Chapter 8Confidence Intervals

8.2

Confidence Intervals About ,

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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.

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

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 N Mean StDev SE Mean 95.0 % CIC3 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)

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)

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)

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)

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)

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)

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.