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In: Statistics and Probability

Concept Check: Test Statistics 1 point possible (graded) Setup: Recall the statistical experiment in which you...

Concept Check: Test Statistics

1 point possible (graded)

Setup:
Recall the statistical experiment in which you flip a coin n times to decide the coin is fair.

You model the coin flips as X1,…,Xn∼iidBer(p) where p is an unknown parameter, and formulate the hypothesis:

H0: p=0.5
H1: p≠0.5,

and design the test ψ using the statistic Tn:

ψn = 1(Tn>C)
where Tn = n−−√∣∣X¯¯¯¯n−0.5∣∣0.5(1−0.5)−−−−−−−−−−√

where the number C is the threshold. Note the absolute value in Tn for this two sided test.

Question:
If it is true that p=1/2, which of the following are true about Tn?
(Choose all that apply.)

Tn is a consistent estimator of the true parameter p=1/2.

limn→∞Tn−→−−n→∞(d)|Z| where Z∼N(0,1) is a standard Gaussian.

Tn involves a shift and rescaling of the sample average so that as n→∞, this random variable will converge in distribution.

The limiting distribution of Tn can be understood using computational software or tables.

unanswered

Solutions

Expert Solution

orchestra answered 1 year ago
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