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MATH3426 Assignment
Question 1:
Derive the following formula for the optimal allocation that minimizes minimum the total cost
for a fixed variance :
≈ [
∑ (√)
=1
2 + ∑ (
2)=1
]
√
= 1, … , .
Question 2:
The following results were obtained from a stratified random sample:
Stratum Population size Sample size Sample mean Sample Variance
1 100 50 10 2800
2 150 50 20 700
3 300 50 30 600
(i) Estimate the mean for the whole population.
(ii) Find a 95% C.I. for the mean. [Use 0.025 = 1.96]
Question 3:
Allocate a total sample size of = 100 between male group and female group having their
respective sizes 1 = 200 and 2 = 300, and their respective variances 1
2 = 81 and 2
2 = 16
using
(i) Equal Allocation
(ii) Proportional Allocation
(iii) Optimal Allocation with costs (1 = 64 2 = 25)
(iv) Neyman Allocation
Question 4:
A researcher working with a group of people desires to estimate their average reaction time to a
certain stimulus. S/He thinks that men and women probably will show a difference in reaction
times, so s/he wants to stratify on gender. The group of 96 people contains 43 men. In previous
studies of this type, researchers have found that the times range from 5 to 20 seconds for men
and from 3 to 14 seconds for women, The costs of sampling are the same for both strata.
(i) Which allocation should be used? Why?
(ii) Use the appropriate allocation to determine the sample size, and stratum sample
sizes to estimate the average reaction time for the group to within 1 second with a
probability of 0.95. [Use 0.025 = 2] (Hint: Use the range ≈ 4 for each stratum)