Question

**a) What is the Central Limit Theorem? It is always true
that as the sample size, n, increases, the distribution of the
sample means will be approximately normally distributed.
Explain**

**b) If the underlying population of study is not normally
distributed, how large should the sample size be? What if the
population is normally distributed ?**

Answer #1

**(a)** The central limit theorem says that the
mean of samples are normally distributed no matter what the
distribution they are taken from.

Mathematically,

Let Y = X1 + X2 + X3 +.........

Define,

As n approaches infinity, Zn approaches the standard normal distribution.

By the definition, As n approaches infinity, Zn approaches the standard normal distribution. Hence it is always true (and in fact desirable) that as the sample size increases, the distribution of sample means will be approximately normally distributed.

**(b)** As a rough guideline, if the population is
not normally distributed then it is suggested to have a sample size
of at least 30 for a good approximation.

If the population is normally distributed then all the sample points would be on a normal distribution curve. So regardless of sample size, sample would be normally distributed.

Which one of the following statements is
true?
A. The Central Limit Theorem states that the sampling
distribution of the sample mean, y , is approximately
Normal for large n only if the distribution of the population is
normal.
B. The Central Limit Theorem states that the sampling
distribution of the sample mean, y , is approximately
Normal for small n only if the distribution of the population is
normal.
C. The Central Limit Theorem states that the sampling
distribution...

True or False.
The central limit theorem states that as the number of sample
size increases, the distribution of the sample means approximates
to a normal distribution.

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.

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.

Which of the following is an appropriate statement of the
central limit theorem? Select just one.
(1) The central limit theorem states that if you take a large
random sample from a population and the data in the population are
normally distributed, the data in your sample will be normally
distributed.
(2) The central limit theorem states that if you take a large
random sample from a population, the data in your sample will be
normally distributed.
(3) The...

The Central Limit Theorem suggests that as the sample size
increases the distribution of the sample averages approaches a
normal distribution, regardless of the nature of the distribution
of the variable itself.
true or false

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.

Using the central limit theorem, what is the distribution of
sample means when the population distribution is the
following?
Part (a)
rectangular
Part (b)
normally distributed
Part (c)
positively skewed
Part (d)
nonmodal
Part (e)
multimodal
Part (f)
negatively skewed

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.

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