STAT7120 Analysis of Scientific Data - Summer Semester, 2020 Week 3 Tutorial Part A A paper appearing in a 2015 issue of Biological Conservation used a randomised response technique to investigate rates of illegal fishing of red abalone in Northern California. The study involved interviews with people at various sites. Once verbal consent to participant was obtained, each respondent was given a coin and an envelope containing two cards. On one card was written the question In the past year have you ever taken abalone under the minimum legal-size limit? while on the other card was written the question Did you get heads on the coin toss? Without the interviewer watching, each respondent first tossed the coin, noting the outcome. They then chose a card at random and answered the question on it to the interviewer. Out of 279 respondents, 102 answered 'Yes'. a) Denoting the unknown proportion by p, draw a tree diagram showing the conditional probabilities of answering 'Yes' following this procedure. b) Based on your tree diagram in (a), what is the estimated proportion of people who have actually taken abalone under the minimum legal-size limit in the last year?
Part B An expensive piece of equipment in a laboratory is starting to show signs of age. Let X be the number of days in any week that the equipment is working and suppose that X has the following probability distribution: x 0 1 2 3 4 5 P(X = x) 0.01 0.09 0.25 0.34 0.24 0.07 a) What is the expected number of days in a week that the equipment is working? b) What is the standard deviation of the number of days in a week that the equipment is working? c) What is the expected value of the total number of days that the equipment is working over a 40-week period? d) Suppose the number of days in a week that the equipment is working is independent from week to week. What is the standard deviation of the total number of days that the equipment is working over a 40-week period?
Part C - Binomial Distribution a) A couple decides that they will have four children. Let X be the number of girls they will have. Assuming that the probability of a girl is 0.50, independent across births, what is the distribution of X? b) Let X be the number of towns in which it will rain tomorrow among five neighbouring towns. Is X a Binomial random variable? c) Suppose 10% of people are left-handed and let X be the number of left-handed people in sample of 20 individuals. What is the probability of at least one left-handed person in the sample? d) Suppose a drug has a 20% chance of making a person drowsy. Out of a sample of 80 people who each take the drug, what is the probability that no more than 10 of them experience drowsiness? e) In Week 4, we looked at the expected value and standard deviation of X,