For all parts requiring calculations, show how you got ...



For all parts requiring calculations, show how you got your answer and indicate clearly what your final answer is.

(1) (25 points)

The following are pulmonary compliance values for 8 random subjects.

180 208 190 228 259 232 206 200

a) Find the sample mean and sample standard deviation.

(You may use a calculator and just write your answer.)

b) What is the value of the standard error of the mean?

c) You are consulted in planning a study of compliances for normal subjects and for

subjects exposed to asbestos to determine if asbestos exposure decreases pulmonary

compliance. About how many subjects would you need in each group in order to

be 95% certain to detect a decrease of 5 in the average compliance if you are willing to

have only a 5% chance of claiming a difference between the groups when in fact

asbestos exposure has no effect on pulmonary compliance?

(1)

a) [pic]

b) SE = [pic]

c) For a 2-sided test of [pic]

SE([pic]) = [pic]

[pic]

(2) (20 points)

In a study of the effect of an oral antiplaque rinse, 14 subjects whose teeth were thoroughly cleaned and polished were randomly assigned to two groups of seven subjects each. Both groups were assigned to use oral rinses (no brushing) for a two week period. Group 1 used a rinse that contained an antiplaque agent. Group 2, the control group, received a similar rinse except that, unknown to the subjects, the rinse contained no antiplaque agent. After the two week period, the plaque index values for the two groups gave the following results.

| |n |Sample mean |Sample standard deviation |

|Control group |7 |1.26 |0.32 |

|Antiplaque group |7 |0.78 |0.40 |

Assume that both populations have the same population variances.

a) Test the null hypothesis that the two groups have the same population means against

the alternative that the groups are not the same. Give a p-value for the test. Is there a

significant effect at the 0.05 level? What do you conclude about the antiplaque agent

(at the α = 0.05 level)?

b) What does it mean to say we tested at the α = 0.05 level in this instance?

c) What is meant by the power of the test in this instance? What two things can the

experimenter do to have a more powerful test?

(2)

a) [pic]

0.02 < p < 0.05 Reject at 0.05 level

b) There is a 5% chance of claiming an effect when there is no difference.

c) The power is the probability of claiming an effect when there is in fact

an effect. To increase power, increase sample size or decrease σ.

(3)

To test a potential anticancer drug, 7 pairs of mice were used, with each pair of mice being from the same litter. For each pair of mice, one was assigned to the drug and the other mouse in that pair was left untreated (control). The resulting tumor weights (in grams) were as follows.

|Pair |1 |2 |3 |4 |5 |6 |7 |

|Control |1.3 |1.4 |2.7 |0.9 |1.2 |1.7 |3.2 |

|Drug |0.8 |0.9 |2.0 |0.8 |1.0 |1.1 |2.3 |

a) Test whether the drug affects tumor size using a two-sided test at the 0.05 level.

b) What is the advantage of running this experiment with pairs of mice rather than 14

independent mice? (No calculations required, just words.)

c) A new study is being planned with a similar drug, again using pairs of mice. The

experimenter wants to be 95% certain of claiming an effect of the drug if in fact the

drug reduces tumor weight by 0.2 grams, but also wants to be 95% certain of not claiming an effect if the drug in fact has no effect on tumor weight. How many pairs of mice are required?

(3)

a) t = 4.78 [pic]= 2.447 [pic]

b) With correlated values, we have a more powerful test and narrower confidence

intervals.

c) [pic]pairs

(4) (14points)

A test is being planned to see if a side effect of a drug is increased blood pressure.

a) What is meant by the power of this test? Explain in terms of this drug test. (Don’t

just give a generic definition of power.)

b) What can we do in planning the experiment to make our test powerful enough to be a

useful study for our purposes?

(4) drug → blood pressure

a) Power = probability of claiming drug has a harmful side effect when it does have a

harmful effect.

b) Have n large enough to have high power to detect a given increase in BP. Keep σ

small – Good experimental technique, pairing if possible.

(5) (20 points)

A study is done comparing FEV values (forced expiratory volume, a measure of lung function) for two groups of children:

|Group |Sample size |Sample mean |Sample standard deviation |

|1: Both parents smoke |23 |2.11 |0.71 |

|2: Neither parent smokes |20 |2.31 |0.41 |

Test [pic]against the alternative[pic]. Do not assume that both populations have the same variances. Give the p-value for the test (to a range determined from the appropriate table). What do you conclude?

(5)

[pic]

0.25 < p-value < 0.15 Don’t reject with α = 0.05

(6) (18 points)

An experiment is conducted to test the effect of 3 levels of a pesticide on alkaline phosphotase in fish. For each level of the pesticide, the experimenter sets up 5 tanks of fish with 10 fish in each tank. In all, there are 15 tanks and 150 fish. Each tank is kept (as nearly as possible) at the appropriate pesticide concentration by mixing the pesticide into the tank’s water. The alkaline phosphotase is measured in each of the 150 fish after a given exposure time.

What is the experimental unit in this design?

(6)

The tanks.

