Section 8.2
Student’s t-Distribution
With the usual enthralling extra
content you’ve come to expect,
by D.R.S., University of Cordele
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Student’s t-Distribution
1.
2.
3.
4.
Properties of a t-Distribution
A t-distribution curve is symmetric and bell-shaped,
centered about 0.
A t-distribution curve is completely defined by its
number of degrees of freedom, df.
The total area under a t-distribution curve equals 1.
The x-axis is a horizontal asymptote for a
t-distribution curve.
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6.5 Finding t-Values Using the
Student t-Distribution
Comparison of the Normal and Student t-Distributions:
A t-distribution is
pretty much the
same as a normal
distribution!
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There’s this additional little
wrinkle of “d.f.”, “degrees of
freedom”. Slightly different
t distributions for different
d.f.; higher d.f. is closer &
closer to the normal
distribution.
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Why bother with t ?
• If you don’t know the population standard
deviation, σ, but you still want to use a sample to
find a confidence interval.
• t builds in a little more uncertainty based on the
lack of a trustworthy σ.
The plan:
1. This lesson – learn about t and areas and critical
values, much like we have done with z.
2. Next lesson – doing confidence intervals with t.
Why bother with t ?
• Observe in the picture how t isn’t quite as high and
bold in the middle part of the bell curve.
• The uncertainty shows up as extra area in the tails of
the bell curve.
• As the sample size n gets larger,
the degrees of freedom d.f. gets larger,
and the uncertainty becomes less uncertain,
and the t bell curve gets very much closer to the
normal distribution bell curve we use in z problems.
• History of t : Q.A. at an Irish brewery circa 1900.
See textbook or internet for all the details.
Example 8.9: Finding the Value of tα
Find the value of t0.025 for the t-distribution with 25
degrees of freedom.
Solution
The number of degrees of freedom is listed in the first
column of the t-distribution table. Since the
t-distribution in our example has 25 degrees of
freedom, the value we need lies on the row
corresponding to df = 25.
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Example 8.9: Finding the Value of tα (cont.)
Looking at the value of the subscript on t, which is the
area in the right tail, 0.025, tells us to use the column
for an area of 0.025 in one tail.
This row and column intersect at 2.060.
Thus, t0.025 = 2.060.
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Example 8.9: Finding the Value of tα (cont.)
*
Area in One Tail
0.100
0.050
0.010
0.005
23
0.200
1.319
Area in Two Tails
0.100
0.050
0.020
1.714
2.069
2.500
0.010
2.807
24
1.318
1.711
2.064
2.492
2.797
25
1.316
1.708
2.060
2.485
2.787
26
1.315
1.706
2.056
2.479
2.779
27
1.314
1.703
2.052
2.473
2.771
28
1.313
1.701
2.048
2.467
2.763
df
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0.025
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Example 8.10: Finding the Value of t Given the
Area to the Right
Find the value of t for a t-distribution with 17 degrees
of freedom such that the area under the curve to the
right of t is 0.10.
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Example 8.10: Finding the Value of t Given the
Area to the Right (cont.)
Solution
Note that according to the picture, the area under the
curve to the right of t is 0.10. This means that α= 0.10.
We are told that the distribution has 17 degrees of
freedom. Looking across the row for df = 17 and down
the column for an area in one tail of 0.100 we see that
t0.10 = 1.333.
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Example 8.10: Finding the Value of t Given the
Area to the Right (cont.)
*
Area in One Tail
0.100
0.050
0.010
0.005
df
0.200
Area in Two Tails
0.100
0.50
0.020
0.010
15
16
17
18
19
20
1.341
1.337
1.333
1.330
1.328
1.325
1.753
1.746
1.740
1.734
1.729
1.725
2.947
2.921
2.898
2.878
2.861
2.845
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0.025
2.131
2.12
2.110
2.101
2.093
2.086
2.602
2.583
2.567
2.552
2.539
2.528
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Example 8.11: Finding the Value of t Given the
Area to the Left
Find the value of t for a t-distribution with 11 degrees
of freedom such that the area under the curve to the
left of t is 0.05.
