STAT 718
due: September 25, 1997

1.	Why is the following procedure not suitable for drawing a simple 
random sample of addresses in Lockhart City?
	a. Randomly select a district between 51 and 75.
	b. Randomly select a house from those in the chosen district.
	c. Reject both district and house selection if the house is 
already in the sample.
	d. Repeat a - c until the desired sample size is achieved.
	
2.  Simple random sampling occurs when all subsets of size n from the
population of size N are equally likely to be the sample.
    a. Show, by example, that it is not sufficient that each element
of the population have probability n/N of appearing in the sample.
    b. Prove that the following sequential procedure does lead to a
simple random sample:  The sample is drawn one element at a time.  At
each stage, all elements of the population not already in the sample
have an equal chance of being the next element chosen to be in the
sample.  (Try to write down this proof as rigorously as you can.)
 
3.	Notice that no district in Lockhart City has more than 1313 
houses.  Prove that the following procedure does produce a simple random 
sample of houses in Lockhart City:
	a. Randomly select a district between 51 and 75.
	b. Randomly select a random number (the potential house selection) 
between 1 and 1313.
	c. Reject both district and house selection if the house number 
exceeds the number of houses in the district or if the house is already 
in the sample.
	d. Repeat a-c until the desired sample size is achieved.

4.	Let N=6, n=3.  For purposes of studying sampling distributions, we 
assume that all population values are known:
	y1 = 98	y2 = 102	y3 = 154	y4 = 133	y5 = 190
	y6 = 175
We are interested in mu, the population mean.  What is the value of mu?
	For each of the following sampling plans, find (i) E[ybar]; (ii) 
V[ybar]; (iii) Bias(ybar); (iv) MSE(ybar).  Which sampling plan do you 
think is the best?  Why?

Plan 1.  8 possible samples
sample	Units	Prob.
1	(1,3,5)	1/8
2	(1,3,6)	1/8
3	(1,4,5)	1/8
4	(1,4,6)	1/8
5	(2,3,5)	1/8
6	(2,3,6)	1/8
7	(2,4,5)	1/8
8	(2,4,6)	1/8

Plan 2.  Three possible samples
sample	Units	Prob
1	(1,4,6)	1/4
2	(2,3,6)	1/2
3	(1,3,5)	1/4

5.	Select a simple random sample of size 10 from Lockhart City.  Use 
any random number table.  Hand in a list of the random numbers you 
selected and the addresses to which they correspond.  Describe exactly 
how you converted a random number to an address.

6.  The cable TV company wants to charge $10 per month.  Estimate from
your (admittedly tiny) sample:
(a) The proportion of households in Lockhart City who are willing to
pay $10 per month for cable TV service.
(b) The average number of TV's in households willing to pay $10 per
month for cable TV service.
(c) The total number TV's in households willing to pay $10 per month.