1) Consider a sequence of coin tosses, and let be the probability of heads at each toss.
b.
ML estimator of
be
For some fixed
, let
th head occurs. Find the
a.
For some fixed
, let
be the number of tosses until the
estimator of
based on .
the number of heads observed in tosses. Find the ML based on .
2) Let = {6.5, 8.8, 7.5, 9.2, 9.9, 12.4} be a random sample 0of.8size 6 from ( , 0.8), i.e. all data points are drawn from Normal distribution with variance .
a. Compute the maximum likelihood estimate for the parameter .
b. Generate %95 confidence interval for the estimated parameter in part (a).
c. At least how many samples more are needed in order to estimate our population mean with a distance of 0.5 with %99 confidence?
3) Let denote the true proportion of students who prefers C# programming language to C++. Let denote the number of students out of , who prefers C#.
b.
Let’s say
for which the probability is 80% that the difference
a.
Compute the smallest
between
and
is less than 0.02.
have a margin
of error equal to 0.06, how many more observations are
required to have a margin half the size?
value is
0: = 120
1
: ≠
120
=10, =16
be rejected?
is tested against
̅= 123.3
what P-
4) Suppose
where
0
associated with the sample mean
. Under what circumstances will
5) Let’s say we take a sample of size 1 from
( ) =
−
,
= 0,1,2…
and want to test:
!
0
: = 6 vs 1
: < 6
by rejecting 0 if ≤ 2.
a. Calculate the probability of committing a Type I error.
b. Calculate the probability of committing a Type II error.
6) Check the following measurements for a predictor ( ) and response variable ( ) :
1 12,6
2 11,6
3 6,8
4 9,2
Find the slope and intercept using least squares estimation.
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