Midterm sta304 fall2021 PDF

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midterm exam of STA304 fall 2021 without solutions...

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UNIVERSITY OF TORONTO MISSISSAUGA Midterm Exam STA304H5F Surveys, Sampling and Observational Data Dr. Luai Al Labadi Duration – 110 Minutes Instructions      

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This test contains 4 short answer questions, Allowed Aids: Textbook, Lecture notes, Formula sheet, and Scientific calculators. Solutions MUST be presented neatly, completely, and with logical flow. Do not skip steps in your solutions or fail to describe what you are doing. Unless otherwise is stated, you may use any fact/formula that have been proved/utilized in lectures. Report the final answers correct to THREE decimals. Solutions will be submitted through crowdmark, meaning you will need to upload PDF, PNG or JPEG versions of your answers. Upload your solution to crowdmark, one question at a time. Be sure to submit your work before it is due (i.e., 6:00 pm). Late submissions will not be accepted.

With regard to remote learning and online courses, UTM wishes to remind students that they are expected to adhere to the Code of Behaviour on Academic Matters regardless of the course delivery method. By offering students the opportunity to learn remotely, UTM expects that students will maintain the same academic honesty and integrity that they would in a classroom setting. Potential academic offences in a digital context include, but are not limited to: Remote assessments: 1. Accessing unauthorized resources (search engines, chat rooms, Reddit, etc.) for assessments. 2. Using technological aids (e.g. software) beyond what is listed as permitted in an assessment. 3. Posting test, essay, or exam questions to message boards or social media. 4. Creating, accessing, and sharing assessment questions and answers in virtual “course groups.” 5. Working collaboratively, in-person or online, with others on assessments that are expected to be completed individually. All suspected cases of academic dishonesty will be investigated following procedures outlined in the Code of Behaviour on Academic Matters. If you have questions or concerns about what constitutes appropriate academic behaviour or appropriate research and citation methods, you are expected to seek out additional information on academic integrity from your instructor or from other institutional resources .

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STA304H5

Midterm Exam

Nov. 2, 2021

Problem 1. (6 marks: 1, 1, 2, 2) A researcher from UTM is interested in estimating 𝑝, the proportion of UTM students who are in favor of in-person classes in winter 2022. He divided the students into two strata: International and Domestic. The summary data are given in the following table.

Domestic International

Population Size

Sample Size

Sample Proportion: In favor of in-person classes

𝑁 = 10000

𝑛 = 1000

𝑝 = 0.64

𝑁 = 5000

𝑛 = 500

𝑝  = 0.52

(a) Find a point estimate for 𝑝. Show all the steps. Circle the final answer. (b) Place a bound on the error of estimation found in part (a). Show all the steps. Circle the final answer. (c) Estimate the total number of UTM students who are in favor of in-person classes in winter 2021. Place a bound on the error of this estimation. Hint: Use parts (a) and (b). Show all the steps. Circle the final answer. (d) Construct 95% confidence interval for the true proportion of UTM Domestic students who are in favor of in-person classes in winter 2021. Show all the steps. Circle the final answer.

Page 2 of 5

STA304H5

Midterm Exam

Nov. 2, 2021

Problem 2. (6 marks: 1, 1, 3, 1) A student from the department of Mathematical and Computational Sciences (MCS) is working on his project. He is interested in estimating 𝜇, the average study hours per week of MCS students. He divided the students into three strata: Mathematics, Computer Science, and Statistics. The summary data are given in the following table.

Mathematics Computer Science Statistics

Population Size

Sample Mean

Sample Standard Deviation

𝑁 = 1500

25

2

18

4

16

5

𝑁 = 3000 𝑁 = 2000

(a) Find a point estimate for 𝜇. Show all the steps. Circle the final answer. (b) Find a point estimator for the total study hours per week for statistics students. Show all the steps. Circle the final answer. (c) Find the overall sample size 𝑛 to estimate μ with a bound on the error of estimation equal to 2 hours using Neyman allocation. Show all the steps. Circle the final answer. (d) Use part (c) to find the sample size allocated to Computer Science? Show all the steps. Circle the final answer.

Page 3 of 5

STA304H5

Midterm Exam

Nov. 2, 2021

Problem 3. (6 marks: 1, 1, 2, 2) “STA3000Y - Advanced Theory of Statistics” is a graduate course taught at UTSG, which can only be taken by PhD students in the Department of Statistical Sciences. Assume the class consists of 𝑁 = 8 students. The final marks in the course are: 95, 66, 71, 93, 75, 88, 68, 100 (a) Find 𝜎  , the population variance. Show all the steps. Do not use any statistical package/excel here. For example, do not copy the final answer from R or from excel. Circle the final answer. (b) Based on a simple random sample (without replacement) of size 4 selected from this class, find the exact variance of the sample total. Show all the steps. