Chapter 8 Confidence Intervals 8 1 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 q The point estimate of the population q The level of confidence q 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 t 0. 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/m. L) 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/m. L

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/m. L

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 C 3 C 4 C 5 C 6 C 7 C 8 C 9 C 10 C 11 C 12 C 13 C 14 C 15 N 6 6 6 6 Mean St. Dev 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)

C 16 C 17 C 18 C 19 C 20 C 21 C 22 C 23 C 24 C 25 C 26 C 27 C 28 C 29 C 30 C 31 C 32 C 33 6 6 6 6 6 1. 833 2. 500 2. 167 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 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. 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)

C 34 C 35 C 36 C 37 C 38 C 39 C 40 C 41 C 42 C 43 C 44 C 45 C 46 C 47 C 48 C 49 C 50 C 51 6 6 6 6 6 3. 833 2. 000 2. 167 2. 000 1. 833 2. 167 2. 833 3. 167 2. 000 3. 333 1. 667 3. 167 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 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. 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, 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) 2. 939) 2. 524)

C 52 C 53 C 54 C 55 C 56 C 57 C 58 C 59 C 60 C 61 C 62 C 63 C 64 C 65 C 66 C 67 C 68 6 6 6 6 6 3. 000 1. 833 2. 000 2. 333 3. 333 2. 667 2. 333 2. 167 2. 667 2. 000 3. 167 2. 000 1. 667 1. 549 1. 169 1. 095 1. 033 1. 506 1. 751 1. 211 1. 033 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. 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. 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) 4. 247) 3. 328) 4. 712) 2. 957) 3. 756) 2. 209) 2. 524)

C 69 C 70 C 71 C 72 C 73 C 74 C 75 C 76 C 77 C 78 C 79 C 80 C 81 C 82 C 83 C 84 C 85 6 6 6 6 6 2. 500 1. 667 2. 500 3. 333 2. 167 2. 500 1. 833 2. 167 3. 000 1. 833 3. 333 2. 667 4. 333 3. 17 1. 049 0. 428 1. 378 0. 563 1. 225 0. 500 0. 816 0. 333 1. 378 0. 563 1. 506 0. 615 0. 983 0. 401 1. 378 0. 563 0. 983 0. 401 1. 602 0. 654 1. 897 0. 775 0. 753 0. 307 2. 160 0. 882 1. 633 0. 667 1. 211 0. 494 2. 71 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. 066, 5. 601) 0. 953, 4. 381) 3. 062, 5. 604) 0. 32, 6. 02)

C 86 C 87 C 88 C 89 C 90 C 91 C 92 C 93 C 94 C 95 C 96 C 97 C 98 C 99 C 100 C 101 C 102 6 6 6 6 6 2. 500 2. 333 3. 500 2. 500 1. 833 2. 333 3. 333 2. 667 1. 667 2. 833 2. 500 2. 667 2. 167 2. 833 2. 000 2. 167 1. 378 1. 506 1. 761 1. 643 0. 983 1. 211 0. 516 1. 506 1. 751 0. 516 0. 983 1. 378 1. 366 1. 169 0. 983 0. 000 1. 169 0. 563 ( 1. 053, 3. 947) 0. 615 ( 0. 753, 3. 913) 0. 719 ( 1. 652, 5. 348) 0. 671 ( 0. 775, 4. 225) 0. 401 ( 0. 801, 2. 865) 0. 494 ( 1. 062, 3. 604) 0. 211 ( 1. 791, 2. 875) 0. 615 ( 1. 753, 4. 913) 0. 715 ( 0. 829, 4. 505) 0. 211 ( 1. 125, 2. 209) 0. 401 ( 1. 801, 3. 865) 0. 563 ( 1. 053, 3. 947) 0. 558 ( 1. 233, 4. 101) 0. 477 ( 0. 940, 3. 394) 0. 401 ( 1. 801, 3. 865) 0. 000 ( 2. 00000, 2. 00000) 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.