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. 23, 2011

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
• 0.1
• 0.5
• 0.6
• 0.7
• 1.0
• Use Binomial Coin Experiment to run 10 experiments each consisting ot tossing 20 coins (with p=0.7).
• 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(mean=3,000 mLi, variance=4002 mLi). Find P(E), if:
• n=20
• n=80 (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: {29  26   30   31   23   28}. For these data, the mean is 27.8 and the standard deviation is 2.9.
• 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.
• Use the SOCR Confidence Interval Analysis applet to compute the interval for the population mean difference between January and May (paste in all 4 columns in the data tab, choose mean unknown variance in teh CI Settings tab, map the 4th column, difference, as the data column in the Mapping tab, and finally click Calculate button). Compare these results to your manual calculations.

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