Let\'s examine the mean of the numbers 1, 2, 3, 4, 5, and 6 by drawing samples f

Question

Let's examine the mean of the numbers 1, 2, 3, 4, 5, and 6 by drawing samples from these values, calculating the mean of each sample, and then considering the sampling distribution of the mean. To do this, suppose you perform an experiment in which you roll a regular die (six-sided) two times (or equivalently, roll two dice one time) and calculate the mean of your sample. Remember your population is the numbers 1, 2, 3, 4, 5, and 6. The true mean (mu) of the numbers 1, 2, 3, 4, 5, and 6 is and the true standard deviation (sigma) is The number of possible different samples (each of size n = 2) is the number of possibilities on the first roll (6) times number of possible on the second roll (also 6), or 6(6) = 36. If you collected all of these possible samples, the mean of your sampling distribution of means (mu_M) would equal and the standard deviation of your sampling distribution of means (that is, the standard error or sigma_M) would be The following chart shows the sampling distribution of the mean (M) for your experiment. Suppose you do this experiment once (that is, you roll the die two times). Use the chart to determine the probability that the mean of your two rolls is equal to the true mean, your two rolls is equal to the true mean, or P(M = mu), is The probability that the mean of your two rolls is greater than 1.5, or P(M > 1.5), is

Explanation / Answer

a) Mean = (1+n)/2 = (1+6)/2 = 7/2 =3.5 Correct answer: Option (A) 3.5

SD = sqrt((n2-1)/12)) = sqrt(*(62-1))/12 = 1.7078 Correct Answer: Option (D) 1.71

b) Sampling distribution of sample mean = population mean = 3.5 correct answer: Option (d)

SD of sampling distribution of sample mean = SD/sqrt(n) = 1.71/sqrt(2)=1.21 Correct answer: Option (d)

c)

P(M=mu) = 6/36 = 0.16667 correct Answer: option (B) 0.17

P(M>1.5) = 1 - (1/36 + 2/36) = 33/36 = 0.916 Correct Answer; Option (B) 0.92

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