Question

Which of the following statements are true?

I. The sampling distribution of ¯xx¯ has standard deviation σ√nσn
even if the population is not normally distributed.

II. The sampling distribution of ¯xx¯ is normal if the population
has a normal distribution.

III. When nn is large, the sampling distribution of ¯xx¯ is
approximately normal even if the the population is not normally
distributed.

I and II

I and III

II and III

I, II, and III

None of the above gives the complete set of true responses.

Answer #1

From The central limiting theorem

We know that the standard deviation of the distribution of is given by

So statement I is correct

If the population has normal distribution then is also normally distributed.

So statement II is also correct.

From central limiting theorem if population doesn't have normal distribution and the sample size is large enough then distribution of is normal

Hence statement III is also correct.

Therefore all the three statements are correct.

*Fourth
Option*

Which of the following statements are true?
I. The sampling distribution of ¯xx¯ has standard deviation σ√nσn
even if the population is not normally distributed.
II. The sampling distribution of ¯xx¯ is normal if the population
has a normal distribution.
III. When nn is large, the sampling distribution of ¯xx¯ is
approximately normal even if the the population is not normally
distributed.
I and II
I and III
II and III
I, II, and III
None of the above gives...

Which of the following statements about the sampling
distribution of the sample mean, ¯xx¯ , is the correct?
1The distribution is normal regardless of the shape of the
population distribution, as long as the sample size, n, is large
enough.
2The distribution is normal regardless of the sample size, as
long as the population distribution is normal.
3The distribution's mean is the same as the population mean.
4The distribution's standard deviation is smaller than the
population standard deviation.
5All of...

Which of the following statements is not true for sampling
distributions?
a. A sampling distribution is necessary for making confidence
statements about an unknown population parameter.
b. A sampling distribution depends on the nature of the
population being sampled.
c. When sampling at random from a normal population, the
sampling distribution for the sample average is a normal
distribution.
d. None of the above.
AND
TRUE or FALSE
Assume a random sample of size n is from a normal population....

Which of the following statement about the sampling distribution
of proportion is true?
Select one:
a. Its mean equals to the population proportion.
b. Its standard error equals to the population standard
deviation.
c. It is approximately normally distributed if n ≥ 30.
d. It is approximately normally distributed if the population is
normally distributed.

Which of the following is NOT true regarding the sampling
distribution of the mean (Sample mean X-bar) for a large sample
size (sample size is greater than 36)? A. It has the same mean as
the population. B. It has a smaller standard deviation than the
population standard deviation. C. Approximately it follows a normal
distribution. D. It has the same distribution as the
population.

Question 2: Which of the following statements about the
sampling distribution of means is not true?
A. A sample distribution's mean will always equal the parent
population distribution's mean
B. The sampling distribution of means approximates the normal
curve.
C. The mean of a sampling distribution of means is equal to the
population mean.
D. The standard deviation of a sampling distribution of means is
smaller than the standard deviation of the population.

Which of the following is true?
The shape of the sampling distribution of sample proportion is
always bell-shaped.
As n increases, the mean of the sampling distribution of sample
proportion gets closer to the population proportion.
The shape of the sampling distribution of sample proportion
becomes approximately normal as n gets large.
The shape of the sampling distribution of sample proportion gets
closer to the shape of the population distribution as n gets
large.

Which of the following is true regarding the sampling
distribution of the mean for a large sample size?
Select one:
a. It has the same shape, mean, and standard deviation as the
population.
b. It has the same shape and mean as the population, but has a
smaller standard deviation.
c. It has a normal distribution with the same mean as the
population but with a smaller standard deviation.
d. It has a normal distribution with the same mean and...

Which of the following is true?
Data sets with the same mean and the same standard deviation
must have the same shape.
The mean is less affected by extreme observations than the
median.
Chebyshev's Rule gives us an idea what proportion of the data
set will fall within certain bounds, regardless of the shape of the
distribution.
i. only
ii. only
iii. only
i. and iii.
None of the above.

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...

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