Consider a random sample X1,.., X, 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 : p2 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 v(X1,..., Xn) = 1 (Ex;< for some integer c. Calculate the power of this test and show that it is a decreasing function of p for any c € {0, .,n – 1}. (2) Suppose that n = 20. Find e 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 r = 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?
Consider a random sample X1,.., X, 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 : p2 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 v(X1,..., Xn) = 1 (Ex;< for some integer c. Calculate the power of this test and show that it is a decreasing function of p for any c € {0, .,n – 1}. (2) Suppose that n = 20. Find e 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 r = 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?
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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Question
![Consider a random sample X1,...,X, 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
Họ :p2 0.5 vs. H1 : 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 v 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 r = 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?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe5046824-623a-41fb-a6e0-643b469382f0%2F5f508580-e451-4fdf-a991-22e92302a596%2F197lv5_processed.png&w=3840&q=75)
Transcribed Image Text:Consider a random sample X1,...,X, 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
Họ :p2 0.5 vs. H1 : 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 v 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 r = 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?
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