(7) (20 points)

Stomach concentrations of DDT in juvenile and adult meadow voles were compared in the paper “Species and Age Differences in Accumulation of C1-DDT by Voles and Shrews in the Field” (Environ. Pollution, 1984). Summary values for the meadow voles are given below.

| | | |Sample standard deviation |

|Group |n |Sample mean | |

|Adult |28 |5.6 |0.4 |

|Juvenile |22 |7.8 |0.9 |

Test the null hypothesis that both populations have the same mean DDT content against the alternative that they are different. Do not assume that the two populations have the same standard deviations. Give (as near as you can from the table) the p-value for the test. State whether to reject or not reject the null hypothesis at the 0.05 level. Briefly state what this means about these populations of meadow voles.

(7) Voles

[pic]

Will come back to the p-value Reject [pic] at the 0.05 level.

Rejecting the null hypothesis means that juvenile and adult voles do not have the same DDT concentration.

(8)

Suppose from past experience, it is known that 30 day old channel catfish raised in a laboratory have lengths with standard deviation about 2 mm. An experiment is being planned with n fish raised under control conditions and n more fish raised exposed to a particular level of a potentially damaging pesticide. If the researchers are willing to accept an α = 0.05 chance of claiming the pesticide has an effect when the pesticide in fact has no effect, how many fish need to be raised in each group to have a 99% chance of detecting a decrease of 0.5 mm in the average length of 30 day old catfish? Assume that the standard deviation of the lengths will still be about 2 mm if the pesticide reduces the average length by 0.5 mm.

(8) σ = 2 [pic]

α = 0.05

ß = 0.01

(9) (20 points)

Suppose that the body temperature of 5 randomly chosen people who have contracted a particular virus were (at the time of diagnosis)

99.2 98.9 99.6 99.9 100.4

a) Find the sample mean and sample standard deviation for these data. (You do not have

to show how you find them. You may use your calculator.)

b) Find the standard error of the mean.

c) Briefly describe what the meaning of the standard error is in this example.

d) Find a 95% confidence interval for the mean body temperature of all people who

contract the virus.

(9)

a) [pic]

b) SE = [pic]

c) Means of n = 5 values from this population will vary about the population mean with a

standard deviation of about 0.26.

d) [pic]

(10) (10 points)

Test the null hypothesis that the population average for the temperatures in problem (9) is 98.6 Give the p-value for your test (as close as you can tell from the table without interpolation). Do you reject the null hypothesis at the 0.05 level? What if [pic]?

What if [pic]?

(10)

[pic]

[pic]

[pic]

0.005 < p-value < 0.01 Reject [pic] at α = 0.05

If [pic] 0.99 < p-value < 0.995

If [pic] 0.01 < p-value < 0.02

(11) (20 points)

Suppose that heights of men in a certain population have a population average of 68 inches and a standard deviation of 2.5 inches.

a) What percent of the population is less than 65 inches tall?

b) What is the 10th percentile of the population?

c) Suppose that as a class project, each student randomly chooses 100 men from the

population and finds the average height of the 100 men. If the class draws a histogram

of the resulting averages, about what would you expect the mean and standard

deviation of the histogram to be?

d) If 100 men are chosen at random, what is the chance (probability) that their average

height will be 68.5 inches or less?

(11)

a) 11.5%

b) 64.8 inches

c) 0.25

d) 97.7%

(12)

The number of stomata per mm were counted on the inner faces and outer faces of five slash fine needles. Test whether there is a significant difference in the average densities of stomata on inner and outer faces of slash pine needles. State your null and alternative hypotheses. Give a p-value for your result. State whether there is a significant difference at the 0.05 level. State what your test shows concerning stomata on slash pine needles.

|Needle # |Inner face |Outer face |

|1 |7.52 |7.80 |

|2 |8.44 |8.57 |

|3 |7.29 |8.35 |

|4 |7.74 |7.97 |

|5 |8.56 |8.72 |

(12) Stomata paired t-test

[pic] [pic]

[pic]

0.05 < p-value < 0.10 Don’t reject [pic]

(13) (12 points)

A population has an average blood pressure of μ = 120 mm. The standard deviation is

σ = 12 mm. As a class project, each student in a class of 200 students sample 100 random people and find the sample average of the blood pressures of his or her 100 people. If the class draws a histogram of the resulting 200 averages, about what would you expect the average standard deviation of the histogram to be?

(13) blood pressures μ = 120 σ = 12

[pic]

(14) (12 points)

Suppose blood pressures of 20-year-olds have an average of 110 mm and a standard deviation of 8 mm, and 40-year-olds’ blood pressures have an average of 130 mm with a standard deviation of 10 mm. For another project, each student randomly selects 100 twenty-year-olds and 100 forty-year-olds and finds the resulting sample averages, [pic]and [pic] for the people in his or her sample. Each student then finds the difference between his or her sample averages, [pic]. If the class draws a histogram of the resulting differences between 40-year-old and 20-year-old averages, about what would you expect the average and standard deviation of this histogram to be?

(14)

[pic]

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