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[answered] 2 [22] You are interested in the coffee habits of undergrad


please help me with all the questions, it is about statistics, sampling, need steps to steps answers


Q. 2 [22] You are interested in the coffee habits of undergraduate U of T students and want to estimate

 

the percentage of students who drink coffee regularly (and some other characteristics). You stay at the

 

front of a large auditorium and quickly interview students who leave after the end of the class (you may

 

have help from an assistant). You stop interviews once you collect a desired number of responses from

 

U of T students.

 

(a) (i) Briefly describe how you would select your subjects, (ii) Assuming that your selection procedure

 

produces an SRS of students from the class, would this sample be representative of U of T students?

 

Explain in short why, or why not.

 

(b) Assume all is nice with the sampling design, (i) What would be your sample size (intended number

 

of interviewed U of T students), if you want to estimate the percentage of students who drink coffee

 

regularly, with an error bound of 5%? From your experience, you have a reasonable guess that this

 

percentage is at least 70%. (ii) How would you choose your sample size, if you have no idea what

 

this percentage might be? Would it significantly affect your sample size, compared with (i)? (iii) Do

 

you have any comment on practicality (or impracticality) of the obtained sample sizes (hopefully,

 

you did it correctly)?

 

(c) Assume again all is nice with the sampling design. You want to estimate the average number of cups

 

of coffee a student drinks per day (including students that don't drink coffee at all). What would you

 

propose as the sample size, if you want to estimate this average with an error bound of 0.2 cups? (to

 

properly answer the question, you have to make some reasonable guess).

 

(continued) 3/12 Assume all is nice with the sampling design, and that the sample obtained is an SRS. You performed

 

your interviewing until you collected answers from (regardless of the sample sizes you calculated

 

previously) 120=? ??students, and obtained the following results:

 

Drink coffee: 85 (females 45, males 40), don't drink: 35 (females 20, males 15).

 

Drink one cup a week: 50, two cups: 25, three cups: 10.

 

(d) (i) Estimate the percentage of students drinking coffee, (ii) the total number of student drinking

 

coffee, assuming there are 60,000 students at U of T, and (iii) place a bound on the error of

 

estimation in (ii).

 

(e) (i) Estimate the average number of cups of coffee a student drinks per day (for all U of T students).

 

(ii) Estimate the average number of cups of coffee a student drinks per day only for students that

 

drink coffee, (iii) What kind of estimators you are using in (i) and (ii)?

 

(f) Estimate the percentage of female students at U of T from the sample. Is your estimate compatible

 

with the known fact that 58% of U of T undergraduate students are females?

 

(g) (bonus) Place a bound on the error of estimation in (e) (ii) [this part may be tricky]. 4/12 Q. 3 [18] At a cafeteria inside a school center, it was of interest to estimate the average amount of

 

money (in $) per week students spend in the cafeteria. Since the money spent may differ greatly from

 

student to student, it was decided to use stratification by schooling, primary (1300 students enrolled),

 

secondary (450 students enrolled), and seniors program (250 students enrolled). Since there was no

 

preliminary information about the variable of interest, it was decided to use proportional allocation of

 

the total sample size and SRS from each stratum. A budget of $150 was provided for the study. The

 

overhead (preparation) cost was $24, and the cost of contacting the students, and collecting information

 

was estimated at $3 per student. After completing the sampling and summarizing data, the following

 

results were obtained: N i n, y, Primary

 

1,300 Secondary

 

450 Seniors

 

250 27 10 5 5.185

 

10.851 10.200 17.600 15.289 23.300 (a) Show how the total sample size and the allocation were calculated.

 

(b) (i) Estimate the required average amount, (ii) Find the 95% confidence interval for the average

 

amount.

 

(c) It is of interest to investigate whether using the optimal allocation would improve estimation of

 

the average. Still assuming the same budget of the study, find the optimal allocation that would

 

minimize the variance. For the strata variances use the estimates obtained from the previous

 

study.

 

(continued) 5/12 (d) (i) How much more efficient would the allocation found in (c) be than the proportional allocation?

 

Would it be worthwhile? (ii) Could you predict the previous conclusion without performing the

 

calculation (still assuming you have the data from the above sample)?

 

(e) (i) If you simplify your sampling by considering an SRS of the same size as above, could it be as

 

efficient as the proportional allocation (approximately), somewhat less efficient, or much less efficient?

