(1 point) A variable of a population has a mean of =300 and a standard deviation

Question

(1 point) A variable of a population has a mean of =300 and a standard deviation of =35.

1. The sampling distribution of the sample mean for samples of size 49 is approximately normally distributed with mean and standard deviation?

2. For part (1) to be true, what assumption did you make about the distribution of the variable under consideration?

A. No assumption was made.

B. Normal distribution.

C. Uniform distribution.

3. Is the statement in part (1) still true if the sample size is 16 instead of 49? Why or why not?

A. No, the sampling distribution of the sample mean is never normal for sample size less than 30.

B. Yes, the sampling distribution of the sample mean is always normal.

C. No. Because the distribution of the variable under consideration is not specified, a sample size of at least 30 is needed for part (1) to be true.

Explanation / Answer

2 > A :no assumptions were made

as we know by lindeberg levy clt for large sample size the sample mean follows normal dist with the same mean and a standard deviation of sigma/sqroot(n), whatever be the distribution of Xi s.

3> C: No. Because the distribution of the variable under consideration is not specified, a sample size of at least 30 is needed for part (1) to be true.

this again speaks of the central limit theorem (lindeberg levy) which we have discussed earlier, the sample mean tends to a normal distribution only when the sample size is sufficiently large(here >30) whatever be the distribution under consideration, but when the sample size is small(here 16) the sample mean may not have a normal distribution .

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