Transcript Understanding Basic Statistics By Brase and Brase

Chapter 7
Estimation
Chapter 7
ESTIMATION
Section 2
Estimating m
When s Is Unknown
What if it is impossible or
impractical to use a large
sample?
Apply the
Student’s t distribution.
The shape of the t distribution
depends only the sample size, n,
if the basic variable x has a
normal distribution.
When using the t distribution,
we will assume that the
x distribution is normal.
Confidence Interval
for the Mean of Small
Samples (n < 30)
Table 6 in Appendix II gives
values of the variable t
corresponding to the number
of degrees of freedom (d.f.)
Degrees of Freedom
d.f. = n – 1
where n = sample size
The t Distribution has a
Shape Similar to that of the
the Normal Distribution
A Normal
distribution
A “t”
distribution
Find the critical value tc for a
95% confidence interval if n = 7.
c
0.900
0.050
0.100
0.950
0.025
0.050
0.980
0.010
0.020
0.990
0.005
0.010
6
...1.9432
2.4469
3.1427
3.7074
7
...1.8946
2.3646
2.9980
3.4995
8
...1.8595
2.3060
2.8965
3.3554
’
 ’’
d.f.
...
Confidence Interval for the Mean
of Small Samples (n < 30) from
Normal Populations
x-E<m<x+E
where x = Sample Mean
E = tc
s
n
c = confidence level (0 < c < 1)
tc = critical value for confidence level c, and
degrees of freedom = n - 1
The mean weight of eight fish
caught in a local lake is 15.7
ounces with a standard
deviation of 2.3 ounces.
Construct a 90% confidence
interval for the mean weight
of the population of fish
in the lake.
Key Information
• Mean = 15.7 ounces
• Standard deviation = 2.3 ounces
• n = 8, so d.f. = n – 1 = 7
• For c = 0.90, Table t chart
• gives t0.90 = 1.8946
The 90% confidence interval is:
2.3
2.3
15.7 - 1.8946
< m < 15.7 + 1.8946
8
8
4.3576
4.3576
15.7 < m < 15.7 +
2.8284
2.8284
15.7- < 1.5406 < m < 15.7 + 1.5406
14.1594 < m < 17.2406
We can say with 90% confidence that
the population mean
weight of the fish in the lake is
between 14.1594 and 17.2406 ounces.
The 90% confidence interval is:
Calculator Computation
VARS
Statistics
TEST
H: lower 14.1594
I: upper 17.2406
We can say with 90%
confidence that the
population mean
weight of the fish in the lake
is between 14.1594 and
17.2406 ounces.
THE
END
OF
SECTION 2