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STAT 3507
Assignment 2
Due Fri. Feb. 11
1.
Consider a small population of N = 4 units with y-values: 3, 2, 2, 5.
c)
Find the population mean μ, and variance σ2.
List all possible simple random samples of size n = 2 that can be selected from the
population, along with their probability of selection.
For each sample calculate the sample mean y and the sample variance s2 . Demonstrate
d)
numerically that for this example that y is an unbiased estimator for the population mean
N
and that E( s 2 ) =
2.
N -1
Is S2 an unbiased estimator of σ2 ? Why or why not?
a)
b)
2.
Text #4.5 (ed 6 only) Do parts (a), (c), (e)
3.
Text # 4.11
4.
For the data in problem #4.16 (#4.7 ed 5) in your text,
a)
b)
5.
a)
b)
find a point estimate for the true average daily water consumption per household and
construct an approximate 95% confidence interval estimate for the true average daily
consumption.
estimate the total number of gallons of water, τ, used daily during the dry spell, and construct
an approximate 95% confidence interval for τ.
For problem #4.20 in your text (#4.12 ed 5): Construct a 90% confidence interval estimate
for the parameter of interest.
Using the data in (a), find the sample size needed to estimate the proportion of game bird
hunters to within 0.02 with 99% confidence.
6.
Using the results of the survey described in #4.27 (#4.20 ed 5), find the sample size required to
estimate the total number of trees on the plantation to within 1500 with 95% confidence.
7.
The Special Census of Maricopa County gave 1995 populations for the following cities:
Buckeye
Gilbert
4,847
59,338
Suppose you want to estimate the percentage of persons who have been immunized against polio in
each city and can take a SRS of persons.
a)
What should be your sample size in each city to have a margin of error of 4 percentage
points?
b)
c)
8.
Recalculate the sample size necessary to have a margin of error of 4 percentage points,
without using the finite population correction factor.
Does the finite population correction make a difference for both cities?
Signatures to a petition were collected on 676 sheets. Each sheet had enough space for 42
signatures, but on many sheets a smaller number of signatures had been collected. The numbers of
signatures per sheet were counted on a random sample of 50 sheets, with the results shown below.
Number of
Signatures
42
41
36
32
29
27
23
19
16
15
Number of
Sheets
23
4
1
1
1
2
1
1
2
2
Number of
Signatures
14
11
10
9
7
6
5
4
3
Number of
Sheets
1
1
1
1
1
3
2
1
1
Note
a)
b)
c)
d)
e)
9.
 f i yi2 = 54,497
Plot the data using a histogram (using SAS as in 9(a) below). Describe the shape of the data.
Estimate the total number of signatures and find the standard error for you estimate (i.e. the
square root of the variance).
Do you think that the sample total from part (b) will be approximately normally distributed?
Why or why not?
Would you recommend giving a confidence interval for your estimate of the total number of
signatures? Why or why not?
Estimate the proportion of signature sheets with 42 signatures and find the standard error
for your estimate.
SAS Question
Use SAS to
a)
b)
Plot the histogram of the data.
Find the sample mean (mean number of signatures per sheet) and standard deviation.
Remember to put the following statements at the top of all your programs:
dm output 'clear';
options linesize=80;
options pagesize=40;
footnote 'yourname, stud#';
You will need the procedures PROC CHART (for histograms) and PROC MEANS (for mean and
standard deviation. These procedures are described in sections 6.3 and 5.3 respectively of Learning
SAS Version 8 (manual available as course pack for $15 from Bookstore).