Question

Given a population with mean *μ=100* and variance
*σ**2**=81*, the Central Limit Theorem applies
when the sample size *n≥30*. A random sample of size
*n=30* is obtained.

- What are the mean, the variance, and the standard deviation of the sampling distribution for the sample mean?
- Describe the probability distribution of the sample mean and draw the graph of this probability distribution with its mean and standard deviation.
- What is the probability that
*x**<101.5*? - What is the probability that
*x**>102*? - What is the probability that
*98<**x**<101*?

SHOW WORK

Answer #1

Given a population with a mean of µ = 100 and a
variance σ2 = 12, assume the central limit
theorem applies when the sample size is n ≥ 25. A random
sample of size n = 52 is obtained. What is the probability
that 98.00 < x < 100.76?

Given a population with a mean of µ = 100 and a variance σ2 =
13, assume the central limit theorem applies when the sample size
is n ≥ 25. A random sample of size n = 28 is obtained. What is the
probability that 98.02 < x⎯⎯ < 99.08?

What is wrong with the following statement of the central limit
theorem?
Central Limit Theorem. If the random variables X1,
X2, X3, …, Xn are a random sample of size n from any distribution
with finite mean μ and variance σ2, then the distribution of will
be approximately normal, with a standard deviation of σ / √n.

Question Central Limit Theorem
a)According to the Central Limit Theorem, what
are the mean and standard deviation of the sampling distribution of
sample means?
b)A population has a mean ?=1800 and a standard
deviation ?=40. Find the mean and standard deviation of the
sampling distribution of sample means when the sample size
n=100.

According to the central limit theorem, if a sample of size 81
is drawn from a population with a variance of 16, the standard
deviation of the distribution of the sample means would equal
_______.
.98
.44
.68
.87
.75

Which of the following statements is not consistent with
the Central Limit Theorem?
1. The Central Limit Theorem applies to non-normal population
distributions.
2. The standard deviation of the sampling distribution will be
equal to the population standard deviation.
3. The sampling distribution will be approximately normal when
the sample size is sufficiently large.
4. The mean of the sampling distribution will be equal to the
population mean.

The Central Limit Theorem says that when sample size n is taken
from any population with mean μ and standard deviation σ when n is
large, which of the following statements are true?
The distribution of the sample mean is approximately
Normal.
The standard deviation is equal to that of the population.
The distribution of the population is exactly Normal.
The distribution is biased.

Apply the Central Limit Theorem for Sample
Means
A population of values has a normal distribution with μ=77 and
σ=9.2. You intend to draw a random sample of size n=30.
Find the probability that a sample of size n=30n=30 is randomly
selected with a mean less than 76.8.
P(M < 76.8) =
Enter your answers as numbers accurate to 4 decimal places.
Answers obtained using exact z-scores or
z-scores rounded to 3 decimal places are accepted.

The Central Limit Theorem allows us to make predictions about
where a sample mean will fall in a distribution of sample means.
One way it does this is by explaining (using a formula) how the
shape of the distribution will change depending on the sample size.
What part of the Central Limit Theorem tells us about the shape of
the distribution?
The part that explains that there is no standardized table you
can use to find probabilities once you use...

Given a population with a mean of µ = 230 and a standard
deviation σ = 35, assume the central limit theorem applies when the
sample size is n ≥ 25. A random sample of size n = 185 is obtained.
Calculate σx⎯⎯

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