The two sample t-test is based on the t distribution and is exact when sigma_1 =

Question

The two sample t-test is based on the t distribution and is exact when sigma_1 = sigma_2. However, when sigma_1 notequalto sigma_2 the use of a t-distribution with upsilon degrees of freedom is an approximation. Here, upsilon = (s_1^2/n_1 + s_2^2/n^2)^2 upsilon =/(s_1^2/n_1)^2/n_1 - 1 + (s_2^2/n_2)^2/n_2 - 1) In this question, you are to carry out simulation experiments that explore the validity of this approximation. take n_1 = 10, n_2 = 10, and assume (under H_0) that X_i ~ N (0, 0.7^2) and Y_i ~ N(0, 1.3^2). (a) Calculate, upsilon = (sigma_1^2/n_1 + sigma_2^2/n_2)^2/(sigma_162/n_1)^2/n_1 - 1 + (sigma_2^2/n_2)^2/n_2 - 1 (b) Generate 10^6 replicates of the random variable T = X - Y/Squareroot s_1^2 + S_2^2/n_2 Note the each generation of T required 10 X's and 10 T's (c) Compare the 0.8, 0.9 and 0.95 quantiles of a redistribution with upsilon defers of freedom and the empirical quantiles obtained from the simulation. (d) Explain your results and comment about this simulation experiment. Are there other ways to carry it out?

Explanation / Answer

Ans:a) v=0.047524/0.00344022=13.81

b) Now generate the 10 X's and Y's

t=-1.343

df=13.8=14

p-value=0.200640

we have to compare it with 0.2,0.1 and 0.05 significance levels

p-value>=0.05, 0.1,0.2 so we can accept at 0.05,0.1 and 0.2 significance levels that x and y have equal means.

z x y 1 0.25 0.175 0.325 2 0.5 0.35 0.65 3 0.75 0.525 0.975 4 1 0.7 1.3 5 1.25 0.875 1.625 6 1.5 1.05 1.95 7 1.75 1.225 2.275 8 2 1.4 2.6 9 2.25 1.575 2.925 10 2.5 1.75 3.25 std dev 0.529839 0.983986 mean 0.9625 1.7875

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