UCI Introduction to Probability Statistics III Worksheet

STATS 120C: Introduction to Probability andStatistics III
Homework # 1
due Friday April 10, 2020 at 5:00PM
1. Remember the definition of t and F distribution (see Rice pages 192-194).
Show that if T ∼ tn , then T 2 ∼ F1,n .
2. Let X1 , X2 , . . . , Xm and Y1 ,Y2 , . . . ,Yn be independent random samples from
N(µ1 , σ21 ) and N(µ2 , σ22 ), respectively. We wish to test the null hypothesis
H0 : σ21 = σ22
vs.
H1 : σ21 6= σ22
(a) Derive the form of the generalized likelihood ratio Λ.
(b) Calling f =
2
∑m
i=1 (xi −x̄) /(m−1)
n (y −ȳ)2 /(n−1) ,
∑i=1 i
espress Λ as a function of f .
(c) Under the null hypothesis H0 , what distribution does
2
∑m
i=1 (Xi − X̄) /(m − 1)
∑ni=1 (Yi − Ȳ )2 /(n − 1)
follow?
3. Verify that the two-sample t test at level α of H0 : µX = µY versus H1 : µX 6=
µY rejects if and only if the confidence interval for µX − µY does not contain
zero.
4. Male fiddler crabs solicit attention from the opposite sex by standing in front
of their burrows and waving their claws at the females who walk by. If a
female likes what she seees, she pays the male a brief visit in his burrow.
If everything goes well and the crustacean chemistry clicks, she will stay
a little longer and mate. In what maybe a ploy to less the risk of spending
the night alone, some of the males build elaborate mud domes over their
burrows. Do the following data suggest that a male’s time spent waving
to a females is influenced by whether his burrow has a dome? Answer the
question by carrying out the appropriate hypothesis test (you can use R if
you want), and by constructing and interpreting a 95% confidence interval.
State what hypothesis test you are using, and, if using R, please include code
and output.
% of Time spent waving to females
Males with domes, xi Males without domes, yi
100.0
76.4
58.6
84.2
93.5
96.5
83.6
88.8
84.1
85.3
79.1
83.6