Hypothesis Testing with t Tests
A l Cl
Arlo
Clark-Foos
kF
Using Samples to Estimate
Population Parameters
Acknowledge
g error
y Smaller samples, less spread
y
Σ( X − M )
s=
N −1
2
The t Statistic
y
Indicates the distance of a sample
p mean
from a population mean in terms of the
standard error s = s
M
N
Hypothesis Tests:
Single--Sample t Tests
Single
“Guinness is the best beer available, it does
not need
d advertising
d
as its quality
l willll sellll it,
and those who do not drink it are to be
sympathized with rather than advertised to.
to”
Hypothesis Tests:
Single--Sample t Tests
Single
y
Hypothesis test in which we compare data from
one sample to a population for which we know the
mean but not the standard deviation.
y
Degrees of Freedom:
◦ The number of scores that are free to vary when
estimating a population parameter from a sample
◦ df = N – 1 (for a Single-Sample t Test)
One Tailed vs.
vs Two Tailed Tests
Six Steps for Hypothesis Testing
1. Identifyy
2. State the hypotheses
3 Characteristics of the comparison
3.
distribution
4 Critical values
4.
5. Calculate
6. Decide
Single--Sample t Test: Example
Single
Participation
p
in therapy
py sessions
y Contract to attend 10 sessions
y μ = 4.6
46
y
Sample:
y 6, 6, 12, 7, 8
y
Single--Sample t Test: Example
Single
Identifyy
1.
◦
◦
Pop 1: All clients who sign contract
g contract
Popp 2: All clients who do not sign
◦
Distribution of means
◦
Test: Population mean is known but not
standard deviation Æ single-sample t test
x
Assumptions
Single--Sample t Test: Example
Single
State the null and research hypotheses
yp
2.
◦
H0: Clients who sign the contract will attend
the same number of sessions as those who
do not sign the contract.
x
◦
μ1 = μ2
H1: Clients who sign the contact will attend
a different number of sessions than those
who do not sign the contract.
contract
x
μ1 ≠ μ2
Single--Sample t Test: Example
Single
Determine characteristics of
comparison distribution
2
μM = μ = 4.6,
Σ( X − M )
s=
= 2.490
sM = 1.114
N −1
3.
s
2.490
sM =
=
= 1.114
N
5
Single--Sample t Test: Example
Single
Determine critical values or cutoffs
4.
◦
◦
df = N-1 = 5-1 = 4, p = .05
t = ± 2.776
Single--Sample t Test: Example
Single
Calculate the test statistic
5.
( M − μ M ) (7.8 − 4.6)
t=
=
= 2.873
sM
1.114
M k a decision
Make
d ii
6
6.
◦
◦
2.873 > 2.776, we reject the null
Clients who sign a contract will attend more
sessions than those who do not sign a
contract.
contract
Reporting Results in APA Format
1.
2.
3.
4.
Write the symbol for the test statistic (e.g., z
or t)
Write the degrees of freedom in parentheses
Write an equal sign and then the value of the
test statistic (2 decimal places)
Write a comma and then whether the p value
associated with the test statistic was less than
or greater than the cutoff p value of .05
05
t(4) = 22.87,
87 p < .05
05
Hypothesis tests for two samples
y
Conditions must be equivalent
q
(controlled) for a fair test
y
Beer Tasting: Between-groups vs. Withingroups
Paired--Samples t Test
Paired
y
Used to compare
p 2 means ffor a withingroups design, a situation in which every
pparticipant
p
is in both samples
p
(paired/dependent)
y
Difference Scores: X1 – Y1, X2 – Y2, …
Paired--Samples t Test
Paired
Identifyy
2. State null & research hyp.
3 Determine characteristics of
3.
comparison distribution
H 0 : μ1 = μ2
1.
Paired--Samples t Test
Paired
Determine cutoffs
5. Calculate test statistic
(M − μM )
4.
t=
6
6.
M k a decision
Make
d ii
sM
Independent Samples t Test
y
Used to compare
p 2 means ffor a betweengroups design, a situation in which each
pparticipant
p
is assigned
g
to onlyy one condition.
Independent Samples t Test
y
What p
percentage
g of cartoons do men and
woman consider funny?
Women:
y Men:
y
84 97 58 90
88 90 52 97
86
Independent Samples t Test
Identifyy
1.
◦
Population 1: Women exposed to humorous
cartoons
Population 2: Men exposed to humorous
cartoons
Distribution: Differences between means
◦
x
◦
Not mean differences
Assumptions are the same
x
We meet one here: DV is interval/scale
Independent Samples t Test
State null and research
hypotheses
2.
◦
◦
◦
◦
H 0 : μ1 = μ2
Women will categorize the
same number of cartoons as
funny as will men.
men
H 1 : μ1 ≠ μ2
Women
Wo
e will
w categorize
catego e a
different number of cartoons
funny as as will men.
Independent Samples t Test
Determine characteristics of
comparison distribution
3.
◦
◦
H0: μ1 = μ2
Measure of spread:
a) Calculate variance for each sample
b) Pool variances,
variances accounting for sample size
c) Convert from squared standard deviation to
squared standard error
d) Add the two variances
e) Take square root to get estimated standard error
for distribution of differences between means.
Independent Samples t Test
a) Calculate variance for each sample
2
Σ
(
X
−
M
)
868.752
s X2 =
=
= 289.584
N −1
4 −1
2
Σ
(
Y
−
M
)
1314 4
1314.4
sY2 =
=
= 328.6
N −1
5 −1
Independent Samples t Test
y
Degrees
g
of Freedom
dfX = N – 1 = 4 – 1 = 3
dfY = N – 1 = 5 – 1 = 4
dfTotal = dfX + dfY = 3 + 4 = 7
Pooled Variance
b)
y
Pool variances, accounting for sample size
Weighted average of the two estimates of variance
– one from each sample – that are calculated when
conducting
d i an iindependent
d p d samples
pl t test.
s
2
Pooled
s
2
Pooled
⎛ df X
=⎜
⎝ dfTotal
⎞ 2 ⎛ dfY
⎟ sX + ⎜
⎠
⎝ dfTotal
⎞ 2
⎟ sY
⎠
⎛3⎞
⎛4⎞
= ⎜ ⎟ 289.584 + ⎜ ⎟ 328.6
⎝7⎠
⎝7⎠
2
sPooled
= 124.107 + 187.771 = 311.878
Independent Samples t Test
c)) Convert from squared
q
standard deviation to
squared standard error
2
sPooled
= 311.878
s
2
MX
2
sPooled
311 878
311.878
=
=
= 77.970
N
4
s
2
MY
2
sPooled
311 878
311.878
=
=
= 62.376
N
5
d) Add the two variances
2
sDifference
= sM2 X + sM2 Y = 77.970 + 62.376 = 140.346
e) Take square root to get estimated standard error for
distribution of differences between means.
2
sDifference = sDifference
= 140.346 = 11.847
Independent Samples t Test
Determine critical values
4.
◦
5.
dfTotal = 7
p = .05
t = ± 2.365
Calculate a test statistic
⎡⎣( M X − M Y ) − ( μ X − μY ) ⎤⎦ ( M X − M Y )
t=
=
sDifference
sDifference
82.25 − 82.6 )
(
t=
= −.03
11 847
11.847
Independent Samples t Test
6.
Make a decision
y
Fail to reject null hypothesis
◦
Men and women find cartoons equally
humorous