Download T-Test Notes

T-Test
The t-test, also known as the student t test, is a test of significance that can be used to
determine whether a significant difference exists or does not exist between two groups.
There are three kinds of t-tests:
1. t-test for one Sample
2. t-test for Dependent Samples
3. t-test for Independent Samples
When a t-test is used to determine whether the two sample means of two independent
samples come from the same population, we use the statistical test called the t-test for
Independent Means. This is the most common t-test used in science. The formula for
calculating “t” depends upon whether the two samples being compare have equal
variances. The null and alternative hypotheses for this type of test are:
H0: m 1 = m 2 ; the means are equal
H1: m1 </> m2 ; the means are different
This is a two tailed test because the Null Hypothesis does not specify a direction, only the
condition of equality.
The assumptions are:
1. the data are Normal
2. the two samples come from distributions that may differ in their mean value, but not in
the standard deviation
3. the observations are independent of each other.
Calculate a t-test value, and compare the value with a critical value of t (given in a book).
If the t value calculated from the data is equal to or larger than the critical value, you
reject the Null hypothesis of H0: m 1 = m 2.
Before we determine the calculated t (tc), look at the variances. If the variances are equal,
calculate t using the formula below:
t=
| m1 - m2 |
_______________________________________________________
If the variances are unequal, calculate t as follow:
t=
| m1 - m2 |
________________________
m = mean
n = sample size
s2=variance
Using the data below:
State your hypotheses; Set a probability level, called the alpha level  = .05; Calculate
‘t’ ; compare the numerical value calculated from the statistical test to a critical value or
tabled value (tt) (obtained from a table/book). In this case tt = 2.000
Reject the Null hypotheses if your calculated value is greater than or equal to the critical
or tabled value.
Population/sample NS1
Sample 1
Gender height (inches)
male 65.00
male
male
male
male
male
male
male
male
male
male
male
male
male
male
male
male
male
male
male
male
male
male
male
male
male
male
70.00
69.00
72.00
70.00
72.00
69.00
68.00
68.00
70.00
68.00
71.00
71.00
71.00
74.00
67.00
76.00
73.00
71.00
68.00
71.00
71.00
71.00
72.00
70.00
71.00
73.00
Sample 2
Gender height (inches)
female 66.00
female 68.00
female 63.00
female 60.00
female 64.00
female 63.00
female 68.00
female 63.00
female 68.00
female 63.00
female 67.00
female 63.00
female 68.00
female 64.00
female 64.00
female 63.00
Using StatView:
Des criptive Sta tistic s
Split By: Popula tion/sample#
H…
Heig ht, male
Heig ht, fema le
Mean
#
70 .444
64 .688
Std . Dev.
#
2.293
2.469
Std . Error
#
.4 41
.6 17
Coun t
#
27
16
Mini mum
#
65 .000
60 .000
Maximu m
#
76 .000
68 .000
# Mi ssin g
0
Va riance
#
Coef. Va r.
#
.0 33
.0 38
Rang e
#
11 .000
8.000
Sum
#
19 02.0 00
10 35.0 00
Sum Squa res #
13 4122 .000
67 043.000
0
5.256
0
6.096
Geom . Mean #
70 .408
64 .643
Harm. Mean
#
70 .372
64 .599
Skewness
#
-.0 15
.1 24
Kurtosis
#
.4 51
-1.000
Medi an
#
71 .000
64 .000
IQR
#
2.750
4.500
Mode
•
71 .000
63 .000
10 % Tr. Mean #
70 .435
64 .786
1.000
1.000
MAD
#
His togram
Split By: Popula tion/sample#
Ce ll: male
16
14
Count
12
10
8
6
4
2
0
58
60
62
64
66
68
70
Heig ht
72
74
76
78
His togram
Split By: Popula tion/sample#
Ce ll: female
8
7
Count
6
5
4
3
2
1
0
58
60
62
64
66
68
70
Heig ht
72
74
76
78
Using Statview: compare the calculated P-value with the probability level, alpha
level  = .05. It is smaller than  = .05; meaning that the means are different.
Unpaired t-test for Height
Grouping Variable: Population/sam ple#
Hypothesized Differenc e = 0
Mean Diff.
ma le, fema le
5.757
DF
41
t-Value
P-Va lue
7.736
<.0001
Group Info for Height
Grouping Variable: Population/sam ple#
Coun t
Mean
Va riance
Std . Dev.
Std . Err
ma le
27
70 .444
5.256
2.293
.4 41
fe male
16
64 .688
6.096
2.469
.6 17