  # Select all of the following that are true for CLT: A) The sampling distribution gets narrower and more normal as sample size increases.B) The sampling distribution of any continuous population distribution will be approximately normal given a sufficiently large sample size (n>30).C) The sampling distribution will have a bigger mean than the population distribution.D) The standard deviation of the sampling distribution is identical to the standard deviation of the population distributionE) The CLT can ONLY be used if the original population distribution is normal.

Question

Select all of the following that are true for CLT:

A) The sampling distribution gets narrower and more normal as sample size increases.

B) The sampling distribution of any continuous population distribution will be approximately normal given a sufficiently large sample size (n>30).

C) The sampling distribution will have a bigger mean than the population distribution.

D) The standard deviation of the sampling distribution is identical to the standard deviation of the population distribution

E) The CLT can ONLY be used if the original population distribution is normal.

check_circleExpert Solution
Step 1

Central Limit Theorem for mean:

If a random sample of size n is taken from any population having mean μ and standard deviation  σ then, as the sample size increases, the sample mean approaches the normal distribution with mean μ  and standard deviation σ/ sqrt(n).

Step 2

Concept of sampling distribution of sample mean:

Let a particular characteristic of a population is of interest in a study. Denote µ as the population mean of that characteristic and σ as the population standard deviation of that characteristic.

Now, it is not always possible to study every population unit. So, a sample of size n is taken from the population.

Let X denotes the random variable that measures the particular characteristic of interest. Let, X1, X2, …, Xn be the values of the random variable for the n units of the sample.

Then, the sample mean has a sampling distribution that has population mean or expected value same as that of X and population standard deviation, called the standard error, which is the population standard deviation of X divided by the square root of the sample size n. The parameters are as follows:

Step 3

Points on sampling distribution of the sample mean:

Carefully read the above concept. Then, note the following points:

• If the true population distribution of a random variable, say, x, is normal with parameters, mean μ and standard deviation σ, then, whatever be the size (n) of the sample taken from the population, the distribution of the sample mean is also normal, with parameters, mean μ and standard deviation σ/√n.
• Even if the true population distribution of a random variable, say, x, is not normal and has population mean μ, standard deviation σ, then, for a large size (n ≥ 30) of a sample taken from the population, the distribution of the sample mean is approximately normal, with parameters, mean μ and standard deviation σ/√...

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