Chapter 8 Confidence Intervals

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Chapter 8Confidence Intervals

• 8.2
• Confidence Intervals About ?,
• ? Unknown

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Histogram for z
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Histogram for t
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• Properties of the t Distribution
• 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.

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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.
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Properties of the t Distribution
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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.
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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.
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NOTE Each observation is in tens of thousand.
So, 9.06 represents 9.06 x 104.
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Boxplot of CFU/mL
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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?
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Boxplot of CFU/mL
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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.
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Variable N Mean StDev SE Mean
95.0 CI C3 6 1.667 0.816
0.333 ( 0.810, 2.524) C4 6
2.333 1.862 0.760 ( 0.379, 4.287) C5
6 2.667 1.366 0.558 ( 1.233,
4.101) C6 6 2.500 1.378
0.563 ( 1.053, 3.947) C7 6
1.667 0.816 0.333 ( 0.810, 2.524) C8
6 2.667 2.066 0.843 ( 0.499,
4.835) C9 6 1.500 0.548
0.224 ( 0.925, 2.075) C10 6
1.833 0.983 0.401 ( 0.801, 2.865) C11
6 3.500 1.761 0.719 ( 1.652,
5.348) C12 6 2.167 1.169
0.477 ( 0.940, 3.394) C13 6
2.000 0.894 0.365 ( 1.061, 2.939) C14
6 2.833 2.137 0.872 ( 0.591,
5.076) C15 6 2.500 1.643
0.671 ( 0.775, 4.225)
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C16 6 1.833 1.169 0.477 (
0.606, 3.060) C17 6 2.500 1.517
0.619 ( 0.908, 4.092) C18 6
2.167 1.169 0.477 ( 0.940, 3.394) C19
6 2.500 1.643 0.671 ( 0.775,
4.225) C20 6 2.500 0.837
0.342 ( 1.622, 3.378) C21 6
1.833 0.753 0.307 ( 1.043, 2.623) C22
6 2.667 1.862 0.760 ( 0.713,
4.621) C23 6 3.333 1.211
0.494 ( 2.062, 4.604) C24 6
1.500 0.837 0.342 ( 0.622, 2.378) C25
6 2.667 2.422 0.989 ( 0.125,
5.209) C26 6 1.833 1.169
0.477 ( 0.606, 3.060) C27 6
2.167 0.753 0.307 ( 1.377, 2.957) C28
6 2.833 0.983 0.401 ( 1.801,
3.865) C29 6 2.000 1.095
0.447 ( 0.850, 3.150) C30 6
2.667 1.033 0.422 ( 1.583, 3.751) C31
6 1.667 1.033 0.422 ( 0.583,
2.751) C32 6 2.167 0.983
0.401 ( 1.135, 3.199) C33 6
2.500 1.225 0.500 ( 1.215, 3.785)
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C34 6 3.833 1.722 0.703 (
2.026, 5.641) C35 6 2.000 1.265
0.516 ( 0.672, 3.328) C36 6
2.167 0.983 0.401 ( 1.135, 3.199) C37
