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𝟐𝟐 2. [2 points] Show that the population mean is 𝝁𝝁 = 𝟔𝟔.
Test #1
PART A [60 points] Suppose that a universe consisting of 𝑵𝑵 = 𝟏𝟏𝟏𝟏 individuals has variable values𝒚𝒚𝒊𝒊 = 𝒊𝒊(𝒊𝒊 = 𝟏𝟏,𝟐𝟐,…,𝟏𝟏𝟏𝟏);thus,𝒚𝒚𝟏𝟏 = 𝟏𝟏,𝒚𝒚𝟐𝟐 = 𝟐𝟐,…,𝒚𝒚𝟏𝟏𝟏𝟏 = 𝟏𝟏𝟏𝟏,andthepopulationunder
studyis𝓟𝓟 = {𝟏𝟏,𝟐𝟐,…,𝟏𝟏𝟏𝟏}.
1. [3 points] Compute the population total 𝑻𝑻. (Hint: recall 𝟏𝟏 + 𝟐𝟐 + ⋯ + 𝑵𝑵 = 𝑵𝑵(𝑵𝑵+𝟏𝟏) ).
3. [5 points] Show that the survey definition of population variance is 𝝈𝝈𝟐𝟐 = 𝟏𝟏𝟏𝟏 . (Hint: recall 𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔
𝟏𝟏𝟐𝟐+𝟐𝟐𝟐𝟐+⋯+𝑵𝑵𝟐𝟐 =𝑵𝑵(𝑵𝑵+𝟏𝟏)(𝟐𝟐𝑵𝑵+𝟏𝟏)and∑𝑵𝑵 (𝒚𝒚 −𝝁𝝁)𝟐𝟐 =∑𝑵𝑵 𝒚𝒚𝟐𝟐−𝑵𝑵𝝁𝝁𝟐𝟐,where𝝁𝝁=𝟏𝟏∑𝑵𝑵 𝒚𝒚. 𝟔𝟔 𝒊𝒊=𝟏𝟏 𝒊𝒊 𝒊𝒊=𝟏𝟏 𝒊𝒊 𝑵𝑵 𝒊𝒊=𝟏𝟏 𝒊𝒊
Suppose that a simple random sample of size 𝒏𝒏 = 𝟑𝟑 is drawn without replacement from the population. Let 𝒀𝒀𝟏𝟏, 𝒀𝒀𝟐𝟐, 𝒀𝒀𝟑𝟑 denote the sample values, so 𝒀𝒀�𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺 = (𝒀𝒀𝟏𝟏 + 𝒀𝒀𝟐𝟐 + 𝒀𝒀𝟑𝟑)⁄𝟑𝟑 is the
�
sample mean.
4. [1 point] Give 𝑬𝑬(𝒀𝒀𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺), the expected value of the sample mean.
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5. [3 points] Compute 𝑽𝑽𝑽𝑽𝒔𝒔(𝒀𝒀�𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺), the variance of the sample mean.
Let 𝑻𝑻�𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺 be an unbiased estimator of 𝑻𝑻 based on the simple random sample of size 𝒏𝒏 = 𝟑𝟑
6. [3 points] Give a formula for 𝑻𝑻�𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺.
7. [3 points] Compute 𝑽𝑽𝑽𝑽𝒔𝒔�𝑻𝑻�𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺�, the variance of 𝑻𝑻�𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺.
Suppose that, when a simple random sample of size 𝒏𝒏 = 𝟑𝟑 is drawn without replacement from the population, the sample values 𝒀𝒀𝟏𝟏 = 𝟏𝟏, 𝒀𝒀𝟐𝟐 = 𝟒𝟒, 𝒀𝒀𝟑𝟑 = 𝟏𝟏𝟏𝟏 are obtained.
8. [2 points] Compute the value of 𝒀𝒀�𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺 based on this sample.
9. [2 points] Compute the value of 𝑻𝑻�𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺 based on this sample.
10. [5 points] Compute the value of an unbiased estimator of 𝝈𝝈𝟐𝟐 based on this sample. 𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔
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11. [3 points] Compute the value of an unbiased estimator of 𝑽𝑽𝑽𝑽𝒔𝒔(𝒀𝒀�𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺) based on this sample.
Suppose that a simple random sample of size 𝒏𝒏 = 𝟑𝟑 is drawn with replacement from the population, and let 𝒀𝒀�𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺 denote the sample mean.
