Question

Let X1,...,Xn be a sample drawn from a normal population with mean μ and standard deviation σ. Find E[X ̄S2].

Answer #1

A random sample is drawn from a population with mean μ = 72 and
standard deviation σ = 6.0. [You may find it useful to reference
the z table.]
a. Is the sampling distribution of the sample mean with n = 17
and n = 45 normally distributed?
Yes, both the sample means will have a normal distribution.
No, both the sample means will not have a normal
distribution.
No, only the sample mean with n = 17 will have...

Let X1, X2,...,Xn represent n random draws from a population
with standard deviation σ and variance σ^2 , so that V ar[X1] = V
ar[X2] = ... = V ar[Xn] = σ^ 2 . Define the sample average taken
from a sample of size n as follows: X¯ n ≡ (X1 + X2 + ... + Xn)/ n
.
a) Derive an expression for the standard deviation of X¯ n.
[Hint: Your answer should depend only on σ and n]...

A random sample is drawn from a population with mean μ
= 53 and standard deviation σ = 4.4. [You may find
it useful to reference the z table.]
a. Is the sampling distribution of the sample
mean with n = 13 and n = 38 normally
distributed?
Yes, both the sample means will have a normal distribution.
No, both the sample means will not have a normal
distribution.
No, only the sample mean with n = 13 will have...

A random sample X1, X2, . . . , Xn is drawn from a population
with pdf. f(x; β) = (3x^2)/(β^3) , 0 ≤ x ≤ β 0, otherwise
(a) [6] Find the pdf of Yn, the nth order statistic of the
sample.
(b) [4] Find E[Yn].
(c) [4] Find Var[Yn].
(d)[3] Find the mean squared error of Yn when Yn is used as a
point estimator for β
(e) [2] Find an unbiased estimator for β.

A random sample is drawn from a population with mean μ
= 52 and standard deviation σ = 4.3.
a. Is the sampling distribution of the sample
mean with n = 13 and n = 39 normally
distributed?
Yes, both the sample means will have a normal distribution.
No, both the sample means will not have a normal
distribution.
No, only the sample mean with n = 13 will have a normal
distribution.
No, only the sample mean with n...

A random sample is selected from a population with mean
μ = 102 and standard deviation σ = 10. Determine
the mean and standard deviation of the x sampling
distribution for each of the following sample sizes. (Round the
answers to three decimal places.)
(a) n = 12
μ =
σ =
(b) n = 13
μ =
σ =
(c) n = 37
μ =
σ =
(d) n = 70
μ =
σ =
(f) n = 140...

A sample of size 15will be drawn from a population with mean 135
and standard deviation 36
.
(a) Is it appropriate to use the normal distribution to find
probabilities for x?
(b) If appropriate find the probability that X will be
less than 118
.
(c) If appropriate find the
90th percentile of x
A sample of size 15will be drawn from a population with mean 135
and standard deviation 36
.
(a) Is it appropriate to use the normal...

Given a normal distribution with population standard deviation
of 2 and a mean of μ = 10. If a random sample of size 69
is drawn, find P(10 ≤ x ≤ 12).
Round to three decimal places.

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.

A simple random sample of size 11 is drawn from a normal
population whose standard deviation is σ=1.8. The sample mean is
¯x=26.8.
a.) Construct a 85% confidence level for μ. (Round answers to two
decimal place.)
margin of error:
lower limit:
upper limit:
b.) If the population were not normally distributed, what
conditions would need to be met? (Select all that apply.)
the population needs to be uniformly distributed
σ is unknown
simple random sample
large enough sample size...

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