GE_210_Final_Exam_2021

1 GE 210 — Probability and Statistics FINAL EXAM Date: Saturday, December 11, 2021 (9:00 am – 12:30 pm) Instructions: ▪ Total Marks: 105 (8 questions) (the 5 marks are a bonus) ▪ Time allotted: 3.5 hours (questions are designed for a maximum of 3 hours; the extra half-hour is offered so you can relax, scan and upload the files). ▪ To obtain full marks, you need to clearly write the steps of your work and highlight your answers. You also need to draw any diagrams neatly. Use a “reasonable” number of significant digits (with units, if appropriate). ▪ You need to submit three files: (1) an Academic integrity document, (2) a PDF with answers, and (3) an excel file for Questions 3, 6, and 8 only . Q1 (4 Marks: 4 + 4) Use the standard normal distribution tables from your textbook (also provided at the end of this document) to find: a. P(? ≤ 28.56) if ? follows the Lognormal distribution ℒ𝒩(μ = 6, σ = 0.8) b. Find ? such that P(? > ?) = 0.496 if ? follows the Lognormal distribution ℒ𝒩(μ = 5, σ = 0.5) Please show the calculation steps. Q2 (5 Marks: 2 + 2 + 2 + 4) Answer briefly and show calculations when needed: a. You double the value of the location parameter in a Normal distribution. How much the standard deviation will change? b. You decrease the value of the scale parameter in a distribution. How will the skewness be affected? c. You have a Normal distribution 𝒩(μ = 5, σ = 0.5) and an Exponential distribution ℰ(λ = 1/5) . Which one has a larger mean? d. You have a Gamma distribution 𝒢(? = 0.8, ? = 1.5) and an Exponential distribution ℰ(λ = 2) . Which one has a larger standard deviation? Q3 (7 Marks: 2 + 2 + 2 + 2 + 2 + 4) - Excel Use excel to estimate the probability of: a. P(? < 5) if ? follows a Weibull distribution 𝒲(β = 5, γ = 1.2) b. P(? > 4) if ? follows a Gamma distribution 𝒢(β = 2, γ = 1.5) c. P(2 < ? ≤ 5) if ? follows a Lognormal distribution ℒ𝒩(μ = 0.5, σ = 1.5) Find the ? value that corresponds to d. probability P(? < ?) = 0.95 if ? follows a Gamma distribution 𝒢(β = 2.5, γ = 5)

2 e. Probability P(? > ?) = 0.3 if ? follows an Exponential distribution ℰ(λ = 10) . f. The Probability Density Function (PDF) of the Weibull distribution 𝒲(β = 2, γ = 2) from 0 to 5 (x-axis will range from 0 to 5). Create a graph to show (Note: provide your answers in the empty sheet named Q3 in GE210_Final.xlsx). Q4 (6 Marks: 5 + 5) 75 water samples are collected to check for a particular pollutant. There is a probability of 0.25 that each sample contains the pollutant. Assume that these samples are independent, and ? is the number of samples that contain the pollutant. a. Estimate P(? < 3) using the Binomial distribution. The scientist that collects the water samples reports to you with an average of 4 emails per week. If ? denotes the number of emails per week b. what is the probability that you get more than 2 emails per week (use the Poisson distribution to estimate P(? > 2) )? Q5 (6 Marks: 5 + 5) The annual precipitation in a city has mean 𝜇 = 500 mm and standard deviation 𝜎 = 50 mm . a. Estimate the parameters of the Gamma distribution using the methods of moments. b. Estimate the parameters of the Lognormal distribution using the methods of moments. Please show the calculation steps. Q6 (7 Marks: 8 + 5) - Excel You are given a sample of 70 values (see sheet Q6 in excel file: GE210_Final.xlsx). a. Fit the Gamma distribution using the method of maximum likelihood and Excel’s Solver (the minimum and maximum values of the parameters to use as constraints in Solver are given in the excel - do not change them). b. Plot both the empirical probability and the fitted probability distribution (CDF) on the same plot. (Note: provide your answers in the sheet named Q6 in GE210_Final.xlsx). Q7 (10 Marks: 2 + 2 + 8 + 8) a. Define the P-value and describe when will the null hypothesis be rejected at a fixed significance level. b. How can the type I error be reduced? c. Old records in a town show that teenagers have an average weight of 60 kgs. A claim was made that teenagers now have a lower weight. A committee visited the town and weighed 30 teenagers, here is the sample: 73.2 64.4 59.1 58.5 69 48.7 46.7 45 57.7 70.9 58.7 73.3 65.3 63 48.2 62.1 49.3 68.3 64.6 46 65.2 35 56.4 48.5 55.8 65.4 56 58.3 63.8 47.1 Is there evidence to claim that teenagers have a lower weight than 60 kgs? Test at 0.05 significance level.