Two Tailed test
Ho
H1
H1
Z=0
Z score where 2.5% of the
distribution lies in the tail:
Z = + 1.96
Critical value for a two tailed
test.
One tailed test
5%
85
One tailed test
5%
85
Z score where 5% of the distribution lies
in the tail:
Z = + 1.65
Critical value for a one tailed test.
Z scores require  and

What happens when the population
mean is known but the standard
deviation is unknown?
T test for single samples
X 
t
SX
S X is the standard error of the
mean, estimated.
SX 
S
n
S
SX 
n
S is representing the standard
deviation of the sample, as an
estimate of the population.
 X  X 
2
S
n 1
n - 1 represents degrees of
freedom. (df)
Degrees of freedom: the
number of values that are
free to vary.
Computational Formula
Standard deviation of a sample as an
estimate of the population.
S
X

X


2
2
n 1
n
Situation that calls for a t test for single
samples:
1) Population mean is know n.
2) The standard deviation of the population is
unknown.
3) A sample of a specific size is available.
Steps for a single sample t test:
1) compute the standard deviation as an
estimate of the population.
2) compute the standard error estimated
3) compute t
4) evaluate the computed t value to
determine if it is statistically significant.
The t distribution:
- Symmetric
- Flatter middle than a normal
distribution. Fatter tails.
X 
t
sX
Assume:
 = 80
Sample data:
X
79
81
63
42
70
50
61
48
74
69

X

X  n
2
2
s
n 1
Assume:  = 80
Sample data:
X
X2
79
81
63
42
70
50
61
48
74
69
6241
6561
3969
1764
4900
2500
3721
2304
5476
4761
637
42197
637
42197 
10
s 
10  1
2
405769
42197 
10
s 
10  1
42197  40576.9
s 
9
1620.1
s 
9
s  13.42
S
SX 
n
13.42
SX 
10
13.42
SX 
3.16
S X  4.24
63.7  80
t
4.24
 16.3
t
4.24
t  3.84
Critical value = 2.262