Transcript Chapter 8 Confidence Intervals

Chapter 8
Confidence Intervals
8.1
Confidence Intervals about a
Population Mean,  Known
A point estimate of a parameter is
the value of a statistic that estimates
the value of the parameter.
A confidence interval estimate of a
parameter consists of an interval of
numbers along with a probability that the
interval contains the unknown parameter.
The level of confidence in a confidence
interval is a probability that represents the
percentage of intervals that will contain if a
large number of repeated samples are
obtained. The level of confidence is denoted
For example, a 95% level of confidence
would mean that if 100 confidence
intervals were constructed, each based
on a different sample from the same
population, we would expect 95 of the
intervals to contain the population
mean.
The construction of a confidence
interval for the population mean
depends upon three factors
 The point estimate of the population
 The level of confidence
 The standard deviation of the sample
mean
Suppose we obtain a simple random sample
from a population. Provided that the
population is normally distributed or the
sample size is large, the distribution of the
sample mean will be normal with
95% of all sample means are in the
interval
With a little algebraic manipulation, we
can rewrite this inequality and obtain:
Chapter 8
Confidence Intervals
8.2
Confidence Intervals About ,
 Unknown
Histogram for z
Histogram for t
Properties of the t Distribution
1. The t distribution is different for different
values of n, the sample size.
2. The t distribution is centered at 0 and is
symmetric about 0.
3. The area under the curve is 1. Because of
the symmetry, the area under the curve to
the right of 0 equals the area under the
curve to the left of 0 equals 1 / 2.
Properties of the t Distribution
4. As t increases without bound, the graph
approaches, but never equals, zero. As t
decreases without bound the graph
approaches, but never equals, zero.
5. The area in the tails of the t distribution is
a little greater than the area in the tails of
the standard normal distribution. This result
is because we are using s as an estimate of 
which introduces more variability to the t
statistic.
Properties of the t Distribution
EXAMPLE Finding t-values
Find the t-value such that the area under the t
distribution to the right of the t-value is 0.2
assuming 10 degrees of freedom. That is, find
t0.20 with 10 degrees of freedom.
EXAMPLE
Constructing a Confidence Interval
The pasteurization process reduces the amount of
bacteria found in dairy products, such as milk. The
following data represent the counts of bacteria in
pasteurized milk (in CFU/mL) for a random sample
of 12 pasteurized glasses of milk. Data courtesy of
Dr. Michael Lee, Professor, Joliet Junior College.
Construct a 95% confidence interval for the
bacteria count.
NOTE: Each observation is in tens of thousand.
So, 9.06 represents 9.06 x 104.
Boxplot of CFU/mL
EXAMPLE The Effects of Outliers
Suppose a student miscalculated the amount of
bacteria and recorded a result of 2.3 x 105. We
would include this value in the data set as 23.0.
What effect does this additional observation
have on the 95% confidence interval?
Boxplot of CFU/mL
What if we obtain a small sample from a population
that is not normal and construct a t-interval? The
following distribution represents the number of
people living in a household for all homes in the
United States in 2000.
Obtain 100 samples
of size n = 6 and
construct 95%
confidence for each
sample. Comment
on the number of
intervals that contain
the population mean,
2.564 and the width
of each interval.
Variable
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N
6
6
6
6
6
6
6
6
6
6
6
6
6
Mean StDev SE Mean
95.0 % CI
1.667 0.816 0.333 ( 0.810, 2.524)
2.333 1.862 0.760 ( 0.379, 4.287)
2.667 1.366 0.558 ( 1.233, 4.101)
2.500 1.378 0.563 ( 1.053, 3.947)
1.667 0.816 0.333 ( 0.810, 2.524)
2.667 2.066 0.843 ( 0.499, 4.835)
1.500 0.548 0.224 ( 0.925, 2.075)
1.833 0.983 0.401 ( 0.801, 2.865)
3.500 1.761 0.719 ( 1.652, 5.348)
2.167 1.169 0.477 ( 0.940, 3.394)
