Answers for the questions and analysis
School of Mathematics and Statistics Te Kura Mātai Tatauranga MATH 177 Assignment 3 Due: 11:59pm, 25 August 2022 Note: Your assignment can be typed or handwritten and then scanned. Be sure to submit your assignment as a PDF file and follow the submission instructions specified on the course Blackboard page. 1 (a) For the situations described below state, after consideration, whether X is a binomial random variable or not. Justify your answer in each case. (i) A survey is conducted to determine the proportion of New Zealand voters who cur- rently intend to vote Labour at the next election. A sample of 1,000 names are drawn at random from the NZ electoral roll. Each individual is interviewed and the number X that intend to vote Labour is recorded. (ii) A biologist randomly selects 100 portions of water from the local reservoir, each equal to 10 ml in volume, and counts the number of bacteria present in each portion. The numbers of bacteria for the 100 portions are then summed, to obtain an estimate X of the number of bacteria per litre present in the reservoir water. (b) A fair coin is tossed 15 times. Assuming that tosses are independent, either calculate or determine from tables, the following quantities. (i) the probability that 7 of the coins land heads; (ii) the probability that at least 4 of the coins land heads; (iii) the mean and standard deviation of the number of coins that land heads; (iv) the mean and standard deviation of the proportion of coins that land heads. 2 The behaviour of a certain stock market index is considered over the next (n+1) consecutive trading days starting from tomorrow. For each day a ‘+’ will be recorded if the index rises above the previous day’s index and a ‘−’ will be recorded otherwise (you may assume that the probability that the index will be the same on two consecutive days is zero). Assume that rises and falls are independent and equally likely and that today’s index exceeds yesterday’s index. (a) Let N be the number of days of the next n for which a ‘+’ will be recorded. (i) Is N a binomial random variable? Justify your answer. (ii) What are the mean and standard deviation of N? (iii) Determine the probability that rises outnumber falls over the next 5 days. (b) The index is said to have a trend on day i if +,+,+ or −,−,− is recorded on days i− 1, i, i+1 respectively. Given n ≥ 2, let Z be the total number of days of the next n for which a trend will be recorded. (i) Is Z a binomial random variable? Justify your answer. (ii) Show that E(Z) = n4 [Hint: let Xi = 1 if there is a trend on day i, Xi = 0 otherwise and note that Z = X1 +X2 + · · ·+Xn] MATH 177, 2022 1 Assignment 3 3 A multiple choice test paper consists of 20 questions. In each question the candidate has to choose the correct answer from a list of four alternative answers. Suppose the candidate answers each question by selecting an alternative at random. (a) Give an expression for the probability function of X, the number of correctly answered questions. What are the mean and standard deviation of X? (b) If each correct answer is awarded 4 marks, and each incorrect answer carries a penalty of 1 mark, determine the mean and standard deviation of the total mark obtained by the candidate. What is the probability that the total mark is more than zero? (c) Suppose now the test paper contains n questions. Consider the probability that the can- didate answers at least one question incorrectly. What is the least number of questions that the test paper would have to contain in order for this probability to exceed 99%? 4 A survey was carried out concerning the top 7 attractions in Sydney. A sample of 330 visitors were asked how many of the 7 attractions they had visited. The results were as follows: Number of Attractions 0 1 2 3 4 5 6 7 Number of Visitors who visited that many of 17 25 150 83 39 11 3 2 the Attractions (a) Test the hypothesis that the number of attractions visited is binomially distributed with n = 7 and some unknown parameter p, which you should estimate first (i.e. do a goodness- of-fit test). (b) Is it likely that the assumptions of the binomial distribution will be valid in this example? Give reasons for your answer. 5 The number of species of plants per square metre in a certain region of bush is 3.8 on average. (a) List the characteristics of this description that would lead you to model the number of species with a Poisson distribution. (b) What is the standard deviation of the number of species per square metre in that region? (c) Find the probability of finding no plants (of any type) in a square metre. (d) Find the probability of finding 10 or more species in a square metre. 6 (a) The number of reported traffic accidents at a dangerous crossroads each month is believed to follow a Poisson distribution, Poi(µ). Six accidents occur during a particular month. Estimate µ, giving approximate error bounds (± 2 × standard error). (b) Suppose that the actual value of µ is 4 accidents/month. Find (i) the probability of obtaining a value as large as, or larger than 6 on any given month. (ii) the mean and variance of the total number of accidents likely to be reported during a full year. MATH 177, 2022 2 Assignment 3 7 A gambler pays $10 to enter a game in which two coins are tossed. The gambler receives $30, $5 or nothing if two heads, one head or no heads appear, respectively. (a) Determine the gambler’s expected winnings per game (i.e. the expected return less the cost of playing). (b) The gambler decides to play until winning the $30 prize. Letting X denote the number of games played until two heads appear (the final game is the Xth game), find the probability function f(x) of X, show that its sum over all possible values of X is one, and find the expected value µ of X. (Note: this is not the expected return on this version of the game. Note too that X here is not quite the same as a geometric random variable, as it was defined in the lecture notes; however, thinking about such rvs will be helpful here). 8 Spectacled weavers are birds that use grass to weave nests that look like hanging baskets. The male bird weaves a nest and then invites the female to inspect the nest. If no female accepts the nest, the male will build another nest; the male continues to build new nests until a female accepts. 200 male birds were observed, and X, the number of nests that were rejected by the females before a nest was accepted, was recorded for each male bird. The results are given in the first two columns of Table 1. Table 1: Number of rejected nests of 200 male weaver birds No. rejected (x) No. male birds (fi) pi ei 0 126 0.625 1 46 0.234 2 13 0.0879 3 12 0.03296 4 3 0.02014 Total 200 (a) Write down a possible distribution for these data, giving reasons for your choice. (b) Calculate the sample mean and standard deviation for these data. Use your answers, as appropriate, to estimate the parameter(s) for the distribution. (c) Write down the formula to calculate P (X = 2) = p2 = 0.0879 (4 dp). Show clearly how to use the answer to (b) to calculate the value given here. (d) Use the pi-values given in the table {which were obtained as in part (c), other than p4 = 1 − P (X ≤ 3)} to calculate the corresponding expected values (ei). Then do a goodness-of-fit test on these data, using a 5% level of significance. MATH 177, 2022 3 Assignment 3