Example one
A researcher believes that in recent years women have been
getting taller. She knows that 10 years ago the average height
of young adult women living in her city was 63 inches. The
standard deviation is unknown. She randomly selects eight
young adult women currently residing in her city and finds
that the average height of these women is 64.25 inches. The
standard deviation of this sample is 2.55.
What are the critical pieces that will tell us
which t-test to use?
A researcher believes that in recent years women have been
getting taller. She knows that 10 years ago the average height
of young adult women living in her city was 63 inches. The
standard deviation is unknown. She randomly selects eight
young adult women currently residing in her city and finds
that the average height of these women is 64.25 inches. The
standard deviation of this sample is 2.55.
• µ=63 (population average)
• n=8 (number in this sample)
•
x =64.25 (average of this sample)
• S=2.55 (standard deviation of this sample)
Which test to do?
• We don’t know the population SD so we can’t find a
z-score.
• We have only one sample, so we need to use a t-test
for single samples.
t
X 
Sx
Sx 
S
n
Statistical Hypothesis Statements
• H0 : u = 63
(This sample is from the population whose
mean is 63).
• H1 : u  63
(This sample is not from the population
whose mean is 63).
t-test for single samples
x  64.25
t
X 
Sx
64 .25  63
t
.902
t  1.39
  63
S  2.55
n8
S
2.5 5  .902
Sx 

n
8
t-test for single samples
• df = n-1
n=8
df = n-1
= 8-1
=7
critical value=2.365
(from t-table)
t-test for single samples
• Since computed t is less than critical t,
(1.39 < 2.365),
We accept H0
(H0: u = 63)
Women continue to be the same height as
they were ten years ago.
Ex.2
A psychologist is interested in determining whether memory is
affected by sleep loss. 12 normal subjects are randomly
selected and assigned to two groups of 6 each. Group one gets
a normal amount of sleep (7-8 hours). This group’s mean
score is 70.17 with a standard deviation of 3.18. Group two is
sleep deprived for 24 hours. They have an average score of
65.33 and a standard deviation of 4.18.
• A psychologist is interested in determining whether memory
is affected by sleep loss. 12 normal subjects are randomly
selected and assigned to two groups of 6 each. Group one
gets a normal amount of sleep (7-8 hours). This group’s
mean score is 70.17 with a standard deviation of 3.18.
Group two is sleep deprived for 24 hours. They have an
average score of 65.33 and a standard deviation of 4.18.
X1  70.17
X 2  65.33
S1=3.18
S2=4.18
Which test to do?
• Can’t do a z-score because we don’t know
the standard dev. of the population
• Can’t do a t-test for single samples because
we have two samples.
• Because we have two samples, we have to
do a t-test for 2 independent samples
Statistical hypothesis statements:
• H0: u1 = u2
The two samples are from the same
population.
• H1: u1 ≠ u2
The two samples are not from the same
population.
t-test for independent samples
•Can use shortened formula because the two samples are of equal size.
t
X1  X 2
S12  S 22
n
t-test for independent samples
t

X1  X 2
S12  S 22
n
70.17  65.33
10.11  17.47
6

4 .8 4
4 .6 0
4.84

2.14
 2.26
t-test for independent samples
• t= 2.26 (computed t value)
• For independent samples t-test
df= n1 + n2-2
=6+6-2
=10
critical t value: 2.228 (from t-table)
t-test for independent samples
• Since computed t is greater than critical T,
(2.26 > 2.228),
We reject H0
Accept H1: u1  u2
The two groups are significantly different
from each other. There is an effect of sleep
deprivation on memory.
Example 3
• To motivate citizens to conserve fuel, the
government would like to mount a
nationwide campaign. Before doing so
however, they wish to test the effectiveness
of the campaign. Twelve families are
selected and their consumption of fuel
before the campaign and afterward is
monitored.
What kind of t-test should be
done here?
• To motivate citizens to conserve fuel, the
government would like to mount a
nationwide campaign. Before doing so
however, they wish to test the effectiveness
of the campaign. Twelve families are
selected and their consumption of fuel
before the campaign and afterward is
monitored.
t-test for matched samples
• Something in the problem talking about
subjects matched or that the same subjects
took a test twice (before and after) or same
subjects taking test at time 1 and time 2
t-test for matched samples
• From raw data (Time 1 and Time 2)
1. Find difference between those measurements
for each subject.
2. Find average of the differences.
3. Find the standard deviation of the
differences.
4. Use this S to find the standard error of the
differences,
s
SD 
D
n
t-test for matched samples
• (continued)
5. Solve for t.
D
t
SD
6. Compare this computed t to critical t
(df=n-1) from t-table to see if significant
difference between before and after scores.