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Unformatted text preview: Last name: First name: Student #: UNIVERSITY OF TORONTO
Faculty of Arts and Science APRIL/MAY 2013 EXAMINATIONS STA 304H1 S/1003H S Duration  3 hours Examination Aids: NonProgrammable Calculator; aidsheet, one sided, or twosided, with
theoretical formulas only, as posted on the website. You may use any back side, with clear
indication of Question . art. ___—m Q. 1. [21] A tourist company organizes various types of sight tours, offering some discount rates for certain types
of customers, among them children (under 12) and seniors (over 65 years of age). The company wants
to estimate the total number of seniors on its tours on the basis of a random sample of site tours
recorded in their books. Each tour record shows the number of seniors, the number of children, and
some other data. There were 4,000 tours organized during the last year, listed as they appeared in
calendar time. The records show seasonal variations in numbers and types of people on the tours. Most
of the tours were organized in summer and least in winter. (a) Describe brieﬂy how you would select an SRS of 100 tours from the last year. (b) The company’s manager prefers a simpler method, using one—in40 systematic sampling of tours.
Describe brieﬂy how you would select this sample. (c) Can the sample obtained in (b) be treated as an SRS for the purpose of estimation? Explain why, or
why not. (continued) l/ll (d) The sample in (a) was selected and the average number of seniors per tour was 20 seniors, with the
sampling standard deviation of 5 seniors. Estimate the total number of seniors taking tours last year,
and place a bound on the error of estimation. (e) The company also wants to estimate the percentage of seniors out of all tourists on the tours last
year. (i) What information should be included in the sample to be able to estimate this percentage, and
to place a bound on the error of estimation? (ii) Exactly what values should be calculated from the
sample to be able to complete the required tasks? (f) It is estimated that the total number of tours next year will be 10% higher than this year. How large
should next year’s SRS sample be so that the estimated total number of seniors will be within i 2,000
of the true total with probability 95%? (g) What sampling design would you suggest that you think would produce better results than one in
(a) or (b)? Explain your choice in some details. 2/11 Q. 2. [18] For a purpose of planning power production Ontario Hydro selected an SRS of 500 residencies from
the population of 2.8 million electricity—using residences in Ontario. Among several other characteristics, the following responses were obtained:
Own a PC: 400; do not own: 100. Total number of PCs in use: 600. (some residencies may own several PCs)
Own electric stove: 450; do not own electric stove: 50 (likely, they use gas). Glassceramic cooktop: 200; regular: 250. (a) Estimate the average number of PCs in use per residence owning a PC. (b) Estimate the total number of PCs in use.
(c) Estimate the proportion and number of residences having a glassceramic stove among residences having an electric stove. (continued) 3/11 (d) Are these estimators in (a), (b), and (c) unbiased? Explain why or why not.
(e) Calculate a bound on the error of estimation of the number of residences having a Glassceramic
cooktop stove.
(f) It is known from the sample that 250 residencies own one PC, 100 own two PCs, and 50 own three
PCs. Of what particular use this information can be in your study (in what parts of Question 2)?
Explain and use it only in one part, if it can be used in more than one. 4/11 Q. 3. [21] A market research ﬁrm conducted a survey in 2000 in a city for the purpose of estimating the total monthly household expenditures on compact discs (CDs) and the total number of households owning a compact disc player (CDP). The city was divided into four geographical areas and a random sample of
, households was selected from each area. The results of the surve are as follows: Sample Sample
Number of Number Average Monthly Proportion
Area Households Samled Ex.enditure $ Ownin_ a CDP 20,000 2080
10,000 1220
35,000
15,000 1648
80,000 _—— (a) (i) Estimate the average monthly household expenditure on CDs in the city, and (ii) the proportion of households in the city owning a CDP.
(b) (i) Estimate the total monthly household expenditure on CDs in the city, and (ii) the total number of households owning a CDP in the city.