6 2.167 1.329 0.543 ( 0.772,
3.562) C38 6 2.000 0.894
0.365 ( 1.061, 2.939) C39 6
1.833 0.983 0.401 ( 0.801, 2.865) C40
6 2.167 2.401 0.980 ( -0.354,
4.687) C41 6 2.833 2.317
0.946 ( 0.402, 5.265) C42 6
2.833 2.137 0.872 ( 0.591, 5.076) C43
6 3.167 1.602 0.654 ( 1.485,
4.848) C44 6 2.000 1.095
0.447 ( 0.850, 3.150) C45 6
3.333 2.066 0.843 ( 1.165, 5.501) C46
6 1.667 0.816 0.333 ( 0.810,
2.524) C47 6 3.167 2.041
0.833 ( 1.024, 5.309) C48 6
2.000 1.095 0.447 ( 0.850, 3.150) C49
6 2.000 1.095 0.447 ( 0.850,
3.150) C50 6 2.000 0.894
0.365 ( 1.061, 2.939) C51 6
1.667 0.816 0.333 ( 0.810, 2.524)
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C52 6 3.000 1.549 0.632 (
1.374, 4.626) C53 6 1.833 1.169
0.477 ( 0.606, 3.060) C54 6
2.000 1.095 0.447 ( 0.850, 3.150) C55
6 2.333 1.033 0.422 ( 1.249,
3.417) C56 6 3.333 1.506
0.615 ( 1.753, 4.913) C57 6
2.667 1.751 0.715 ( 0.829, 4.505) C58
6 2.667 1.211 0.494 ( 1.396,
3.938) C59 6 2.333 1.033
0.422 ( 1.249, 3.417) C60 6
2.167 0.983 0.401 ( 1.135, 3.199) C61
6 2.167 0.983 0.401 ( 1.135,
3.199) C62 6 2.667 1.506
0.615 ( 1.087, 4.247) C63 6
2.000 1.265 0.516 ( 0.672, 3.328) C64
6 3.167 1.472 0.601 ( 1.622,
4.712) C65 6 2.167 0.753
0.307 ( 1.377, 2.957) C66 6
2.000 1.673 0.683 ( 0.244, 3.756) C67
6 1.667 0.516 0.211 ( 1.125,
2.209) C68 6 1.667 0.816
0.333 ( 0.810, 2.524)
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C69 6 2.500 1.049 0.428 (
1.399, 3.601) C70 6 2.500 1.378
0.563 ( 1.053, 3.947) C71 6
2.500 1.225 0.500 ( 1.215, 3.785) C72
6 1.667 0.816 0.333 ( 0.810,
2.524) C73 6 2.500 1.378
0.563 ( 1.053, 3.947) C74 6
3.333 1.506 0.615 ( 1.753, 4.913) C75
6 2.167 0.983 0.401 ( 1.135,
3.199) C76 6 2.500 1.378
0.563 ( 1.053, 3.947) C77 6
1.833 0.983 0.401 ( 0.801, 2.865) C78
6 2.167 1.602 0.654 ( 0.485,
3.848) C79 6 3.000 1.897
0.775 ( 1.009, 4.991) C80 6
1.833 0.753 0.307 ( 1.043, 2.623) C81
6 1.833 0.753 0.307 ( 1.043,
2.623) C82 6 3.333 2.160
0.882 ( 1.066, 5.601) C83 6
2.667 1.633 0.667 ( 0.953, 4.381) C84
6 4.333 1.211 0.494 ( 3.062,
5.604) C85 6 3.17 2.71
1.11 ( 0.32, 6.02)
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C86 6 2.500 1.378 0.563 (
1.053, 3.947) C87 6 2.333 1.506
0.615 ( 0.753, 3.913) C88 6
3.500 1.761 0.719 ( 1.652, 5.348) C89
6 2.500 1.643 0.671 ( 0.775,
4.225) C90 6 1.833 0.983
0.401 ( 0.801, 2.865) C91 6
2.333 1.211 0.494 ( 1.062, 3.604) C92
6 2.333 0.516 0.211 ( 1.791,
2.875) C93 6 3.333 1.506
0.615 ( 1.753, 4.913) C94 6
2.667 1.751 0.715 ( 0.829, 4.505) C95
6 1.667 0.516 0.211 ( 1.125,
2.209) C96 6 2.833 0.983
0.401 ( 1.801, 3.865) C97 6
2.500 1.378 0.563 ( 1.053, 3.947) C98
6 2.667 1.366 0.558 ( 1.233,
4.101) C99 6 2.167 1.169
0.477 ( 0.940, 3.394) C100 6
2.833 0.983 0.401 ( 1.801, 3.865) C101
6 2.000 0.000 0.000 ( 2.00000,
2.00000) C102 6 2.167 1.169
0.477 ( 0.940, 3.394)
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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.