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2B03 Assignment 3 Sampling Distributions and Statistical Inference (Chapters 5, 6 & 7) Ryan Pollard 400341499 March 10 2022 Table of Contents Instructions You are to use R Markdown for generating your assignment output file. You begin with the R Markdown script downloaded from A2L, and need to pay attention to information provided via introductory material posted to A2L on working with R, R Markdown. Having added your answers to the R Markdown script, you then are to generate your output file using “Knit to PDF” and, when complete, upload both your R Markdown file and your PDF file to the appropriate folder on A2L. Questions Question 1 Define the following terms in a sentence (or short paragraph) and state a formula if appropriate (this question is worth 5 marks). i. Hypothesis test ii. Point estimate iii. Interval estimate iv. Sampling distribution v. Level of confidence Answer i.Hypothesis testing is a statistical procedure in which a choice is made between a null hypothesis and an alternative hypothesis based on information in a sample. i. Estimating a population parameter by a single number represented by a statistic based on the parameters of u. ii. An interval within which the value of a parameter of a population has a stated probability of occurring.

iii. Sampling distribution is a statistic computed from the sample that shows the distribution of different outcomes. iv. Level of confidence is a percentage showing how certain you can be of an outcome. Question 2 If the income in a community is normally distributed, with a mean of $38,000 and a standard deviation of $6,000, what maximum income does a member of the community have to earn in order to be in the bottom 5%? What is the maximum income one can have and still be in the middle 50% (this question is worth 4 marks)? Answer mu=38000 sd=6000 z<- qnorm(0.5) bottom= z*sd + mu show(bottom) Answer 28130.88 Question 3 Suppose that the number of hours per week of lost work due to illness in a certain automobile assembly plant is approximately normally distributed, with a mean of 40 hours and a standard deviation of 15 hours. For a given week, selected at random, what is the probability that (this question is worth 3 marks): i. The number of lost work hours will exceed 70 hours? ii. The number of lost work hours will be between 30 and 45 hours? iii. The number of lost work hours will be exactly 50 hours? Answer i. ##x follow normal(40,15) #p(x>70) 1-pnorm(70,40,15) [1] 0.02275013 ii. ##p(30<x<45) pnorm(45,40,15)-pnorm (30,40,15) [1] 0.3780661 iii. ##p(x=50) answer 0 as x continuous Question 4 A senator claims that 58% of her constituents favour her voting policies over the past year. In a random sample of 50 of these people, the sample proportion of those favoured her voting policies was only 0.4. Is this enough evidence to make the senator’s claims strongly suspect? (Hint: Use a normal approximation to the binomial distribution then construct a confidence interval - this question is worth 2 marks). Answer p =0.58 sample proportion = 0.4 n = 50 The standard error is given by SE = √ pq n SE = √ 0.4 ∗ 0.6 50 SE = 0.06928 The confidence interval is given by

CI = z_{0.05}SE(p) CI = 0.4 { CI = [0.2642,0.5358]} pnorm(0.4,0.58,0.0698) [1] 0.004957258 Question 5 I wish to estimate the proportion of defectives in a large production lot with plus or minus D = 0.02 of the true proportion, with a 99% level of confidence. From past experience it is believed that the true proportion of defectives is π = 0.02 . How large a sample must be used? (Hint: Use a normal approximation for the sample proportion ^ P - this question is worth 2 marks). Answer p = 0.02 E = 0.02 z <- qnorm(.995) n = (z/E)^2 p (1 - p) n= 326 Question 6 A cereal company checks the weight of its breakfast cereal by randomly checking 62 of the boxes. This particular brand is packed in 20-ounce boxes. Suppose that a particular random sample of 62 boxes results in a mean weight of 20.02 ounces. How often will the sample mean be this high, or higher if μ = 20 and σ = 0.10 (this question is worth 4 marks)? Answer x = 20.02 mu = 20 sigma = 0.10 n = 62 se = si gm a / s qr t ( n ) z = ( x−mu ) / se pnorm(z, lower.tail = FALSE) probability =0.005

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Grading instructions (These are the instructions the TA receives to grade the assignment (along with the correct answers). I am leaving them here for you (the student) to see, so that you have a good idea about how we are assigning scores to questions.) First, a note about formatting. Poorly formatted .pdf s are extremely difficult to grade. It is also very hard to grade fairly across submissions when some submissions are not properly formatted. For this reason, we have special rules in place for formatting. If the pdf file is not well-formatted, for example because: • the table of contents is incorrect • incorrect indentation, so that subquestions are not appropriately numbered • printing huge amounts of code or data to the screen unnecessarily • weird math/LaTeX stuff • code presented outside of code chunks • no title/author at top, i.e. incorrect YAML • … • (any other reasons: use common sense, and think of the .pdf as a written assignment, like a pdf/Word document you’d hand in for a writing course) then: • subtract 8 points (Assignment 3 only, penalty will continue to increase as we go) • write at the top of the feedback: “The formatting in this document is off. I am assigning a penalty of 8 points to draw your attention to it. The penalty will be much higher in future assignments. Please go and talk to Chris during OO or after class at your earliest convenience.” Some other meta-penalties, before we get to the specific answers for this assignment: • if there is code but no output: 0 points for that subquestion • if a subquestion requires the use of code, and an answer is provided without using code: 0 points • if the assignment is late, assign 0 points and tell them to see me during OO if they would like to discuss • if there is no .pdf , assign 0 points and tell them to see me during OO if they would like to discuss • same is there is no .Rmd • for incorrect answers, briefly describe what they got wrong. rather than providing the correct answer or a long explanation, it is more efficient and leads to better learning outcomes to provide a bit of feedback and then chat with the students who want to understand better.

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