**HW #6 – Introduction to Confidence Intervals**

1. (IN CLASS) Most people think that the average body temperature in adult humans is 98.6. However, this figure is based on data from the 1800’s. In a 1992 article in the *Journal of the American Medical Association, *it is reported a more accurate figure is 98.2. Assume a normal model is appropriate and that the standard deviation is 0.7. Assume the standard deviation is from the population.

A) Assume the 98.2 was obtained from a sample of size 258. Give a 95% confidence interval for the mean body temperature for all adult humans.

B) How large a sample is needed to ensure that a 95% confidence interval will have a margin of error of only 0.05 degrees?

2. (ANSWER GIVEN)Suppose the measurements on the stress needed to break a type of bolt follow a Normal distribution with a mean of 75 kilopounds per square inch(ksi) and a standard deviation of 8.3 ksi. Assume the standard deviation is from the population.

A) Assume the estimate of the mean of 75 came from a sample of size 410. Give a 90% confidence interval for the mean of all such bolts.

B) How large a sample is needed to ensure that a 90% confidence interval will have a margin of error of only 0.5 ksi?

3. (SOLUTION GIVEN) Assume the cholesterol levels of adult American women can be described by a Normal model with a mean of 188 mg/dL and standard deviation of 24. Assume the standard deviation is from the population.

A) Assume this mean of 188 came from a sample of size 508. Give a 99% confidence interval for the mean cholesterol level of all adult American women.

B) How large a sample is needed to ensure that a 99% confidence interval will have a margin of error of only 1 mg/dL?

4. (HOMEWORK) Biological measurements on the same species often follow a Normal distribution quite closely. The weights of seeds of a variety of winged bean are approximately Normal with a mean of 525 mg and a standard deviation of 110 mg. Assume the standard deviation is from the population.

A) Assume that this mean of 525 came from a sample of size 120. Give a 95% confidence interval for the mean weights of all seeds.

B) How large a sample is needed to ensure that a 95% confidence interval will have a margin of error of 10 mg?

5. (ALTERNATE HW) The heights of women aged 20-29 follow approximately a Normal distribution with a mean of 64 inches and a standard deviation of 2.7 inches. Assume the standard deviation is from the population.

A) Assume the mean of 64 inches came from a sample of size 911. Give a 95% confidence interval for the mean height of all such women.

B) How large a sample is needed to ensure that a 95% confidence interval will have margin of error of 0.5 inches?

6. (IN CLASS)Given are the differences in the times a person could run 1 mile before and after an intense fitness class. Times are in seconds.

Person / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10After / 580 / 611 / 542 / 570 / 542 / 540 / 490 / 490 / 488 / 490

Before / 630 / 660 / 560 / 542 / 580 / 585 / 500 / 522 / 533 / 544

Person / 11 / 12 / 13 / 14 / 15 / 16 / 17 / 18 / 19 / 20

After / 600 / 465 / 455 / 710 / 600 / 510 / 510 / 480 / 480 / 489

Before / 520 / 470 / 460 / 700 / 820 / 600 / 610 / 500 / 544 / 566

Assume the standard deviation of the improvements in the population is 70 seconds.

A) Give a 95% CI for the mean improvement in times (before-after) for all possible people.

B) How large a sample is needed so that the margin of error of a 95% CI will be 15 seconds?

7. (ANSWER GIVEN) The weights of 21 randomly selected cans of peaches are weighed on two scales.

Can / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10 / 11Scale A / 11.83 / 12.46 / 11.87 / 12.99 / 12.33 / 13.30 / 12.73 / 11.55 / 13.31 / 12.26 / 12.13

Scale B / 11.71 / 12.44 / 11.91 / 12.58 / 11.88 / 13.49 / 13.11 / 11.02 / 12.99 / 11.58 / 12.07

Can / 12 / 13 / 14 / 15 / 16 / 17 / 18 / 19 / 20 / 21

Scale A / 12.41 / 12.51 / 12.14 / 12.17 / 12.80 / 12.27 / 11.57 / 12.57 / 11.59 / 11.64

Scale B / 12.78 / 12.38 / 11.68 / 11.95 / 12.81 / 12.38 / 11.36 / 11.48 / 11.50 / 11.45

Assume the standard deviation of the differences in the population is .15.

A) Give a 95% CI for the mean A-B for the population.

B) How large a sample is needed so that the margin of error of a 95% CI is .05?

8. (SOLUTION GIVEN) Given are the number of sit-ups people could do in 5 minutes before and after an intense fitness class designed especially abs.

Person / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10 / 11After / 164 / 142 / 154 / 143 / 157 / 147 / 174 / 174 / 163 / 165 / 168

Before / 150 / 150 / 150 / 94 / 95 / 156 / 160 / 180 / 177 / 99 / 86

Person / 12 / 13 / 14 / 15 / 16 / 17 / 18 / 19 / 20 / 21

After / 192 / 182 / 195 / 176 / 193 / 184 / 152 / 142 / 190 / 183

Before / 162 / 162 / 133 / 165 / 165 / 165 / 166 / 166 / 172 / 172

Person / 22 / 23 / 24 / 25 / 26 / 27 / 28 / 29 / 30 / 31

After / 175 / 146 / 161 / 182 / 183 / 178 / 176 / 192 / 192 / 173

Before / 153 / 166 / 144 / 153 / 144 / 99 / 80 / 138 / 130 / 111

Assume the standard deviation of after – before in the population is 40.

A) Give a 95% CI for the mean after – before for the population.

B) How large a sample is needed so that the margin of error of a 95% CI is 3?

9. (HOMEWORK) The table gives data on the absorption into the blood taken on 20 healthy female subjects for a pair of drugs, one generic and the other the reference name brand drug. Half were picked at random and received the generic drug first and the rest took the reference drug first. In all cases, a washout period separated the two drugs so that the first had disappeared before the subject took the second.

Subject / A / B / C / D / E / F / G / H / I / JReference / 4110 / 2536 / 2769 / 3853 / 1832 / 2436 / 1999 / 1719 / 1829 / 2594

Generic / 1755 / 1148 / 1603 / 2254 / 1309 / 2120 / 1851 / 1878 / 1685 / 2643

Subject / K / L / M / N / O / P / Q / R / S / T

Reference / 2354 / 1864 / 1022 / 2256 / 938 / 1339 / 1262 / 1438 / 1735 / 920

Generic / 2738 / 2202 / 1254 / 3051 / 1287 / 1930 / 1964 / 2549 / 3335 / 3044

Assume the standard deviation of the differences in the population is 1000.

A) Give a 95% CI for the mean difference (Ref – Gen) for all people.

B) How large a sample is needed to ensure the margin of error of a 95% CI for the difference is 50?

10.(ALTERNATE HW) The table gives the high temperature on a random sample of 17 days at the downtown and airport locations in a big city. Assume a normal population and a standard deviation of the differences in the population of .35 degrees.

Day / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10 / 11 / 12 / 13 / 14 / 15 / 16 / 17Downtown / 72 / 74 / 61 / 90 / 88 / 46 / 52 / 60 / 70 / 44 / 32 / 60 / 60 / 45 / 93 / 97 / 80

Airport / 75 / 73 / 61 / 94 / 93 / 45 / 52 / 60 / 68 / 51 / 35 / 58 / 59 / 49 / 93 / 96 / 84

A) Give a 95% CI for the mean difference (downtown – airport) for all days.

B) How large a sample is needed to ensure the margin of error of a 95% CI for the difference is .05 degrees?

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