2.000 0.894 0.365 ( 1.061, 2.939)
2.833 2.137 0.872 ( 0.591, 5.076)
2.500 1.643 0.671 ( 0.775, 4.225)
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6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
1.833
2.500
2.167
2.500
2.500
1.833
2.667
3.333
1.500
2.667
1.833
2.167
2.833
2.000
2.667
1.667
2.167
2.500
1.169
1.517
1.169
1.643
0.837
0.753
1.862
1.211
0.837
2.422
1.169
0.753
0.983
1.095
1.033
1.033
0.983
1.225
0.477
0.619
0.477
0.671
0.342
0.307
0.760
0.494
0.342
0.989
0.477
0.307
0.401
0.447
0.422
0.422
0.401
0.500
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
0.606,
0.908,
0.940,
0.775,
1.622,
1.043,
0.713,
2.062,
0.622,
0.125,
0.606,
1.377,
1.801,
0.850,
1.583,
0.583,
1.135,
1.215,
3.060)
4.092)
3.394)
4.225)
3.378)
2.623)
4.621)
4.604)
2.378)
5.209)
3.060)
2.957)
3.865)
3.150)
3.751)
2.751)
3.199)
3.785)
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6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
3.833
2.000
2.167
2.167
2.000
1.833
2.167
2.833
2.833
3.167
2.000
3.333
1.667
3.167
2.000
2.000
2.000
1.667
1.722
1.265
0.983
1.329
0.894
0.983
2.401
2.317
2.137
1.602
1.095
2.066
0.816
2.041
1.095
1.095
0.894
0.816
0.703
0.516
0.401
0.543
0.365
0.401
0.980
0.946
0.872
0.654
0.447
0.843
0.333
0.833
0.447
0.447
0.365
0.333
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
2.026,
0.672,
1.135,
0.772,
1.061,
0.801,
-0.354,
0.402,
0.591,
1.485,
0.850,
1.165,
0.810,
1.024,
0.850,
0.850,
1.061,
0.810,
5.641)
3.328)
3.199)
3.562)
2.939)
2.865)
4.687)
5.265)
5.076)
4.848)
3.150)
5.501)
2.524)
5.309)
3.150)
3.150)
2.939)
2.524)
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6
6
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6
6
6
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6
6
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6
6
6
3.000
1.833
2.000
2.333
3.333
2.667
2.667
2.333
2.167
2.167
2.667
2.000
3.167
2.167
2.000
1.667
1.667
1.549
1.169
1.095
1.033
1.506
1.751
1.211
1.033
0.983
0.983
1.506
1.265
1.472
0.753
1.673
0.516
0.816
0.632
0.477
0.447
0.422
0.615
0.715
0.494
0.422
0.401
0.401
0.615
0.516
0.601
0.307
0.683
0.211
0.333
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
1.374,
0.606,
0.850,
1.249,
1.753,
0.829,
1.396,
1.249,
1.135,
1.135,
1.087,
0.672,
1.622,
1.377,
0.244,
1.125,
0.810,
4.626)
3.060)
3.150)
3.417)
4.913)
4.505)
3.938)
3.417)
3.199)
3.199)
4.247)
3.328)
4.712)
2.957)
3.756)
2.209)
2.524)
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6
6
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6
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2.500
2.500
2.500
1.667
2.500
3.333
2.167
2.500
1.833
2.167
3.000
1.833
1.833
3.333
2.667
4.333
3.17
1.049
1.378
1.225
0.816
1.378
1.506
0.983
1.378
0.983
1.602
1.897
0.753
0.753
2.160
1.633
1.211
2.71
0.428
0.563
0.500
0.333
0.563
0.615
0.401
0.563
0.401
0.654
0.775
0.307
0.307
0.882
0.667
0.494
1.11 (
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
1.399, 3.601)
1.053, 3.947)
1.215, 3.785)
0.810, 2.524)
1.053, 3.947)
1.753, 4.913)
1.135, 3.199)
1.053, 3.947)
0.801, 2.865)
0.485, 3.848)
1.009, 4.991)
1.043, 2.623)
1.043, 2.623)
1.066, 5.601)
0.953, 4.381)
3.062, 5.604)
0.32, 6.02)
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6 2.500 1.378 0.563 ( 1.053, 3.947)
6 2.333 1.506 0.615 ( 0.753, 3.913)
6 3.500 1.761 0.719 ( 1.652, 5.348)
6 2.500 1.643 0.671 ( 0.775, 4.225)
6 1.833 0.983 0.401 ( 0.801, 2.865)
6 2.333 1.211 0.494 ( 1.062, 3.604)
6 2.333 0.516 0.211 ( 1.791, 2.875)
6 3.333 1.506 0.615 ( 1.753, 4.913)
6 2.667 1.751 0.715 ( 0.829, 4.505)
6 1.667 0.516 0.211 ( 1.125, 2.209)
6 2.833 0.983 0.401 ( 1.801, 3.865)
6 2.500 1.378 0.563 ( 1.053, 3.947)
6 2.667 1.366 0.558 ( 1.233, 4.101)
6 2.167 1.169 0.477 ( 0.940, 3.394)
6 2.833 0.983 0.401 ( 1.801, 3.865)
6 2.000 0.000 0.000 ( 2.00000, 2.00000)
6 2.167 1.169 0.477 ( 0.940, 3.394)
Notice that the width of each interval differs
– sometimes substantially.
In addition, we would expect that 95 out of
the 100 intervals would contain the
population mean, 2.564. However, 90 out of
the 100 intervals actually contain the
population mean.