Assignment03

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(Revision 10) Assignment 3 Overview Total marks: / 75 This assignment covers content from Unit 3 of the course. It assesses your knowledge of random variables, types of random variables and various types of probability distributions, along with their means and standard deviations. Instructions Show all your work and justify all of your answers and conclusions, except for the True/False questions. Keep your work to 4 decimals, unless otherwise stated. (4 marks) 1. Circle True (T) or False (F) for each of the following statements: T (F) The following table, which lists values of x and their probabilities, represents a valid probability distribution: x P(x ) 3 0.32 4 0.54 5 0.24 T (F) The following table, which lists values of x and their probabilities, represents a valid probability distribution: x P(x ) 0 .09 1 0.28 2 0.42 3 0.39 (T) F The speed of a car travelling on the Queen Elizabeth Highway is an example of a continuous variable. T (F) The binomial distribution can be used only when the probabilities of the two possible outcomes are equal. Mathematics 215: Introduction to Statistics Assignment 3 1
(Revision 10) (16 marks) 2. The following table lists the frequency distribution of the number of vehicles owned per household from a sample of 200 households: x 0 1 2 3 4 5 f 33 106 45 10 4 2 (4 marks) a. Construct a probability distribution table for the number of vehicles owned per household. Number of Vehicles Owned (per household) 0 1 2 3 4 5 Probability 33/200 = 0.165 106/200 = 0.53 45/200 = 0.225 10/200 = 0.05 4/200 = 0.02 2/200 = 0.01 (2 marks) b. Calculate the mean of this probability distribution. Hint: Consider adding the appropriate column to the table created in part ( a ). (0)(0.165) + (1)(0.53) + (2)(0.225) + (3)(0.05) + (4)(0.02) + (5)(0.01) = 1.26 (4 marks) c. Calculate the standard deviation of this probability distribution. Hint: Consider adding the appropriate columns to the table created in part ( a ). (0 2 )(0.165) + (1 2 )(0.53) + (2 2 )(0.225) + (3 2 )(0.05) + (4 2 )(0.02) + (5 2 )(0.01) = 2.45 x 2 ¿ ( p ( x ) ) μ 2 ¿ ¿ σ = ¿ = 2.45 1.26 2 = 0.8624 ≈ 0.9287 (4 marks) d. Give a brief interpretation (one or two sentences each) of the values of the mean and the standard deviation. The mean of the probability distribution tells us what the expected value of a discrete random variable, while the standard deviation of the probability distribution helps us to measure the variability of the possible outcomes. The mean also tells us that the average amount of vehicles per household is 1.26 Mathematics 215: Introduction to Statistics Assignment 3 2
(Revision 10) (2 marks) e. What is the probability that a household selected at random will have at least two vehicles? P(x ≥ 2) = 1 - P(x < 2) = 1 - P(x = 0) - P(x = 1) = 1 - 0.165 - 0.53 = 0.305 (13 total marks) 3. When transferring a goldfish to a new water source, such as a different fish tank, there is an 8% chance that the goldfish will die within the first week. If we select at random 5 goldfish that have been transferred to a new water source, what is the probability: (3 marks) a. that exactly one of them will die within the first week? P(x =1) = 0.08% = 0.2866 (6 marks) b. that fewer than three of them will die within the first week? P(<3) = 0.659081523 + 0.286557184 + 0.049836032 = 0.9962 (2 marks) c. that at least one of them will die within the first week? P(at least one of them will die within the first week) = P(x ≥ 1) = 1 - P(x = 0) = 1 - 0.66 = 0.3409 (2 marks) d. Circle True (T) or False (F) for each of the following statements: (T) FIf we randomly select 6 goldfish that have been transferred instead of 5, the experiment continues to satisfy the conditions for a binomial experiment. T (F) John transfers each goldfish to the same bowl. In this case, the chance that a goldfish will die goes up by 1% for each additional goldfish that is selected. This new experiment continues to satisfy the conditions for a binomial experiment. (11 total marks) 4. Thirty percent of students graduate from high school before they reach the age of 18. In a random sample of 16 high-school graduates, what is the probability that: [ Hint: Use binomial table.] (3 marks) a. more than 10 of them graduated before they were 18 years old? Mathematics 215: Introduction to Statistics Assignment 3 3
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