Section 8 – 3B
Finding Critical Values for a T Distribution
A Critical Value for a t Distribution is a t value on the t axis that is the vertical boundary
separating the area in one tail of the graph from the remaining area.
The entire area that is to be used in the tail(s) denoted by α . The entire area denoted by α can placed
in the left tail and produce a Left Tail Critical Value. The entire area denoted by α can be placed in
the right tail and produce a Right Tail Critical Value. The entire area denoted by α can split in half,
denoted by α 2 , and an area of α 2 can be placed in both the left tail and the right tail. This will
produce both a Negative Left Tail Critical Value and a Positive Right Tail Critical Value.
Left Tail Critical t Value
Right Tail Critical t Value
The entire area denoted by α is in the left tail.
The area to the left of the t value is α
The critical value is the negative t score that has
has an area of α to the left of the t value.
area to the
left of t = α
area to the
right of t = 1− α
area to the left
of t = 1− α
area to the
right of t = α
right tail
area = α
left tail
area = α
t=+?
positive
critical
value
t=–?
negitive
critical
value
Section 8–3B
The entire area denoted by α is in the right tail.
The area to the right of the t value is α
The critical value is the positive t score that
has an area of α to the right of the t value.
Page 1 of 12
©2013 Eitel
Two Tail Critical t Values
The total area for both of the tails is denoted by α
The area denoted by α is split in half , denoted by α 2 , and an area of α 2 is be placed in both the left
tail and the right tail. This will produce both a Left Tail Critical Value and a Right Tail Critical
Value.
Left Tail Critical Area
Right Tail Critical Area
The left tail has an area of α 2
The right tail has an area of α 2
The remaining area between the two tails is 1− α
left tail
area = α 2
right tail
area = α 2
1− α
Left Tail Critical Value
Right Tail Critical Value
The negative critical value is the negative t score
that has an area of α 2 to the left of the t value.
The positive critical value is the positive t score
that has an area of α 2 to the right of the t value
area to the right of + t = α 2
left tail
area = α 2
t=–?
negitive
critical
value
Section 8–3B
Page 2 of 12
right tail
area = α 2
t=+?
positive
critical
value
©2013 Eitel
t Distribution: Critical t Values
Degrees of
Area In One Tail (Right Tail)
Freedom
1
2
3
4
5
6
7
8
9
10
11
12
13
14
0.100
3.078
1.886
1.638
1.533
1.476
1.440
1.415
1.397
1.383
1.372
1.363
1.356
1.350
1.345
0.050
6.314
2.920
2.353
2.132
2.015
1.943
1.895
1.860
1.833
1.812
1.796
1.782
1.771
1.761
0.025
12.706
4.303
3.182
2.776
2.571
2.447
2.365
2.306
2.262
2.228
2.201
2.179
2.160
2.145
0.010
31.821
6.965
4.541
3.747
3.365
3.143
2.998
2.896
2.821
2.764
2.718
2.681
2.650
2.624
0.005
63.657
9.925
5.841
4.604
4.032
3.707
3.499
3.355
3.250
3.169
3.106
3.055
3.012
2.977
Distribution
Finding The Right Tail Critical Value for the t Distribution
Example 1
Find the Critical Value for t with an area of α = .05 in the right tail and n = 41
If n = 41 then the Degrees of Freedom = n – 1= 41 – 1= 40
DF = 41–1 =
40
right tail
area
α = .05
t = + 1.684
t
If the area in the right tail is .05 for DF = 40 then
t = 1.684
t Distribution: Critical t Values
Degrees of
Freedom
40
Section 8–3B
Area In One Tail (Right Tail)
0.100
0.050
0.025
0.010
0.005
1.303
1.684
2.021
2.423
2.704
Page 3 of 12
©2013 Eitel
Finding The Right Tail Critical Value for the t Distribution
Example 2
Find the Critical Value for t with an area of α = .01 in the right tail and n = 66
DF = 66–1 =
65
right tail
area
α = .01
t = + 2.385
positive
critical
value
If n = 66 then the Degrees of Freedom = n – 1= 66 – 1= 65
If the area in the right tail is
0.01
for DF =
65
then
t = 2.385
t Distribution: Critical t Values
Degrees of
Freedom
65
Section 8–3B
Area In One Tail (Right Tail)
0.100
0.050
0.025
0.010
0.005
1.295
1.669
1.997
2.385
2.645
Page 4 of 12
©2013 Eitel
Finding The Right Tail Critical Value for the t Distribution
Example 3
Find the Critical Value for t with an area of α = .10 in the right tail and n = 35
DF = 35–1 =
34
right tail
area
α = .10
t
t = + 1.307
If n = 35 then the Degrees of Freedom = n – 1= 35 – 1= 34
If the area in the right tail is
0.1
for DF =
34
then
t = 1.307
t Distribution: Critical t Values
Degrees of
Freedom
34
Section 8–3B
Area In One Tail (Right Tail)
0.100
0.050
0.025
0.010
0.005
1.307
1.691
2.032
2.441
2.728
Page 5 of 12
©2013 Eitel
Finding The Left Tail Critical Value for the t Distribution
The table for the t distribution only contains the positive t values for right tail areas of α = .10, α = .05,
α = .025 , α = .01, and α = .005 α = .05. This means that only positive t values can be read form the
table.
How do I find negative t values for left tail areas?
The t distribution is a normal curve that is symmetrical about the center of the graph.
