Ivo Dinov
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STAT 13 (1a, 1b, 1c)

Introduction to Statistical Methods for the Life and Health Science

Department of Statistics

## Instructor: Ivo Dinov

Homework 2
Due Date: Monday, Oct. 15, 2007

Please, submit your homework right before lecture on the due date. See the HW submission rules. On the front page include the following header.
• (HW_2) Do these problems are variants of problems from pages 87-111, Samuels & Witmer's text Statistics for the Life Sciences, Prentice Hall (2003):
• 3.7: Suppose that a disease is inherited via a sex-linked mode of inheritance, so that a male offspring has a 50% chance of inheriting the disease whereas the female offspring have no chance of getting the disease. Assume that 51.3% of births are male.
• In  a family with one male and one female offsprings, what is the probability that exactly both sibling are affected?
• What is the probability that exactly one sibling is affected?
• What is the probability that neither sibling is affected?
• 3.8: Suppose a student who is about to take a multiple choice test has only learned 40% of the material covered by the exam. Thus, there is a 40% chance that she will know the answer to the question. However if she does not know the answer to a question, she still has a 20% chance of getting the right answer by guessing.
• If we choose a question at random from the exam, what is the probability that she will get it right?
• If we know that she correctly answered a question in the exam, what is the probability that she learned the material covered in the question?
• 3.11: Suppose that a medical test has a 92% chance of detecting a disease if the person has it (i.e. 92% sensitivity) and a 94% chance of correctly indicating that the disease is absent if the person really does not have the disease (i.e. 94% specificity). Suppose that 10% of the population has the disease.
• What is the probability that a randomly chosen person will test positive?
• Suppose that a randomly chosen person does test positive. What is the probability that the person does have the disease?
• 3.14: Suppose that in a certain population of married couples 30% of the husbands smoke, 20% of the wives smoke and in 8% of the couples both the husband and wife smoke. Is the smoking status of the husband independent of that of the wife? Why or why not?
• 3.18 & 3.20: In a certain population of the European starling, there are 5,000 nests with young. The distribution of brood size (number of young in a nest) is given in the accompanying table. Suppose one of the 5,000 broods is chosen at random and let Y be the size of the brood. Find
• P(Y=3)
• P(Y>=7)
• P(4<=Y<=6)
• Calculate the mean of the random variable Y.
 BroodSize BroodNumber 1 90 2 230 3 610 4 1400 5 1760 6 750 7 130 8 26 9 3 10 1 Total 5,000
• 3.21 & 3.22: Consider a population of the fruit fly Drosophila melanogaster in which 30% of the individuals are black because of a mutation, while 70% of the individuals have the normal gray color. Suppose three flies are chosen at random from the population; let Y denote the number of black flies out of the three. Then the probability distribution for Y is given by the following table:
• Find Pr {Y ≥ 2}.
• Find Pr {Y ≤ 2}.
• Calculate the mean of Y.

 Y (No. Black) Probability 0 0.343 1 0.441 2 0.189 3 0.027 Total 1.000
• 3.28: A certain drug treatment cures 90% of cases of hookworm in children.  Suppose that 20 children suffering from hookworm are to be treated, and that the children can be regarded as a random sample from the population.  Find the probability that:
• All 20 will be cured.
• All but 1 will be cured.
• Exactly 18 will be cured.
• Exactly 90% will be cured.
• 3.34: Childhood lead poisoning is a public health concern in the US. In a certain population, one child in eight has a high blood lead level (>30 µg/dLi). Compute the following probabilities for a randomly chosen group fo 16 children from this population:
• P(none have high blood lead)
• P(one has high blood lead)
• P(two have high blood lead)
• P(three or more have high blood lead)

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