have to guess the age of a random person about whom you have no information, you may guess an
interval (0,120) which would probably contain that person’s age. In a confidence interval, this is
obtained by having a larger half interval. Since k (Z in a normal distribution), s and n are the three
variables which give us the half length, and the values of s and n are fixed, we can only change k.
Having a larger k will give us a larger interval. This can be done by increasing the confidence level but
it will not be practical for we want to keep the confidence level fixed too. That means we must use
another probability distribution which looks like the normal distribution but has a larger variance.
Luckily, we have such a distribution, known as
Student’s T distribution
. The distribution is named
after the nickname of William Sealy Gosset who worked for Guinness (Dublin, Ireland) at the time
and did not want to use his real name for proprietary restrictions (or so it is rumored). It is also
known as
T distribution
in short and has a bell shape like the normal distribution. T distribution has
different variances for different sample sizes, and eventually converges to normal distribution for
very large sample sizes. A T table, unlike the Z table, does not give probabilities for every possible T
value, but rather lists T values for popular probabilities (like 0.90, 0.95, 0.99, etc in the middle). Still it
is possible to find every probability (or T value) by integrating the probability distribution function (or
use calculators or programs or websites which do so). T tables list these T values depending on
the
degrees of freedom
which is
(n – 1)
, therefore T tables typically list degrees of freedom from 1 to 30

after. In practice more degrees o
and the standard normal distribu
probability (which correspond to
Sample size (n)
1
T value
12.706
As expected, for very small samp
they get smaller as sample size g
Below is the graph of the t distrib
Let us now solve the previous ex
test statistic become:
g2020 =
So the test statistic is now;
of freedom are seldom required since Central Li
ution is utilized. Below you will find some T valu
o a Z =
±
1.96).
5
10
20
30
120
2.571
2.228
2.086
2.042
1.98
ples the k values (now shown as
t
instead of a
z
gets larger. And about n = 121, t and z are very
bution and the general view of a regular t table.
xample for a small sample of size n = 19. The con
= g1850
g3364
±g1872
(g2869g2879
g3080
g2870
)
g1871
uni221Ag1866
g1853g1866g1856 g1872
g3030g3028g3039
=
g1850
g3364
− g2020
g1871
uni221Ag1866
22
imit Theorem applies
ues for a 0.95 middle
uni221E
1.96
) are very large, and
y close to each other.
nfidence interval and

23
g1852
g3030g3028g3039
=
g1850
g3364
− g2020
g1871
uni221Ag1866
=
489− 500
25.3
uni221A19
=
−11
5.8
= −1.9
To conclude we will need the t
tab
from the table. T tables typically cover the probability from
−∞
to a
positive t value. So we look up (for our two sided test) t
(1-α/2),(n-1)
= t
0.975,18
=
±
2.101. Since the test
statistic is smaller than the critical value, we fail to reject the null hypothesis (μ = 500) at a 95%
confidence level. Obtaining a 95% confidence interval for μ is left as an exercise.

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