DF =
left tail
area = α
t=–A
negitive
critical
value
n – 1
right tail
area = α
t=+A
positive
critical
value
t
This means that for any given value of α in the left tail
the negative critical t value will be the same number as the positive t value for α in the right tail
but with a negative sign.
How do I find negative t values for left tail areas?
For a given value of n
If the positive critical t value for a right tail area of α is +A
then the negative critical t value for a left tail area of α is –A
Section 8–3B
Page 6 of 12
©2013 Eitel
Finding The Left Tail Critical Value for the t Distribution
Example 4
To find the negitive Critical Value for t with an area of α = .01 in the left tail and n = 41
we first find the Positive critical value for α = .01 in the right tail and n = 41
If n = 41 then the Degrees of Freedom = n – 1= 41 – 1=
40
Positive critical value is t = 2.423
The negative critical value for α =
0.01
in the left tail and DF =
40
is t = – 2.423
t Distribution: Critical t Values
Degrees of
Freedom
40
Area In One Tail (Right Tail)
0.100
0.050
0.025
0.010
0.005
1.303
1.684
2.021
2.423
2.704
DF = 41–1 =
40
left tail
area
α = .01
t = – 2.423
Section 8–3B
Page 7 of 12
t
©2013 Eitel
Finding The Left Tail Critical Value for the t Distribution
Example 5
To find the negative Critical Value for t with an area of α = .05 in the left tail and n = 81
we first find the Positive critical value for α = .05 in the right tail and n = 81
If n = 81 then the Degrees of Freedom = n – 1= 81 – 1=
80
Positive critical value is t = 1.664
The negative critical value for α = .05 in the left tail and n = 81 is
t = – 1.664
t Distribution: Critical t Values
Degrees of
Freedom
80
Area In One Tail (Right Tail)
0.100
0.050
0.025
0.010
0.005
1.292
1.664
1.990
2.374
2.639
DF = 81–1 =
80
left tail
area
α = .05
t = – 1.664
Section 8–3B
Page 8 of 12
t
©2013 Eitel
Finding The Left Tail Critical Value for the t Distribution
Example 6
To find the negative Critical Value for t with an area of α = .10 in the left tail and n = 39
we first find the Positive critical value for α = .10 in the right tail and n = 39
If n = 39 then the Degrees of Freedom = n – 1= 39 – 1= 38
Positive critical value is t = 1.304
The negative critical value for α =
0.1
i n the left tail and n =
38
is
is t = – 1.304
t Distribution: Critical t Values
Degrees of
Freedom
38
Area In One Tail (Right Tail)
0.100
0.050
0.025
0.010
0.005
1.304
1.686
2.024
2.429
2.721
DF = 39–1 =
38
left tail
area
α = .10
t = – 1.304
Section 8–3B
Page 9 of 12
t
©2013 Eitel
Finding the negative and positive Critical t Values
for a t Distribution with Two Tails
The t distribution is a normal curve that is symmetrical about the center of the graph. This means that for
any given value of α 2 in the left tai the negative critical t value will be the same number as the positive
t value for α 2 in the right tail but with a negative sign.
DF =
n – 1
left tail
area = α 2
t=–A
negitive
critical
value
right tail
area = α 2
t=+A
t
positive
critical
value
How do I find the negative and positive critical t values for a
t Distribution with Two Tails
For a given value of n
split the value for α in half and put α 2 in the left and right tails.
Find the positive critical t value for a right tail α 2
If the positive critical t value for a right tail area of α 2 is +A
then the negative critical t value for a left tail area of α 2 is –A
Section 8–3B
Page 10 of 12
©2013 Eitel
Finding the negative and positive critical t values
for a t Distribution with two tails
Example 7
Find the negative and positive Critical Values for an area of α = .05
that is equally divided between the left and the right tail
with n = 41
α 2 = .025 in the left tail and α 2 = .025 in the right tail.
α =.05
DF = 41–1 =
40
left tail area
α 2 =.025
right tail area
α 2 =.025
t = – 2.021
negitive
critical
value
t = + 2.021
t
positive
critical
value
To find the negative Critical Value for t with an area of α 2 = .025 in the left tail and n = 41
we first find the Positive critical value for α 2 = .025 in the right tail and n = 41
If α 2 =
0.025
and the Degrees of Freedom
40
the Positive critical value is t = 2.021
The negative critical value for is t = – 2.021
t Distribution: Critical t Values
Degrees of
Freedom
40
Section 8–3B
Area In One Tail (Right Tail)
0.100
0.050
0.025
0.010
0.005
1.303
1.684
2.021
2.423
2.704
Page 11 of 12
©2013 Eitel
Example 8
Find the negative and positive Critical Values for an area of α = .01
that is equally divided between the left and the right tail
with n = 51
α 2 = .005 in the left tail and α 2 = .005 in the right tail.
α =.01
DF = 51–1 =
40
left tail area
α 2 =.005
right tail area
α 2 =.005
t = – 2.678
negitive
critical
value
t = + 2.678
t
positive
critical
value
To find the negative Critical Value for t with an area of α 2 = .005 in the left tail and n = 51
we first find the Positive critical value for α 2 = .005 in the right tail and n = 51
If α 2 =
0.005
and the Degrees of Freedom =
50
the Positive critical value is t = 2.678
The negative critical value is t = – 2.678
Distribution
Degrees of
Freedom
50
Section 8–3B
Area In One Tail (Right Tail)
0.100
0.050
0.025
0.010
0.005
1.299
1.676
2.009
2.403
2.678
Page 12 of 12
©2013 Eitel