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(3) Suppose that 5 voters voted for candidate A and 15 for candidate B. Is it safe to conclude using the test you derived in part (2) at level 10% that candidate A will loose the election? (4) Suppose that 15 voters voted for candidate A and 5 for candidate B. Is it safe to conclude using the test you derived in part (2) at level 10% that candidate A will

(3) Suppose that 5 voters voted for candidate A and 15 for candidate B. Is it safe to conclude using the test you derived in part (2) at level 10% that candidate A will loose the election? (4) Suppose that 15 voters voted for candidate A and 5 for candidate B. Is it safe to conclude using the test you derived in part (2) at level 10% that candidate A will

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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pbinom (2,size=10,prob=D0.3) ## [1] 0.3827828
Transcribed Image Text:pbinom (2,size=10,prob=D0.3) ## [1] 0.3827828
Consider a random sample X1,...,Xn of voters who either voted for Candidate A (“X; = 1") or for Candidate B (“X; = 0"). Denote the probability of voting for Candidate A by p. We wish to test Ho : p 2 0.5 vs. Hị : p< 0.5. (i.e. the hypothesis that Candidate A will not loose the election). (1) Consider a test of the above hypothesis of the form i=1 for some integer c. Calculate the power of this test and show that it is a decreasing function of p for any c e {0, ...,n – 1}. (2) Suppose that n = 20. Find c so that ý in (1) is a test at level 10% with the largest possible power (from among the tests of the same form at level 10%). What is the size of your test? Use that the CDF of the binomial distribution with size 20 and success probability 0.5 evaluated at z = 0,..., 20 equals (rounded to 3 decimal places) round (pbinom(0:20, size=20,p=0.5),3) ## [1] 0.000 0.000 0.000 0.001 0.006 0.021 0.058 0.132 0.252 0.412 0.588 0.748 ## [13] 0.868 0.942 0.979 0.994 0.999 1.000 1.000 1.000 1.000 (3) Suppose that 5 voters voted for candidate A and 15 for candidate B. Is it safe to conclude using the test you derived in part (2) at level 10% that candidate A will loose the election? (4) Suppose that 15 voters voted for candidate A and 5 for candidate B. Is it safe to conclude using the test you derived in part (2) at level 10% that candidate A will not loose the election? For the test that you derived in part (2) (again with (5) n = 20), calculate the propăbility of type II error when p = 0.4. You are allowed and encouraged to use software to calculate binomial probabilities. In R, the CDF F(x) of the binomial distribution with size n and success probability p can be calculated as follows: For example, for x = 2, n = 10 and p = 0.3 we get
Transcribed Image Text:Consider a random sample X1,...,Xn of voters who either voted for Candidate A (“X; = 1") or for Candidate B (“X; = 0"). Denote the probability of voting for Candidate A by p. We wish to test Ho : p 2 0.5 vs. Hị : p< 0.5. (i.e. the hypothesis that Candidate A will not loose the election). (1) Consider a test of the above hypothesis of the form i=1 for some integer c. Calculate the power of this test and show that it is a decreasing function of p for any c e {0, ...,n – 1}. (2) Suppose that n = 20. Find c so that ý in (1) is a test at level 10% with the largest possible power (from among the tests of the same form at level 10%). What is the size of your test? Use that the CDF of the binomial distribution with size 20 and success probability 0.5 evaluated at z = 0,..., 20 equals (rounded to 3 decimal places) round (pbinom(0:20, size=20,p=0.5),3) ## [1] 0.000 0.000 0.000 0.001 0.006 0.021 0.058 0.132 0.252 0.412 0.588 0.748 ## [13] 0.868 0.942 0.979 0.994 0.999 1.000 1.000 1.000 1.000 (3) Suppose that 5 voters voted for candidate A and 15 for candidate B. Is it safe to conclude using the test you derived in part (2) at level 10% that candidate A will loose the election? (4) Suppose that 15 voters voted for candidate A and 5 for candidate B. Is it safe to conclude using the test you derived in part (2) at level 10% that candidate A will not loose the election? For the test that you derived in part (2) (again with (5) n = 20), calculate the propăbility of type II error when p = 0.4. You are allowed and encouraged to use software to calculate binomial probabilities. In R, the CDF F(x) of the binomial distribution with size n and success probability p can be calculated as follows: For example, for x = 2, n = 10 and p = 0.3 we get
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