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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.

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chapter 8 confidence intervals

Chapter 8Confidence Intervals

8.1

Confidence Intervals about a Population Mean,  Known

slide2

A point estimate of a parameter is the value of a statistic that estimates the value of the parameter.

slide4

A confidence interval estimate of a parameter consists of an interval of numbers along with a probability that the interval contains the unknown parameter.

slide5

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

slide6

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.

slide7

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
slide8

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

slide11

95% of all sample means are in the interval

With a little algebraic manipulation, we can rewrite this inequality and obtain:

chapter 8 confidence intervals17

Chapter 8Confidence Intervals

8.2

Confidence Intervals About ,

 Unknown

slide23

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.
slide24

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.

slide27

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.

slide29

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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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?

slide35

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.

slide37

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)

slide38

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)

slide39

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)

slide40

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)

slide41

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)

slide42

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)

slide43

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.