Question

Permutation Test Homework Work on this with at most one partner If you work with a partner, turn in one homework with both names listed You will find the handout "Permutation Test: An Example" helpful in working the below . . Observations: 6, -2, 4 (so n 3) Conduct a (two-sided) permutation test that the population is symmetric about zero; use a significance level of a 0.05: 1. How many ways are there to attach signs to the observations? 2. List all ways of attaching signs to the observations and give, in each case, the associated average. 3. List the averages in ascending order. 4. Determine the p-value. 5. What is your conclusion? Choose one: a. Do not reject the population is symmetric about zero b. Reject the population is symmetric about zero 6. (Bonus/Optional) What is the smallest the p-value can be in the case of n 3 observations? Explain.

Answer 1

Single Sample t-Test:

Ho : Population is symmetric about zero i.e. mean(μ)= 0

Ha : Population is not symmetric about zero i.e. mean(μ) != 0

Sample: 6, -2, 4 ; n = 3

x̅ = [ 6 + (-2) + 4 ] / 3

Sample mean (x̅) = 2.667

Sample Standard Deviation (s) = 4.163

t = (2.667 - 0) / (4.163/√3)

t = 1.109

df = n-1 = 2

p-value = P(t > 1.109) + P(t < -1.109) @ df = 2 and 95% confidence interval

The P-Value is 0.382925.
The result is not significant at p < 0.05

=> It null hypothesis(Ho) can not rejected, means Population is symmetric about zero

Permutation Test:

a. Permutation of sample : Problem is similar to tossing 3 coins simultaneously and outcomes are 8 (HHH,HHT,HTT.......TTT)

b. ample: (6, 2, 4), (6, 2, -4), (6, -2, 4), (-6, 2, 4), (6, -2,- 4), (-6, 2, -4), (-6, -2, 4), (-6, -2, -4)

Sample Average: 4, 1.333, 2.667, 0, 0, -2.667, -1.333, -4

c. -4, -2.667, -1.333, 0, 0, 1.333, 2.667, 4

d.

Ho : All means are equal i.e. mean(μ1= μ2 = μ3.......=μ8)= 0

Ha : At least two means are not equal

$ARLqZ7Bs9ei8AAAAAElFTkSuQmCCAA==$

$A1HbrhXVYPu5AAAAAElFTkSuQmCCAA==$

P-value = 0.386 > alpha(0.05)

Hence Ho is not rejected i.e Population means are equal