Ivo Dinov
UCLA Statistics, Neurology, LONI

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STAT 13

Introduction to Statistical Methods for the Life and Health Science

Department of Statistics

## Instructor: Ivo Dinov

Homework 5
Due Date: Wednesday, Feb. 24, 2010

Please, submit your homework before lecture on the due date. See the HW submission rules. On the front page include the following header.
• (HW_5) Do the following problems:
• Suppose we are to draw a random sample of five individuals from a large population in which 40% of the individuals are mutants. Let p^ represent the proportion of mutants in the sample (sample proportion, an estimate of the population proportion of mutants).
• Use the Binomial Distribution to determine the probability that p^ will be equal to:
• 0
• 0.2
• 0.4
• 0.6
• 0.8
• 1.0
• Display a histogram of the sampling distribution of p^.
• Use Binomial Coin Experiment to run 10 experiments each consisting ot tossing 20 coins (with p=0.4).
• Compare the empirical probabilities (the third column in the results table) to the exact/theoretical probabilities (second column in the results table).
• How close are the values in columns 2 and 3?
• Are they supposed to be similar? Would they become more or less similar if we increase the number of trials to 100, for each experiment?
• Would the values in columns 2 and 3 become more or less similar if we run 100 cumulative experiments (each of 20 coins)?

• An important indicator of lung function is forced expiratory volume (FEV), which is the volume of air that a person can expire in one second. Dr. Jones plans to measure FEV in a random sample of n young women from a certain population and to use the sample-mean as an estimate of the population-mean. Let E={event that Dr. Jones sample-mean is within +/-100 mLi of the true (unknown) population-mean}. Assume that the population is Normal(3,000 mLi, 4002 mLi). Find P(E), if:
• n=15
• n=60 (quadruple expansion of the sample size)
• How does P(E) depend on the sample size?  Does P(E) increase, decrease or stay unchanged with increase of n?

• A certain cross between sweet-pea plants will produce progeny that are either purple flowered of white flowered; the probability of purple-flowered plant is p=9/16. Suppose n progeny are to be examined, and let p^ be the sample proportion of purple-flowered plants. It may happen by chance that p^ would be closer to 1/2 than to 9/16. Use Normal-approximation without continuity correction to find the probability that this misleading event will occur if
• n=1
• n=64
• n=320

• In a certain lab population of mice, the weights at 20 days of age follow approximately Normal distribution with mean weight=8.3g and standard deviation=1.7g. Suppose many litters of 10 mice each are to be weighted. If each litter can be regarded as a random sample from the population, what percentage of litters will have total weight of 90g or more? (Hint: How is the total weight of a litter related to the mean weight of its members?)
• Six healthy three-year-old female Suffolk sheep were injected with the antibiotic Gentamicin at a dosage of 10 mg/kg body weight. Their blood serum concentrations (μg/mLi) of Gentamicin 1.5 hours after injection were as follows: 33  26   34   31   23   25. For these data, the mean is 28.7 and the standard deviation is 4.6.
• Construct a 95% confidence interval for the population mean
• Define in words the population mean that you estimated above.
• Is it typical for the 95% confidence interval constructed in the first part to nearly contain all of the observations?
• Human beta-endorphin (HBE) is a hormone secreted by the pituitary gland under conditions of stress. A researcher conducted a study to investigate whether a program of regular exercise might affect the resting (unstressed) concentration of HBE in the blood. He measured blood HBE levels, in January and again in May, in ten participants in a physical fitness program. The results were as shown in the table.
 HBE Level (pg/mLi) Participant January May Difference 1 42 22 20 2 47 29 18 3 37 9 28 4 9 9 0 5 33 26 7 6 70 36 34 7 54 38 16 8 27 32 -5 9 41 33 8 10 18 14 4 Mean 37.8 24.8 13.0 SD 17.6 10.9 12.4
• Conduct a 95% confidence interval for the population mean difference in HBE levels between January (unstressed level) and May (HBE level possibly perturbed by exercise). (Hint: You need to use only the values in the right-hand column.)
• Interpret the confidence interval from the first part - that is explain what the confidence interval tells you about the HBE levels.

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