 

Discuss it in principle, using the above data, without doing any calculation, (ii) If an SRS of the same

 

sample size as above is to be used, can you estimate the variance of the sample mean, using the data

 

from the sample? If you can, do it. 6/12 Q. 4 [12] An alphabetical list of 300 test marks (on a 0-100 scale) is available.

 

(a) (i) Explain how you would select a systematic sample of size 8 from the list, (ii) If the part 55612

 

78095 83197 33732 05810 of a table of random numbers is used, which elements from the list will

 

be included in the sample? Explain your method of using the table.

 

(b) Suppose the following systematic sample in (a) was obtained (test marks):

 

70, 64, 95, 68, 81, 75, 57, 66.

 

Estimate the average class mark. Can you estimate the variance of that estimator using this

 

systematic sample? Explain,

 

(continued) 7/12 (c) Suppose that marks were listed by magnitude and the systematic sample

 

92, 86, 75, 73, 68, 66, 64, 57 was obtained. Can you estimate the average class mark using this sample? How would you estimate

 

the variance of that estimator? Justify your choice and estimate it.

 

(d) What design would you prefer to use in (c), SRS or systematic sampling? Explain. 8/12 Q. 5 [15] A survey is conducted in a town to estimate the number of residents who visited physician

 

and their expenditures on drugs in the month immediately preceding the survey. The survey made use of

 

the available list of all residential units (643 in number). Five of the residential units were selected from

 

the list at random, and al the residents of the unit were interviewed. The results are the following:

 

Expenditures on

 

Number who

 

Residence Number of

 

drags

 

in residence ($)

 

visited

 

a

 

physician

 

residents

 

No.

 

0

 

0, 0, 65

 

3

 

45

 

400, 350

 

1

 

2

 

131

 

1

 

105

 

1

 

207

 

0

 

0,0

 

2

 

398

 

1

 

50,0

 

2

 

519

 

(Resident may spend money on drugs without visiting a physician)

 

(a) What is the population of interest in this study? What are the sampling units, what is the frame?

 

What kind of sampling design was used here?

 

(b) (i) Estimate the number of residents in the town, (ii) Estimate the proportion and the number of

 

residents in the town who visited a physician in the preceding month, (iii) Which of these estimators

 

are biases, unbiased? Why?

 

(c) Estimate (i) the average expenditure on drugs per residence and per person and (ii) the total

 

expenditure on drugs in the town in the preceding month.

 

(continued) 9/12 (d) Place a bound on the error of estimation of the total expenditure on drugs in (c) (ii).

 

(e) Later was found that the town population was 1248. (i) Use this information to reestimate the total

 

expenditure on drugs in (c). (ii) Which of these two estimators would you prefer in general? Which

 

one in this particular case? (don't calculate anything; take also into account the sample size) 10/12 Q. 6 [15] A city transportation company (CTC) conducted a small sample study to estimate utilization of

 

bus stops in the city. Three out of 20 city areas were sampled at random and then a few bus stops within

 

these areas. The numbers of people using the bust stop over a specified hour in a specified week day

 

(weekend excluded) are given in the following table:

 

Sampled

 

Area

 

1

 

2

 

3 Number of

 

bus stops

 

45

 

36

 

20 Number of

 

bus stops sampled

 

9

 

7

 

4 Sample

 

average

 

variance

 

82

 

30

 

80

 

20

 

56

 

30 (a) (i) Explain what type of the sample design is used here, (ii) Estimate the average number of people

 

using a bus stop in one hour for the week day per bus stop in the city. Is this estimator unbiased? (iii)

 

Place a bound on the error of the estimation (you don't need to complete calculation of the variance:

 

write down the formula, calculate all entries and replace required figures).

 

(b) (i) Estimate the total number of people using the bus stops in the city during one hour in the week

 

day. Is this estimator unbiased? Explain, (ii) Place a bound on the error of estimation (you don't

 

need to complete calculation of the variance: write down the formula, calculate all entries and

 

replace required figures).

 

(continued) 11/12 (c) Using the previous results, how could you estimate the total number of people who used bus stops in

 

the city over entire week (considering only daytime 7AM-7PM, and weekend excluded)? Would this

 

estimator be reasonable? Discus in short possible problems with the estimator.

 

(d) If the study is to be done again, would you recommend that areas be sampled with probabilities

 

proportional to their numbers of bus stops? Why or why not?

 

(e) Assuming the sample was obtained by PPS, estimate again the total number of people using the bus

 

stops in the city during one hour in the week day and the standard deviation of this estimate. Assume

 

there are 700 bus stops in the city. Total marks = 100

 

12/12

 


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