Question

Let ? ̅ and ?2 be the mean and variance of a random sample of size 16 from a normal distribution N(4, 128). Find (a) ?(5 < ? ̅ < 8) (b) ?(200 < ?2 < 262.4)

Answer #1

a. If ? ̅1 is the mean of a random sample of size n from a
normal population with mean ? and variance ?1 2 and ? ̅2 is the
mean of a random sample of size n from a normal population with
mean ? and variance ?2 2, and the two samples are independent, show
that ?? ̅1 + (1 − ?)? ̅2 where 0 ≤ ? ≤ 1 is an unbiased estimator
of ?.
b. Find the value...

Let X1, X2 be a random sample of size 2 from the standard normal
distribution N (0, 1). find the distribution of {min(X1, X2)}^2

Let ??, ??, . . . , ?? form a random sample of size ? from some
probability distribution with mean ? ? and variance ?.If ?
isgreaterthan60, (a) What is the distribution of sample mean?
Explain.

Let X1, X2, . . . , Xn be a random sample of size n from a
distribution with variance σ^2. Let S^2 be the sample variance.
Show that E(S^2)=σ^2.

Let X1,...,Xn be a random sample from a normal distribution with
mean zero and variance σ^2. Construct a 95% lower conﬁdence limit
for σ^2. Your anwser may be left in terms of quantiles of some
particular distribution.

Let X be the mean of a random sample of size n from a N(θ, σ2)
distribution,
−∞ < θ < ∞, σ2 > 0. Assume that σ2 is known. Show that
X
2 − σ2
n is an
unbiased estimator of θ2 and find its efficiency.

Let x bar be the mean of a random sample of n = 36
currents (in milliamperes) in a strip of wire in which each
measurement has a mean of 16 and a variance of 6. X bar has an
approximate normal distribution, find P(12.5 < x bar < 15.6)
.

Let X1, X2, X3 be a random sample of size 3 from a distribution
that
is Normal with mean 9 and variance 4.
(a) Determine the probability that the maximum of X1; X2; X3
exceeds 12.
(b) Determine the probability that the median of X1; X2; X3 less
than
10.
(c) Determine the probability that the sample mean of X1; X2;
X3
less than 10. (Use R or other software to find the
probability.)

Let ?1 and ?2 be a random sample of size 2 from a normal
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size 0.005 for testing the null hypothesis ??: ? = 0 against the
composite alternative ?1: ? ≠ 0?

Let X1,...,Xn be a random sample from a normal
distribution where the variance is known and the mean is
unknown.
Find the minimum variance unbiased estimator of the
mean. Justify all your steps.

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