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Example 8.11: Finding the Value of t Given the
Area to the Left (cont.)
Solution
Because the t-distribution is symmetric, we can look up
the t-value for an area of 0.05 under the curve to the
right of t. Using the table, we get t0.05 = 1.796.
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Example 8.11: Finding the Value of t Given the
Area to the Left (cont.)
*
Area in One Tail
0.100
0.050
0.010
0.005
df
0.200
Area in Two Tails
0.100
0.050
0.020
0.010
9
10
11
12
13
14
1.383
1.372
1.363
1.356
1.350
1.345
1.833
1.812
1.796
1.782
1.771
1.761
3.25
3.169
3.106
3.055
3.012
2.977
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0.025
2.262
2.228
2.201
2.179
2.160
2.145
2.821
2.764
2.718
2.681
2.65
2.624
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Example 8.11: Finding the Value of t Given the
Area to the Left (cont.)
However, since the given area is to the left of t, the
t-value needs to be negative. So, for this example,
-t0.05 = -1.796.
Some TI-84 Plus Silver Edition calculators can also be
used to find the t-value.
• Press
and then
to go to the DISTR
menu.
• Choose option 4:invT(.
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Example 8.11: Finding the Value of t Given the
Area to the Left (cont.)
• Enter the area to the left of t and df in the
parentheses as: invT(area to the left of t, df ).
• Enter invT(0.05,11).
The answer given by the calculator is t ≈ -1.796.
If your calculator doesn’t
have invT, you must use
the printed tables. This is
probably the case with
TI-83/Plus. Sorry!
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Example 8.11: Finding the Value of t Given the
Area to the Left , with Excel
• Excel: T.INV(area to the left of t, df), same thing.
• Excel special if you know area in two tails total:
=T.INV.2T(area in two tails total, df)
Example 8.12: Finding the Value of t Given the
Area in Two Tails
Find the value of t for a t-distribution with 7 degrees of
freedom such that the area to the left of -t plus the
area to the right of t is 0.02, as shown in the picture.
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Example 8.12: Finding the Value of t Given the
Area in Two Tails (cont.)
Solution
This is a two-tailed problem because the given area,
0.02, is divided between both sides of the distribution.
Therefore, when looking up the t-value in the table, we
simply find the given area in the row labeled “Area in
Two Tails” as shown in the following excerpt from the
table. So the value of t for a t-distribution with 7
degrees of freedom such that the total area in the two
tails is 0.02 is t = 2.998.
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Example 8.12: Finding the Value of t Given the
Area in Two Tails (cont.)
Area in One Tail
0.100
0.050
0.010
0.005
df
0.200
Area in Two Tails
0.100
0.050
0.020
0.010
5
6
7
8
9
10
1.476
1.440
1.415
1.397
1.383
1.372
2.015
1.943
1.895
1.860
1.833
1.812
4.032
3.707
3.499
3.355
3.250
3.169
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0.025
2.571
2.447
2.365
2.306
2.262
2.228
3.365
3.143
2.998
2.896
2.821
2.764
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Example 8.12: Finding the Value of t Given the
Area in Two Tails (cont.)
To use a TI-84 Plus calculator to find t given the area in
two tails, you need to enter the area in the left tail only.
Since the problem indicates that the area is divided
between both ends, we must divide the area in half
before we use the calculator. Therefore, we calculate
the area in one tail as follows:
 0.02

 0.01.
2
2
TI-84 invT(area to left, df) does one-tailed only.
You make the adjustment when doing a two-tailed problem.
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Example 8.12: Finding the Value of t Given the
Area in Two Tails (cont.)
• Press
and then
to go to the DISTR
menu.
• Choose option 4:invT(.
• Enter invT(0.01,7).
Notice that the value of t that is returned is negative,
-t 2  -t0.01  -2.998.
If you want the positive value of t,
just ignore the negative sign since
the t-distribution is symmetric.
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Example 8.12: Finding the Value of t Given the
Area in Two Tails (cont.) – with Excel
• Recall: we seek t and –t such that
two tails total area 0.02, d.f. = 7
• Excel with convenient =T.INV.2T(total area, d.f.)
