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381
Inferences About the Mean-II
(Small Samples)
QSCI 381 – Lecture 22
(Larson and Farber, Sect 6.2)
Small Samples and the t-Distribution
381


In many / most real situations, the
population standard deviation will be
unknown and the sample size will be
smaller than 30. The methods of
yesterday’s lecture cannot therefore be
applied.
If the sampling distribution is normally
distributed (or approximately so), the
sample mean, x , is t-distributed.
Side Note – Stout and Statistics.
381


The t-distribution is often referred to as
“Student’s” t-distribution.
“Student” (W.S. Gossett) was an employee of
the Guinness Brewing Company and was
prohibited from publishing papers (after
someone published trade secrets) so he
published under the pseudonym Student for
his publications to avoid detection of his
publications by his employer!
Properties of the t-distribution-I
381
1.
2.
3.
4.
The t-distribution is bell-shaped and
symmetric.
The total area under the curve is 1.
The mean, median and mode are equal to
0.
The t-distribution is a family of curves
(rather than one as was the case for the
normal distribution). Each element of the
family is determined by a parameter called
the degrees of freedom (n-1 for the case of
the sample mean)
x 
t
s/ n
df=5
Density
381
Properties of the t-distribution-II
df=2
-3.5
-1.5
0.5
2.5
Critical Values and the t-Distribution
381

Find the values of t such that there is a
probability of 0.025 beyond t for d.f. =
1, 5, 10, 20, 100. I used the EXCEL
Function:TINV(p,d.f)
d.f.
Critical t
1
12.71
5
2.57
10
2.23
20
2.09
100
1.98
Confidence Interval for the
Population Mean-I
381
1.
2.
3.
4.
5.
Find the sample statistics n, x , and d.f (=n-1).
Calculate the sample standard deviation.
Find the level of t that corresponds to the
confidence level (this depends on the d.f).
Find the maximum error of estimate E:
s
E  tc
n
Construct a confidence interval using the
formula:
x E   x E
Confidence Interval for the
Population Mean-II
381

Find a 95% confidence interval for the
mean when x  50; s  10 for d.f.s of 1, 5,
and 10 (i.e. n=2, 6 and 11).
d.f.
tc
E
Confidence interval
1
12.71
89.84
-39.85; 139.85
5
2.57
10.49
39.51; 60.49
10
2.23
6.72
43.28; 56.72
Is n30?
381
Use the normal distribution
with

Yes
E  zc
No
If  is unknown use s instead
Is the population
normally, or approximately
normally distributed?
Yes
Is  known?
Yes
No
Use t-distribution with
E  tc
n
s
n
No
You cannot use the normal
distribution or the t-distribution
Use normal distribution with
E  zc

n
Examples
381

You take:




24 samples, the data are normally
distributed,  is known.
14 samples, the data are normally
distributed,  is unknown.
34 samples, the data are not normally
distributed,  is unknown.
12 samples; the data are not normally
distributed,  is unknown.
Sample Sizes-I
381



The width of a confidence interval can be
reduced by increasing the size of the sample on
which the mean is based.
Given a c-confidence level and a maximum error
of estimate E, the minimum sample size needed
to estimate  is:
2
z 
n c 
 E 
In EXCEL:

=(STDEV(D2:D51)*NORMINV(0.995,0,1)/Eval)^
2
Sample Sizes-II
381

1.
2.
Find the sample size so that we are 99%
confident that the actual mean density is within
1 unit of the sample mean density if the
standard deviation is 10.66.
The level of confidence is 99% so the critical
value of z is 2.575.
The needed sample size is:
2
 2.575 x10.66 
n
  754 (rounded up)
1

