Question

1.      There is a population of three people: Xerxes, Yasmin, and Zoe. [Note: this is a population, not a sample.] They are asked how many books they read last month. Xerxes read 2 books, Yasmin read 4 books, and Zoe read 12 books. We are too lazy to poll the entire population, so we take a sample of two (with replacement and order matters). Please draw the sampling distribution of the mean of a sample of size two.

2.      Continuing with the above example, what is the probability that the sample average will be within two books of the population average?

3.      Continuing with the above example, what is the standard error of the sampling distribution of the mean for a sample of size two? (In other words, what is the standard error for the distribution drawn in the first question.)

4.      Continuing with the above example, calculate the standard error of the sampling distribution of the mean for a sample of size two in a completely different way than you calculated it in question 3.

Answer 1

1)

Th e books reads by three people: Xerxes(X), Yasmin(Y), and Zoe(Z)

Population	Books read
X	2
Y	4
Z	12

Now, the ordered sampling with replacement of size two are,

Sample	Average Books read
XY	(2+4)/2=3
XZ	(2+12)/2=7
YZ	(4+12)/2=8

The mean and standard deviation of samples,

Sample meanY- 3+7+8 3 6 l

2)

(3 6)2(7 6)2 (8-6)2 Std dev,s 2.64575 1 n 1

Probability that the sample average will be within two books of the population average is,

P(L-2Xu+ 2)= P(X <u2)- P(X I2)

The probability is obtained by converting to standard normal distribution,

X- -)-P( -2 X P( P(-2 X+2) VI VI

P(u 2X 2) P(z)-P(:

P(u-2X 2) P(z 0.7559) P(z-0.7559

P(u-2X < 2)0.7752 0.2248 0.5503

3)

2.645 1.8708 Standard error, ox n