Calculate the Critical t
• df = N -1
NowStep 2: Calculate the Critical t
• df = N1 + N2 - 2
• df = 4 + 4 - 2 = 6 = .05
• t critical = 2.447
Step 3: Draw Critical Region
tcrit = 2.447tcrit = -2.447
Remember before. . .Step 4: Calculate t observed
tobs = (X - ) / Sx
NowStep 4: Calculate t observed
tobs = (X1 - X2) / Sx1 - x2
NowStep 4: Calculate t observed
tobs = (X1 - X2) / Sx1 - x2
NowStep 4: Calculate t observed
tobs = (X1 - X2) / Sx1 - x2
• X1 = 3.75
• X2 = 2.50
NowStep 4: Calculate t observed
tobs = (X1 - X2) / Sx1 - x2
Standard Error of a Difference
Sx1 - x2
When the N of both samples are equal
If N1 = N2:
Sx1 - x2 = Sx12 + Sx2
2
Results
Pill Group
5
3
4
3
No Pill Group
1
2
4
3
Standard Deviation
S =-1
Standard Deviation
Pill Group
5
3
4
3
No Pill Group
1
2
4
3
X1= 15
X12= 59
X2= 10
X22= 30
Standard Deviation
Pill Group
5
3
4
3
No Pill Group
1
2
4
3
S = .96 S = 1.29
X1= 15
X12= 59
X2= 10
X22= 30
Standard Deviation
Pill Group
5
3
4
3
No Pill Group
1
2
4
3
S = .96 S = 1.29
Sx= .48 Sx= . 645
X1= 15
X12= 59
X2= 10
X22= 30
Standard Error of a Difference
Sx1 - x2
When the N of both samples are equal
If N1 = N2:
Sx1 - x2 = Sx12 + Sx2
2
Standard Error of a Difference
Sx1 - x2
When the N of both samples are equal
If N1 = N2:
Sx1 - x2 = (.48)2 + (.645)2
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
Standard Error of a Difference Raw Score Formula
When the N of both samples are equal
If N1 = N2:
Sx1 - x2 =
Sx1 - x2 =
X1= 15
X12= 59
N1 = 4
X2= 10
X22= 30
N2 = 4
Sx1 - x2 =
X1= 15
X12= 59
N1 = 4
X2= 10
X22= 30
N2 = 4
15 10
Sx1 - x2 =
X1= 15
X12= 59
N1 = 4
X2= 10
X22= 30
N2 = 4
15 1059 30
Sx1 - x2 =
X1= 15
X12= 59
N1 = 4
X2= 10
X22= 30
N2 = 4
15 1059 304 4
4 (4 - 1)
Sx1 - x2 =
X1= 15
X12= 59
N1 = 4
X2= 10
X22= 30
N2 = 4
15 1059 304 4
12
56.25 25
X1= 15
X12= 59
N1 = 4
X2= 10
X22= 30
N2 = 4
15 1059 304 4
12
56.25 257.75.80 =
NowStep 4: Calculate t observed
tobs = (X1 - X2) / Sx1 - x2
Sx1 - x2 = .80X1 = 3.75
X2 = 2.50
NowStep 4: Calculate t observed
tobs = (3.75 - 2.50) / .80
Sx1 - x2 = .80X1 = 3.75
X2 = 2.50
NowStep 4: Calculate t observed
1.56 = (3.75 - 2.50) / .80
Sx1 - x2 = .80X1 = 3.75
X2 = 2.50
Step 5: See if tobs falls in the critical region
tcrit = 2.447tcrit = -2.447
Step 5: See if tobs falls in the critical region
tcrit = 2.447tcrit = -2.447
tobs = 1.56
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
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.
SPSS
4 2.5000 1.2910 .6455
4 3.7500 .9574 .4787
PILL.00
1.00
SCOREN Mean
Std.Deviation
Std. ErrorMean
Group Statistics
.500 .506 -1.555 6 .171 -1.2500 .8036 -3.2164 .7164
-1.555 5.534 .175 -1.2500 .8036 -3.2573 .7573
Equalvariancesassumed
Equalvariancesnotassumed
SCOREF Sig.
Levene's Test forEquality of Variances
t dfSig.
(2-tailed)Mean
DifferenceStd. ErrorDifference Lower Upper
95% ConfidenceInterval of the Mean
t-test for Equality of Means
Independent Samples Test
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
Results
Psychology
110
150
140
135
Sociology
90
95
80
98
Step 1: Hypotheses
• Alternative hypothesis– H1: psychology > sociology
• Null hypothesis– H0: psychology = or < sociology
Step 2: Calculate the Critical t
• df = N1 + N2 - 2
• df = 4 + 4 - 2 = 6 = .05
• One-tailed
• t critical = 1.943
Step 3: Draw Critical Region
tcrit = 1.943
NowStep 4: Calculate t observed
tobs = (X1 - X2) / Sx1 - x2
9.38 =
X1= 535
X12=
72425
N1 = 4
X1 = 133.75
X2= 363
X22=
33129
N2 = 4
X2 = 90.75
535 36372425 331294 4
4 (4 - 1)
Step 4: Calculate t observed
4.58 = (133.75 - 90.75) / 9.38
Sx1 - x2 = 9.38X1 = 133.75
X2 = 90.75
Step 5: See if tobs falls in the critical region
tcrit = 1.943
tobs = 4.58
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
Step 7: Put answer into words
• We Reject H0, and accept H1
• Psychology majors have significantly ( = .05) higher IQs than sociology majors.
Practice
SPSS Problem #2
• 7.37
• 7.11