Recall the problem where you Hipped a quarter and a nickel, and depending on the
ID: 3171193 • Letter: R
Question
Recall the problem where you Hipped a quarter and a nickel, and depending on the outcome of the coin tosses, you answered either question A or B. We had n = 127 students respond. Before I tell you how many yesses and how many noes, let us do the following. Let m be the number of students who would say "yes" to the cheating question if asked. We would like to believe that m = 0. Let us believe that m = 0 unless the data strongly suggests otherwise. (We are doing what is called a "hypothesis test, " and our "null hypothesis" is that m = 0. The alternative hypothesis is that m > 0. You will see more of this in 2028.) Assuming m = 0, what is the probability that a student says "yes" to that homework question? (Notice that this just depends on the two coin tossesi when m = 0.) Let X be the number of yesses among the n = 127 answers. Assuming m = 0, what is the distribution of X? Assuming m = 0, what is the mean of X? We will believe that m = 0 unless we get too many yesses, but what is too many? Write a program in Python 3 that generates a table with the following two columns. The first column gives k and the second column gives P{X greaterthanorequalto k}. Generate this table for k from 30 to 60. (When you write this program, please leave n and p as variables since we will be able to use this program in lots of different situations.) At what point would you reject the hypothesis that m = 0. After we collect the homework, I will let you know the value the number of yesses among the 127 answers so that you can decide whether it's unreasonable to believe that m = 0. (In hypothesis testing, we would say that we reject the null hypothesis.) If you like, we could also talk about how you might estimate the value of m.Explanation / Answer
(a) Assuming that m=0 means that no cheating question is asked and thus output (yes or no) depends on the outcome of coin being tossed. We can get head with probability =1/2 and similarly tail with probability = 1/2 . Thus student says "yes" with probability = 1/2.
(b) We have n= 127, where student would say "yes" with probability =1/2 and "no" with probability = 1/2 . Also, all students makes their decisions independently. All the conditions of binomial distribution distribution are satisfied as observations are independent and probabilty of success ( getting "yes" ) is constant, hence X follows binomial distribution with n=127 and p= .5
(c) we know that in binomial distribution, mean of the distribution is given by n.p
so E(X) = 127*.5 = 63.5
ANSWER
TY!
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