• Or Excel with one-tailed version, manually divide
area by 2: = T.INV(one tailed area, d.f.)
Example 8.13: Finding the Value of t Given Area
between -t and t
Find the critical value of t for a t-distribution with 29
degrees of freedom such that the area between −t and
t is 99%.
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Example 8.13: Finding the Value of t Given Area
between -t and t (cont.)
Solution
Since 99% of the area under the curve is in the middle,
that leaves 1%, or 0.01 of the area in the two tails.
Since the t-distribution has 29 degrees of freedom, look
across the row for df = 29 and down the column for an
area in two tails of 0.010. Thus, t = 2.756.
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Example 8.13: Finding the Value of t Given Area
between -t and t (cont.)
Area in One Tail
0.100
0.050
0.010
0.005
df
0.200
Area in Two Tails
0.100
0.050
0.020
0.010
27
28
29
30
31
32
1.314
1.313
1.311
1.310
1.309
1.309
1.703
1.701
1.699
1.697
1.696
1.694
2.771
2.763
2.756
2.750
2.744
2.738
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0.025
2.052
2.048
2.045
2.042
2.040
2.037
2.473
2.467
2.462
2.457
2.453
2.449
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Example 8.13: Finding the Value of t Given Area
between -t and t (cont.)
To use a TI-84 Plus calculator to find t, you need to
enter the area in the left tail only. We have determined
that the area in two tails is 0.01. Thus, we calculate the
area in one tail as follows:
 0.01

 0.005.
2
2
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Example 8.13: Finding the Value of t Given Area
between -t and t (cont.)
• Press
and then
to go to the DISTR
menu.
• Choose option 4:invT(.
• Enter invT(0.005,29).
Notice that the value of t that is returned is negative,
-t 2  -t0.005  -2.756.
If you want the positive value of t,
just ignore the negative sign since
the t-distribution is symmetric.
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Example 8.14: Finding the Critical t-Value for a
Confidence Interval
Find the critical t-value for a 95% confidence interval
using a t-distribution with 24 degrees of freedom.
Convert 95%
Solution
confidence
Interval into
α = area in two
tails
and extra
adjustment of
dividing by 2 if
using invT(
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Example 8.14: Finding the Critical t-Value for a
Confidence Interval (cont.)
Since we are looking for the critical value for a 95%
confidence interval, we want to find the value of t such
that the area between -t 2 and t 2 is 0.95. If the area
under the curve between the two t-values is c = 0.95,
then  = 1 − c = 1 − 0.95 = 0.05 is the area in the two
tails. Since the t-distribution has 24 degrees of freedom
and the area in two tails is 0.05, looking across the row
for df = 24 and down the column for an area in two tails
of 0.050, we find a critical t-value of t 2  t0.025  2.064.
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Example 8.14: Finding the Critical t-Value for a
Confidence Interval (cont.)
Area in One Tail
0.100
0.050
0.010
0.005
df
0.200
Area in Two Tails
0.100
0.050
0.020
0.010
21
22
23
24
25
26
1.323
1.321
1.319
1.318
1.316
1.315
1.721
1.717
1.714
1.711
1.708
1.706
2.831
2.819
2.807
2.797
2.787
2.779
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0.025
2.080
2.074
2.069
2.064
2.060
2.056
2.518
2.508
2.500
2.492
2.485
2.479
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Example 8.14: Finding the Critical t-Value for a
Confidence Interval (cont.)
To use a TI-84 Plus calculator to find t, you need to
enter the area in the left tail only. We have determined
that the area in two tails is 0.05. Thus, we calculate the
area in one tail as follows:
 0.05

 0.025.
2
2
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Example 8.14: Finding the Critical t-Value for a
Confidence Interval (cont.)
• Press
and then
to go to the DISTR
menu.
• Choose option 4:invT(.
• Enter invT(0.025,24).
Notice that the value of t that is returned is negative,
-t 2  -t0.025  -2.064.
If you want the positive value of t,
just ignore the negative sign since
the t-distribution is symmetric.
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