Question

**Explain fully how the average, standard deviation, and
distribution of means of samples changes as the sample size
increases.**

Answer #1

Using the central limit theorem, the distribution for the sample means from any population for sample size n > 30 is given as:

**Therefore the mean remains the same as the sample size
increases.**

The **standard deviation of the sample mean decreases as
the sample size increases** as we can see from the above
notation that the standard error of mean is inversely proportional
to the square root of the sample size n.

The distribution tends to a normal distribution for a larger sample size. The normal distribution approximation is valid only for n > 30 that is the central limit theorem can only be applied to sample size n > 30

3. Describe how the shape and standard deviation of a sampling
distribution changes as sample size increases. In other words,
describe the changes that occur to a sampling distribution
according to the Central Limit Theorem. Make sure you describe what
a sampling distribution is in your answer. Generate
pictures/diagrams to illustrate your thoughts if you would
like.

True or False?
1. σM equals the standard deviation
divided by the square root of the sample size
2. Larger samples more accurately reflect the population than
smaller samples
3. If n = 1, the standard error will equal the standard
deviation of the population
4. If a population is skewed, the distribution of sample means
will never be normal
5. The mean for the distribution of samples means is equal to
the mean of the population
6. As the sample...

Use the formula to find the standard error of the distribution
of differences in sample means, .
Samples of size 120 from Population 1 with
mean 87 and standard deviation 14 and samples
of size 85 from Population 2 with mean 71 and
standard deviation 17
Round your answer for the standard error to two decimal
places.
standard error
=

Suppose x has a distribution with a mean of 70 and a standard
deviation of 4. Random samples of size n = 64 are drawn.
(a) Describe the x distribution and compute the mean and
standard deviation of the distribution. x has distribution with
mean μx = and standard deviation σx = .
(b) Find the z value corresponding to x = 71. z =
(c) Find P(x < 71). (Round your answer to four decimal
places.) P(x < 71)...

Suppose x has a distribution with a mean of 90 and a standard
deviation of 27. Random samples of size n = 36 are drawn. (a)
Describe the x distribution and compute the mean and standard
deviation of the distribution. x has distribution with mean μx =
and standard deviation σx = . (b) Find the z value corresponding to
x = 99. z = (c) Find P(x < 99). (Round your answer to four
decimal places.) P(x < 99)...

Suppose x has a normal distribution with mean μ = 16 and
standard deviation σ = 11. Describe the distribution of x values
for sample size n = 4. (Round σx to two decimal places.)
μx = σx = Describe the distribution of x values for sample size
n = 16. (Round σx to two decimal places.)
μx = σx = Describe the distribution of x values for sample size
n = 100. (Round σx to two decimal places.)
μx...

Suppose x has a normal distribution with mean μ = 26 and
standard deviation σ = 6. Describe the distribution of x values for
sample size n = 4. (Round σx to two decimal places.) μx = σx =
Describe the distribution of x values for sample size n = 16.
(Round σx to two decimal places.) μx = σx = Describe the
distribution of x values for sample size n = 100. (Round σx to two
decimal places.) μx...

Suppose x has a normal distribution with mean μ = 52 and
standard deviation σ = 4. Describe the distribution of x values for
sample size n = 4. (Round σx to two decimal places.) μx = σx =
Describe the distribution of x values for sample size n = 16.
(Round σx to two decimal places.) μx = σx = Describe the
distribution of x values for sample size n = 100. (Round σx to two
decimal places.) μx...

Suppose x has a normal distribution with mean
μ = 57 and standard deviation σ = 7.
Describe the distribution of x values for sample size
n = 4. (Round σx to two
decimal places.)
μx
σx
Describe the distribution of x values for sample size
n = 16. (Round σx to two
decimal places.)
μx
σx
Describe the distribution of x values for sample size
n = 100. (Round σx to two
decimal places.)
μx
σx
How do the...

The standard error of a distribution of sample means is
__________ than the standard deviation of a population of
individuals it came from because most of the sample means will be
__________ to the mean of the population of individuals

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