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Chapter 8 Confidence Intervals

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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

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### Chapter 8Confidence 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 8Confidence Intervals

8.2

Confidence Intervals About ,

 Unknown

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.

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.

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.

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?

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 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)

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)

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)

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)

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)

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)

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.