12. [1 point] Give 𝑬𝑬(𝒀𝒀�𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺), the expected value of the sample mean. 13. [3 points] Compute 𝑽𝑽𝑽𝑽𝒔𝒔(𝒀𝒀�𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺), the variance of the sample mean.
14. [3 points] Which estimator is more efficient, 𝒀𝒀�𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺 or 𝒀𝒀�𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺? Justify your answer briefly.
Suppose that, when a simple random sample of size 𝒏𝒏 = 𝟑𝟑 is drawn with replacement from the population, the sample values 𝒀𝒀𝟏𝟏 = 𝟒𝟒, 𝒀𝒀𝟐𝟐 = 𝟒𝟒, 𝒀𝒀𝟑𝟑 = 𝟏𝟏𝟏𝟏 are obtained.
15. [2 points] Compute the value of 𝒀𝒀�𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺 based on this sample.
16. [3 points] Compute the value of an unbiased estimator of 𝑽𝑽𝑽𝑽𝒔𝒔(𝒀𝒀�𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺) based on this sample.
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Suppose that a simple random sample of size 𝒏𝒏 = 𝟑𝟑 is drawn with replacement from the population, and suppose that the population values for 𝒎𝒎 distinct indeividuals are obtained. Let 𝒀𝒀�𝑯𝑯𝒀𝒀𝑯𝑯𝑺𝑺𝑯𝑯𝑯𝑯 denote the sample mean of the 𝒎𝒎 distinct values.
17. [1 point] Give 𝑬𝑬(𝒀𝒀�𝑯𝑯𝒀𝒀𝑯𝑯𝑺𝑺𝑯𝑯𝑯𝑯), the expected value of the sample mean. 18. [6 points] Compute 𝑽𝑽𝑽𝑽𝒔𝒔(𝒀𝒀�𝑯𝑯𝒀𝒀𝑯𝑯𝑺𝑺𝑯𝑯𝑯𝑯), the variance of the sample mean.
19. [2 points] Which estimator is more efficient, 𝒀𝒀�𝑯𝑯𝒀𝒀𝑯𝑯𝑺𝑺𝑯𝑯𝑯𝑯 or 𝒀𝒀�𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺? Justify your answer briefly.
Suppose that, when a simple random sample of size 𝒏𝒏 = 𝟑𝟑 is drawn with replacement from the population, the sample values 𝒀𝒀𝟏𝟏 = 𝟒𝟒, 𝒀𝒀𝟐𝟐 = 𝟒𝟒, 𝒀𝒀𝟑𝟑 = 𝟏𝟏𝟏𝟏 are obtained.
20. [3 points] Compute the value of 𝒀𝒀�𝑯𝑯𝒀𝒀𝑯𝑯𝑺𝑺𝑯𝑯𝑯𝑯 based on this sample.
21. [4 points] Compute the value of an unbiased estimator of 𝑽𝑽𝑽𝑽𝒔𝒔(𝒀𝒀�𝑯𝑯𝒀𝒀𝑯𝑯𝑺𝑺𝑯𝑯𝑯𝑯) based on this sample.
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PART B [20 points] Suppose that 200 students take a midterm test. The course instructor is
trying to plan how much time should be allotted for regrades of the test. From previous
Let 𝑵𝑵 be number of students out of the 200 who submit their test for a regrade. If it is assumed
experience, the instructor assumes that any student will ask for a regrade with probability 10%.
that the students ask for regrades independently of one another, then 𝑵𝑵 has the binomial
minutes required to regrade all of the tests submitted will be 𝑻𝑻 = 𝑻𝑻 + ⋯ + 𝑻𝑻 , where 𝑻𝑻 is the 𝟏𝟏𝑵𝑵𝒊𝒊
distribution with 200 trials and probability of success 0.1 on each trial. The total number of
time required in minutes to complete regrade 𝒊𝒊 (𝒊𝒊 = 𝟏𝟏, … , 𝑵𝑵). The key feature of 𝑻𝑻 is that it is a
𝑻𝑻 ,…,𝑻𝑻 areindependentrandomvariableshaving𝑬𝑬(𝑻𝑻 ) = 𝟖𝟖and𝑽𝑽𝑽𝑽𝒔𝒔(𝑻𝑻 ) = 𝟒𝟒 (𝒊𝒊 = 𝟏𝟏,…,𝑵𝑵). 𝟏𝟏𝑵𝑵𝒊𝒊𝒊𝒊
sum of a random number of terms. For planning purposes, the instructor assumes that the times
1. [2 points] Compute 𝑬𝑬(𝑵𝑵), the expected value of the number of regrades 𝑵𝑵. 2. [3 points] Compute 𝑽𝑽𝑽𝑽𝒔𝒔(𝑵𝑵), the variance of the number of regrades 𝑵𝑵.
