Answer:

General guidance

Concepts and reason

The concepts used are: normal distribution and central limit theorem.

The sampling distribution of the mean is found out using the central limit theorem. This also helps with the shape of the distribution, its mean and standard deviation.

Fundamentals

If a data is following a normal distribution, then the mean of a sample drawn also follows a normal distribution, according to central limit theorem. The sampling distribution of the mean has mean, equal to the population mean and standard deviation \sigma = \frac{{{\sigma _{population}}}}{{\sqrt n }}σ=n?σpopulation??

If the sampling distribution of the mean is normal in nature, then the shape will be bell-shaped.

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FIRST STEP | ALL STEPS | ANSWER ONLY

Step-by-step

Step 1 of 10

(a)

Options a=skewed left, b=skewed right and d=not enough information are incorrect because according to the central limit theorem, the sampling distribution of the mean is normal.

Explanation | Hint for next step

According to central limit theorem, for a large sample size, the sampling distribution of the mean follows a normal distribution, given that the sample is drawn from a normal distribution. As the mean is normally distributed, the shape of its graph is symmetric rather than skewed left or right.

Step 2 of 10

Option c=mean is normal is correct due to the central limit theorem which states that the sampling distribution of the mean.

Part a

c=mean is normal

Explanation | Common mistakes | Hint for next step

The central limit theorem states that the sampling distribution of the mean is normal, if the sample is taken from a normal distribution or is large enough.

Step 3 of 10

(b)

The mean of the sampling distribution is equal to the population mean which is equal to 210 mg/dL.

Part b

The mean of the sampling distribution is 210 mg/dL.

Explanation | Common mistakes | Hint for next step

The central limit theorem states that the mean of the sampling distribution will be equal to the population mean, if the sample is drawn from a normal population.

Step 4 of 10

(c)

The standard deviation of the sampling distribution is:

\begin{array}{c}\\\sigma = \frac{{{\sigma _{population}}}}{{\sqrt n }}\\\\ = \frac{{20}}{{\sqrt {42} }}\\\\ = 3.09\\\end{array}σ=n?σpopulation??=42?20?=3.09?

Part c

The standard deviation of the sampling distribution is 3.09 mg/dL.

Explanation | Common mistakes | Hint for next step

The central limit theorem states that the standard deviation of the sampling distribution will be equal to the standard deviation of the population divided by the square root of the sample size, if the sample is drawn from a normal population.

Step 5 of 10

(d.a)

Options b = become skewed left and c = become skewed right are incorrect because according to the central limit theorem, the sampling distribution of the mean is normal.

Explanation | Hint for next step

According to central limit theorem, for a large sample size, the sampling distribution of the mean follows a normal distribution, given that the sample is drawn from a normal distribution. As the mean is normally distributed, the shape of its graph is symmetric rather than skewed left or right.

Step 6 of 10

Option a = also be normally distributed is correct due to the central limit theorem which states that the sampling distribution of the mean.

Part d.a

a = also be normally

Explanation | Common mistakes | Hint for next step

The central limit theorem states that the sampling distribution of the mean is normal, if the sample is taken from a normal distribution or is large enough. An increase in the sample size will only strengthen the theorem.

Step 7 of 10

(d.b)

Option a = change to is incorrect because according to the central limit theorem, the mean of the sampling distribution will be equal to the population mean

Explanation | Hint for next step

According to central limit theorem, for a large sample size, the sampling distribution of the mean follows a normal distribution, given that the sample is drawn from a normal distribution and the mean of the sampling distribution will be equal to the population mean

Step 8 of 10

Option b = remain as population mean is correct due to the central limit theorem which states the mean of the sampling distribution will be equal to the population mean.

Part d.b

b = remain as population mean

Explanation | Common mistakes | Hint for next step

The central limit theorem states that the mean of the sampling distribution will be equal to the population mean, if the sample is drawn from a normal population. An increase in the sample size will only strengthen the theorem.

Step 9 of 10

(d.c)

Option b = remain as population mean is incorrect due to the central limit theorem which states the standard deviation of the sampling distribution will decrease with increasing sample sizes.

Explanation | Hint for next step

According to central limit theorem, for a large sample size, the standard deviation of the sampling distribution will be equal to the standard deviation of the population divided by the square root of the sample size. As sample size increases, the denominator increases but the numerator remains the same leading to a decreased standard deviation.

Step 10 of 10

The standard deviation of the sampling distribution with sample size as 100 is:

\begin{array}{c}\\\sigma = \frac{{{\sigma _{population}}}}{{\sqrt n }}\\\\ = \frac{{20}}{{\sqrt {100} }}\\\\ = 2\\\end{array}σ=n?σpopulation??=100?20?=2?

Option a = change to 2 is correct because according to the central limit theorem, the standard deviation of the sampling distribution will be equal to the standard deviation of the population divided by the square root of the sample size.

Part d.c

a = change to 2

Explanation | Common mistakes

The central limit theorem states that the standard deviation of the sampling distribution will be equal to the standard deviation of the population divided by the square root of the sample size, if the sample is drawn from a normal population.

Answer

Part a

c=mean is normal

Part b

The mean of the sampling distribution is 210 mg/dL.

Part c

The standard deviation of the sampling distribution is 3.09 mg/dL.

Part d.a

a = also be normally

Part d.b

b = remain as population mean

Part d.c

a = change to 2