Question

(a) Create a sampling distribution for the sample mean using sample sizes n=2. Take N=2000 repeated sample sizes of 2, and observe the histogram of the sample means. What shape does this sampling distribution have?

The sampling distribution is normal

The sampling distribution is triangular

The sampling distribution is skewed right

The sampling distribution is uniform

(b) Now take N = 2000 repeated samples of size 8. Explain how the variability and the shape of the sampling distribution changes as n increases from 2 to 8.

The sampling distribution is more uniform, and the variability is smaller

The sampling distribution is more uniform, and the variability is larger

The sampling distribution is more normal, and the variability is larger

The sampling distribution is more normal, and the variability is smaller

(c) Now take N= 2000 repeated samples of size 25. Explain how the variability and the shape of the sampling distribution changes as n increases from 2 to 25.

The sampling distribution is more normal, and the variability is much smaller

The sampling distribution is more uniform, and the variability is much smaller

The sampling distribution is more normal, and the variability is much larger

The sampling distribution is more uniform, and the variability is much larger

(d) Compare the results from parts a through c to the displayed example curves.

The distribution from Q1 matches the displayed curve for n=30​, but the distributions in Q1 and Q3 are more uniform than the displayed curves.

The distribution from Q1 matches the displayed curve for n=2, but the distributions in Q2 and Q3 are more uniform than the displayed curves.

The distributions from parts Q1through Q3 go in reverse order from the displayed curves.

The distributions from Q1 through Q3 roughly match the displayed curves.

(e) Explain how the central limit theorem describes what has been observed in this problem.

The sampling distribution of the mean became more and more normal as the sample size decreased from 30 to 8 to 2​, which the central limit theorem says should happen.

The sampling distribution of the mean became more and more uniform as the sample size decreased from 30 to 8 to 2​, which the central limit theorem says should happen.

The sampling distribution of the mean became more and more uniform as the sample size increased from 2 to 8 to 30​, which the central limit theorem says should happen.

The sampling distribution of the mean became more and more normal as the sample size increased from 2 to 8 to 30​, which the central limit theorem says should happen.

The options below the questions are my choices.

Answer 1

Ans. a.) The sampling distribution is skewed right.

Reason: The sampling distribution of the sample mean would have been normal if the sample size was 30 or more. Since the sample size is small enough, the distribution would be skewed right.

Ans. b.) The sampling distribution is more normal and the variability is much smaller.

Reason: When the sample size increases, the skewness is reduced and the data becomes more precise which reduces the variance.

Ans. c.) The sampling distribution is more normal and the variability is much smaller.

Reason: Same as b.)

d.) No curve has been displayed for this answer.

Ans e.) The sampling distribution of the mean became more and more normal as the sample size increased from 2 to 8 to 30​, which the central limit theorem says should happen.

Reason: The central limit theorem states that the sample mean is approximately normally distributed for samples of size equal to or greater than 30 regardless of the distribution of the population from which the sample is drawn.