3. [3 points] Compute 𝑬𝑬(𝑻𝑻|𝑵𝑵), the expected value of the total time 𝑻𝑻 required for the regrades given the number of regrades 𝑵𝑵.
4. [3 points] Compute 𝑽𝑽𝑽𝑽𝒔𝒔(𝑻𝑻|𝑵𝑵), the variance of the total time 𝑻𝑻 required for the regrades given the number of regrades 𝑵𝑵.
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5. [4 points] Compute 𝑬𝑬(𝑻𝑻), the expected value of the total time 𝑻𝑻 required for the regrades. 6. [5 points] Compute 𝑽𝑽𝑽𝑽𝒔𝒔(𝑻𝑻), the variance of the total time 𝑻𝑻 required for the regrades.
PART C [20 points] Suppose that an arborist wishes to design a survey to estimate the total number of birch trees in a certain geographical area. The arborist divides the entire area into 200 sub-regions, and the arborist assumes that the numbers of birch trees per sub-region has population variance 𝝈𝝈𝟐𝟐 = 𝟑𝟑𝟔𝟔.
The arborist intends to use simple random sampling to estimate the total number of trees in the area, and she wants to use a sample size sufficiently large that the estimate of the total is within 300 trees of the true total with probability at least 90%.
1. [5 points] Suppose that the arborist decides to use simple random sampling with replacement. Compute the smallest possible sample size that the arborist can use.
2. [5 points] Suppose that the arborist decides to use simple random sampling without replacement. Compute the smallest possible sample size that the arborist can use.
𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔
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Suppose that the arborist wants to use a sample size sufficiently large that the estimate of the
that the number of birch trees per sub-region has coefficient of variation 𝝈𝝈 ⁄𝝁𝝁 = 𝟏𝟏. 𝟓𝟓. 𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔
3. [5 points] Suppose that the arborist decides to use simple random sampling with replacement. Compute the smallest possible sample size that the arborist can use.
total is within 10% of the true total with probability 90%. For this purpose, the arborist assumes
4. [5 points] Suppose that the arborist decides to use simple random sampling without replacement. Compute the smallest possible sample size that the arborist can use.
PART D [40 points] An economist, who is studying household income in rural communities, investigates a particular region of 𝑵𝑵 = 𝟔𝟔𝟏𝟏𝟏𝟏 houses that is divided into 𝑯𝑯 = 𝟑𝟑 postal areas. The sizes and population means and variances for the three postal areas are as follows: postal area 1
= 𝟑𝟑𝟏𝟏𝟏𝟏,𝝁𝝁 = 𝟑𝟑𝟏𝟏,𝝈𝝈𝟐𝟐 = 𝟏𝟏𝟏𝟏𝟏𝟏;postalarea2has𝑵𝑵 = 𝟐𝟐𝟏𝟏𝟏𝟏,𝝁𝝁 = 𝟔𝟔𝟏𝟏,𝝈𝝈𝟐𝟐 = 𝟗𝟗𝟏𝟏𝟏𝟏; 𝟏𝟏 𝟏𝟏 𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔,𝟏𝟏 𝟐𝟐 𝟐𝟐 𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔,𝟐𝟐
has𝑵𝑵
and postal area 3 has 𝑵𝑵 = 𝟏𝟏𝟏𝟏𝟏𝟏, 𝝁𝝁 = 𝟗𝟗𝟏𝟏, 𝝈𝝈𝟐𝟐 = 𝟗𝟗𝟏𝟏𝟏𝟏. The economist decides to treat the
𝟑𝟑 𝟑𝟑 𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔,𝟑𝟑
postal areas as strata for a survey.
1. [4 points] Compute the overall population mean 𝝁𝝁 for all 600 houses in the region.
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2. [3 points] Compute the within strata variation for the three postal areas.
3. [3 points] Compute the between strata variation for the three postal areas.
