Homework 2 Fall 2018 STAT 305B Due 9/13 (R) NOTE: All work associated with a given problem part MUST be placed DIRECTLY beneath that part.

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1 Homework 2 Fall 208 STAT 305B Due 9/3 (R) Name NOTE: All work associated with a given problem part MUST be placed DIRECTLY beneath that part PROBLEM (20pts) In a recent survey [ ] it is noted that 84% of voting conservatives disagree with the scientific community as to the causes of global warming Also, in [ ] it is noted that 38% of voters disagree with the scientific consensus Finally, among voters, 40% are conservatives Define the random variables: Y The act of recording whether the person is not [=0], or is [=] conservative The act of recording whether the person does not [Y=0], or does [Y=] disagree with the scientific consensus Clearly, the sample space for (, Y) is: S (, Y ) {(0,0), (,0), (0,), (,) } Denote Pr[ x Y y] as The above numbers can be interpreted as the following probabilities Specifically: Pr[ Y ] 084 ; Pr[ Y ] 0 38 ; Pr[ ] 0 40 (a)(0pts) Derive the numerical value for each of the four p( x, y) singleton set probabilities [Since the set includes 4 elements, each single element can be placed in a singleton set] Since you are to solve for 4 unknowns, you will need 4 equations One equation is easy The sum of the 4 probabilities must equal 0 To get the other 3 equations, I would recommend that you begin with sets, not probabilities For example, give the subset of S that corresponds to the event ( Y, ) [ ] ] S( Y, ) p( x, y) (b)(5pts) Among voters who disagree with the scientific consensus, find the percent who are conservative [This is a conditional statement Specifically, the condition is that we restrict our attention to voters who disagree Hence, we have, what I call, a restricted sample space] (c)(5pts) Among non-conservatives, find the percent who do not disagree with the scientific consensus

2 2 PROBLEM 2(30pts) The main journal bearing in a certain John-Deer tractor has a design life of 5,000 hours The claimed probability of failure within this life is 0002 For any randomly selected tractor, define the random variable, via the events: = the event that it will fail within this period, and = the event that it will not fail within this [ ] [ 0] period Let p Pr[ ] denote the probability of failure Clearly, the sample space for is (a)(5pts) Recall that if events A and B are mutually exclusive (ie A B Recall also that for any sample space, S, Pr( ) Use these two facts to explain why Explanation: S S {0,} AB A PR B Pr[ 0] p ), then Pr( ) Pr( ) ( ) (b)(5pts) Company records for the past year show that out of tractors that have hit the 5,000 hr mark, 3 of them had a bearing replacement prior to hitting this mark (i) Use this data to arrive at an estimate, call it of the parameter p Give your answer in decimal notation (ii) Then explain why you believe or do not believe that The Deere claimed specification is accurate Answer: p (c)(5pts) The estimator that gave the numerical result in (b) is Answer: p k k Give the sample space for p (d)(0pts) In this part you will use simulations to better understand its probability structure Each k ~ binomial(, Use the command binornd(,ptrue,,0^4) to run 0 4 simulations of From those simulations estimate the following: (i), (ii) p p, and (iii) compute a histogram-based plot of the probability mass function f p ( [NOTES: In this part, assume the true value for p is the claimed value In relation to (iii), use the hist command and a bin-center array 0:00:0 ] [See (c)] p (i) mu_phat = ; (ii)sigma_phat = Figure 2(d) Simulation-based plot of f p ( (e)(5pts) Use the information in your plot in (d) [or your numerical values for f p ( ] to arrive at the numerical value of Pr[ p 0003] Then, based on this value, discuss whether you might change your belief related to part (b) Discussion:

3 3 PROBLEM 3(25pts) The estimator in (c) of PROBLEM 2, p k, where { k } k ~ iid binomial(,, is one k of the most popular statistics It is the estimator of the population proportion In PROBLEM 2 we used simulations to arrive at its probability structure In this problem we will address its structure using theory We begin with the following FACT: For { k} ~ iid binomial(,, the random variable Y ~ binomial ( n, n k n k k (a)(4pts) For a true proportion p 0002 use the Matlab binopdf to obtain a plot of and a sample size ( y) f Y n, over the y-values 0::0 [See 3(a)] Figure 3(a) Binomial(n=,p=002) pdf (b)(3pts) You should observe that the only notable difference between Figure 2(d) and Figure 3(a) relates to the values on the horizontal axis The probability values should be visually identical Explain why they are identical Explanation: (c)(3pts) Use the information in your plot in (a) [or your numerical values for ( y) ] to arrive at the numerical value of Pr[ Y 3] Then compare this to the value for Pr[ p 0003] that you obtained in 2(e) f Y (d)(3pts) The random variable ~ binomial (, with sample space S {0, } is also called a Bernoulli ( random variable Recall the following: Definition: E [ g( )] g( x) f ( x) dx Use this definition to show that p for ~ Bernoulli( [NOTE: Since the sample space is discrete, the formal integral is simply a summation] S 2 (e)(3pts) Use the definition in (d) to show that p(

4 4 (f)(9pts) Recall that p k To compute the numerical values for p and p using the theoretical expressions k given in (d) and (e), utilize the following two facts for any two random variables and Y: 2 2 FACT : E( a by ) ae( ) be( Y) FACT 2: If and Y are independent, then Var( a by) a Var( ) b Var( Y) Compute these values for Finally, compare these results to your simulation-based results in 2(d) [Note that the given facts are for any two independent rvs But they extend directly to any n independent rvs] p 0002

5 5 PROBLEM 4(25pts) Consider various scenarios related to basketball tryouts Specifically, for any given person trying out, let the event the person makes any free throw attempt be with Pr[ ] p 0 8 Assume that successive attempts are independent and identically distributed (iid) [ ] (a)(5pts) Compute the probability that the shooter makes ten shots in a row (b)(5pts) Compute the probability that the shooter s first missed shot will occur on or before his/her 7 th attempt (c)(5pts) Suppose that the shooter is required to attempt n=5 shots Let Y denote the number of made shots Compute Pr[ Y 0] (d)(5pts) Let Y denote the number of attempts until the shooter makes a total of 0 shots Clearly, S Y { 0,, } Compute Pr[ Y 2] (e)(5pts $$$$ ) Let Y denote the number of shots taken on order to make 0 shots in a row Find Pr[ Y 20]

6 6 Appendix Matlab Code %PROGRAM NAME: hw2m (8/27/8) %PROBLEM 2 %Part(c): p=0002; %True value for p n=; %Sample size nsim=0^4; %Number of simulations x=************; %Data array phat=********; %nsim estimates of p mu_phat = *********** sigma_phat=********* figure(0) bvec=0:00:0; %Bin centers h=hist(phat,bvec); Pr_phat=************ stem(bvec,pr_phat) title('simulation-based pdf for phat for ptrue=0002 & n=') xlabel('p') grid %Part (e): %====================================== %PROBLEM 3 %Part (a): p=0002; n=; y=0:0; fy=***********; figure(30) stem(y,fy) title('binomial(n=,p=002) pdf') xlabel('y') grid %Part (c):