Question

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.

Answer #1

**Solution:**

Given: 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, then **the distribution of the sample mean is
approximately Normal.**

**thus first options is correct.**

In this case standard deviation of sample means = which is not equal to population standard deviation, thus second option is not correct.

Since population distribution is unknown, we can not say distribution is exactly Normal. Thus third option is not correct.

We either say distribution is normally or approximately Normal or Non-normal, so fourth option is not correct.

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.

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.

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.

Given a population with mean μ=100 and variance
σ2=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...

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

The Central Limit Theorem indicates that in selecting random
samples from a population, the sampling distribution of the the
sample mean x-bar can be approximated by a normal distribution as
the sample size becomes large.
Select one: True False

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.

Which of the following is NOT a conclusion of the Central Limit
Theorem? Choose the correct answer below.
A. The distribution of the sample means x overbar will, as the
sample size increases, approach a normal distribution.
B. The mean of all sample means is the population mean mu.
C. The distribution of the sample data will approach a normal
distribution as the sample size increases.
D. The standard deviation of all sample means is the population
standard deviation divided...

When σ is unknown and the sample is of size n ≥ 30, there are
two methods for computing confidence intervals for μ. Method 1: Use
the Student's t distribution with d.f. = n − 1. This is the method
used in the text. It is widely employed in statistical studies.
Also, most statistical software packages use this method. Method 2:
When n ≥ 30, use the sample standard deviation s as an estimate for
σ, and then use the...

According to the central limit theorem, a sample mean distribution
is aproximately a normal distribution , what are the mean and
standard deviation of this normal distribution ?

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