4. [3 points] Compute the overall population variance 𝝈𝝈𝟐𝟐 (at least approximately) for all 600
𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔
The arborist considers taking a simple random sample of size 𝒏𝒏 = 𝟔𝟔𝟏𝟏 drawn without replacement
houses in the region.
from the entire population of all 600 houses. Let 𝒀𝒀�𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺 denote the sample mean. 5. [3 points] Compute 𝑽𝑽𝑽𝑽𝒔𝒔(𝒀𝒀�𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺), the variance of 𝒀𝒀�𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺.
Suppose the arborist considers a total sample size 𝒏𝒏 = 𝟔𝟔𝟏𝟏, achieved by taking simple random
performs stratified sampling with 𝒏𝒏 = 𝒏𝒏 = 𝒏𝒏 = 𝟐𝟐𝟏𝟏. Let 𝒀𝒀� denote the stratified
samples without replacement of size 20 from each of the three postal areas. Thus, the arborist
𝟏𝟏 𝟐𝟐 𝟑𝟑 𝑺𝑺𝑻𝑻𝑺𝑺𝑺𝑺𝑻𝑻,𝑬𝑬𝑬𝑬𝑬𝑬𝑺𝑺𝑬𝑬
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estimator.
6. [4 points] Compute 𝑽𝑽𝑽𝑽𝒔𝒔�𝒀𝒀𝑺𝑺𝑻𝑻𝑺𝑺𝑺𝑺𝑻𝑻,𝑬𝑬𝑬𝑬𝑬𝑬𝑺𝑺𝑬𝑬�, the variance of 𝒀𝒀𝑺𝑺𝑻𝑻𝑺𝑺𝑺𝑺𝑻𝑻,𝑬𝑬𝑬𝑬𝑬𝑬𝑺𝑺𝑬𝑬.
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Suppose the arborist considers a total sample size 𝒏𝒏 = 𝟔𝟔𝟏𝟏, achieved by taking simple random
postal area is chosen proportional to the size of the postal area. Let 𝒀𝒀� denote the
samples without replacement from each of the three postal areas where the sample size for each
𝑺𝑺𝑻𝑻𝑺𝑺𝑺𝑺𝑻𝑻,𝑷𝑷𝑺𝑺𝑺𝑺𝑷𝑷 7. [3 points] Compute the sample sizes 𝒏𝒏𝟏𝟏, 𝒏𝒏𝟐𝟐, 𝒏𝒏𝟑𝟑 for the three postal areas.
stratified estimator.
8. [4 points] Compute 𝑽𝑽𝑽𝑽𝒔𝒔�𝒀𝒀�𝑺𝑺𝑻𝑻𝑺𝑺𝑺𝑺𝑻𝑻,𝑷𝑷𝑺𝑺𝑺𝑺𝑷𝑷�, the variance of 𝒀𝒀�𝑺𝑺𝑻𝑻𝑺𝑺𝑺𝑺𝑻𝑻,𝑷𝑷𝑺𝑺𝑺𝑺𝑷𝑷.
Suppose the arborist knows the individual variances for the three postal regions and considers
using a stratified estimator 𝒀𝒀� based on a total sample size 𝒏𝒏 = 𝟔𝟔𝟏𝟏 with simple random
𝑺𝑺𝑻𝑻𝑺𝑺𝑺𝑺𝑻𝑻
samples taken without replacement from the postal regions where the sample sizes 𝒏𝒏 , 𝒏𝒏 , 𝒏𝒏 are
chosen to minimize the variance 𝑽𝑽𝑽𝑽𝒔𝒔(𝒀𝒀�𝑺𝑺𝑻𝑻𝑺𝑺𝑺𝑺𝑻𝑻).
10. [4 points] Compute 𝑽𝑽𝑽𝑽𝒔𝒔(𝒀𝒀�𝑺𝑺𝑻𝑻𝑺𝑺𝑺𝑺𝑻𝑻), the variance of 𝒀𝒀�𝑺𝑺𝑻𝑻𝑺𝑺𝑺𝑺𝑻𝑻.
𝟏𝟏𝟐𝟐𝟑𝟑 9. [4 points] Compute the sample sizes 𝒏𝒏𝟏𝟏, 𝒏𝒏𝟐𝟐, 𝒏𝒏𝟑𝟑 for the three postal areas.
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Suppose that the cost per house to survey is $4 in the first postal area, $9 in the second postal area, and $16 in the third postal area. The arborist has $400 to spend on surveying the houses and intends to use stratified sampling by choosing sample sizes to minimize the variance of the estimator.
11. [5 points] Compute the overall sample size 𝒏𝒏 that the arborist will use.
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