(c) (i) How many households would be sampled from each area if the sample of 400 were with proportional allocation? (ii) Considering already obtained sample, do you think that the stratiﬁed
sample with proportional allocation would produce better results than an SRS of the same size, for the
both parameters in (a), for only one of them, or none of them? (continued) 5/11 ((1) Explain on which parameter estimated in (a) you can place a bound on the error of estimation, and
then calculate it. (e) (i) Can you ﬁnd the optimal allocation for estimation of the proportion of households in the city that
own a CDP? Explain, but don’t calculate anything. (ii) Would this allocation be likely near to optimal
for estimation of the average monthly household expenditure on CDs? Explain. 6/11 Q. 4. [12]
From a directory of 16 households on a street, the actual numbers of people living in the households
(household size) are as follows: Household “nunM u (a) Calculate the theoretical standard deviations (i) of a systematic sample of one in four households
from the directory and (ii) of an SRS of the same size for estimating the average household size.
Which design is expected to give better result? (b) If the street were much longer would you expect same results from an SRS, as from a systematic
sample of the same size? Explain. (e) If you select an SRS of households ﬁom the street and record the household size and the house age
(in years) for each household, would using a ratio (or regression) estimator be better than just using
an SRS estimator? Explain. Assume that the average age of all houses is known. 7/ll Q. 5. [18] A psychologist wants to determine how many hours per week 810 year old boys in Toronto spend
playing video and computer games. To investigate this she randomly selects three Toronto primary
schools from the population of 230 primary schools in Toronto, and then randomly selects 20 (twenty)
8—1 0 year old boys from each school. With the parents help, she has each of the sampled boys record
their video and computer game times for a calendar week, in hours. The results are presented in the followin table: School Number of 8—10 Sample Sample Sample
ear old be S size Mean St. Dev. Q mm (3) Explain what kind of design is used here. Do you think this design is appropriate? Explain
(ignore the small sample size problem). (b) Comment on the condition of the design “. . . for a calendar week ...”. Would the selection of a
“calendar week” affect results of the study? Q (c) (i) Estimate the total number of hours spent by 810 year old Toronto boys uring a week on
video and computer games, and (ii) the variance of the estimator (you do not need to complete the
calculation of the variance, but you have to calculate all entries). (iii) Is the estimator in (i)
unbiased? Explain. (continued) 8/11 ((1) Do you expect that your standard error in (0) would be larger, smaller or about the same as the
standard error of an estimator based on SRS of the same size (60)? Explain. (e) (i) Estimate the percentage of the free time of 810 year old boys spent during a week on playing
the games. Assume that 8h a day is spent on sleep (7 days a week), and 7h in school and
transportation (5 days a week). (ii) What kind of estimator are you using in (i)? Can you place a
bound on the error of this estimator? Explain, but don’t calculate, if you can. 9/11 Q. 6. [10] (“theoretical” question) In Question 2, an SRS of 500 residencies from the population of 2.8 million electricityusing residences in Ontario was selected. Among several other characteristics, the following responses were obtained:
Own electric stove: 450; do not own electric stove: SO. Glassceramic cooktop: 200; regular: 250. You estimated in (c) the proportion of residences having a glassceramic stove among residences
having an electric stove. This proportion applies, obviously, only to residencies with an electric stove.
Consider now the general case: Given: N  population size, 11 —sample size of an SRS of residencies,
n1 — the number of residencies in the sample having an electric stove, . . . . . . A Tl
n2 — the number of re51denc1es 1n the sample havmg a g1assceram1c electrlc stove (n2 5 n1), p1 = :1,
A __ 112
p2  ”1 (a) 132 is our estimator for the proportion of residences having a glassceramic stove among residences
having an electric stove. Derive a convenient general formula for Var (p‘z) (estimated Var (132)) using
the above notation, where needed. (hint: what kind of estimator are you using?) (b) Calculate Var (152) from the actual sample data using your formula. 10/11 /// an extra page for work 11/11 ...
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