Question

Let ?̅ be the sample mean of a continuous random variable ? for a sample with n ≥ 30 observations. Explain in your words how to compute the probability that the sample mean, ?̅, falls between two real values, ? and ?, with ? < ?, i.e.

Prob(? ≤ ?̅≤ ?)

Specify carefully all the assumptions you make and describe all the steps implemented.

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Answer #1

Let x be a continuous random variable that has a normal
distribution with μ=85 and σ=12. Assuming n/N ≤ 0.05, find the
probability that the sample mean, x¯, for a random sample of
18taken from this population will be between 81.7 and 90.4.
Round your answer to four decimal places.

For a continuous random variable , you are given that
the mean is and the variance is .
Let
Given = 27, = 130 and =
592, use Chebyshev's inequality to compute a lower bound for the
following probability
Lower bound means that you need to find a
value such that
using Chebyshev's inequality.

Let x be a continuous random variable that is normally
distributed with mean 73 and std. deviation 14. Sketch & shade
the curves and find the probabilities that x assumes a value:
a. less than 51
b. greater than 72

Let a random variable X̄ represent the mean of a sample
consisting of 16 observations. The sample mean equals 56 and the
sample standard deviation equals 28.
I. Statistics Calculate the following:
1) Standard Error of the Mean = Answer
II. Probabilities
1) P(42 < X̄ < 56) = Answer %
2) P(X̄>=70) = Answer %
3) P(X̄<=70) = Answer %

1) let X be a continuous random
variable that has a normal distribution with a mean of 40 and a
standard deviation of 5. Find the probability that X
assumes a value:
a. between 32 and
35 b. between 41 and 50
c. greater than
43 d. less than 49

Let X be a continuous uniform (-2,5) random variable. Let W =
|X| Your goal is to find the pdf of W.
a)Begin by finding the sample space of W
b)Translate the following into a probability statement about X:
Fw(w) = P[W <= w] = ....
c) Consider different values of W the sample of W. Do you need
to break up the sample space into cases?
d)Find the cdf of W
e)Find the pdf of W

Let the random variable X follow a distribution with a mean of
μ and a standard deviation of σ. Let X1 be the mean of a sample of
n1 (n1=1) observations randomly chosen from
this population, and X2 be the mean of a sample of n2(
n2 =49) observations randomly chosen from the same
population. Which of the following statement is False? Evaluate the
following statement.
P(μ
- 0.2σ <X 1 < μ + 0.2σ) <
P(μ - 0.2σ <X...

For a continuous random variable XX, the population mean and
standard deviation are 118 and 11, respectively. A sample of 42
observations is randomly selected. The mean of the sampling
distribution of x¯ is:
118
1.18
1.7
11
For the standard normal distribution, the area between z=-1.58
and z=1.57is:
0.9418
0.9936
0.0014
0.8847
For the standard normal distribution, the area to the left of
z=-1.06 is:
0.0082
0.1446
0.8554
0.0764

Let T be a continuous random variable denoting the time (in
minutes) that a students waits for the bus to get to school in the
morning. Suppose T has the following probability density
function:
f ( t ) = 1/10 ( 1 − t/30 ) 2 , 0 ≤ t ≤ 30.
(a) Let X = T/30 . What distribution does X follow? Specify the
name of the distribution and its parameter values.
(b) What is the expected time a...

Let
X be a continuous random variable rv distributed via the pdf f(x)
=4e^(-4x) on the interval [0, infinity].
a) compute the cdf of X
b) compute E(X)
c) compute E(-2X)
d) compute E(X^2)

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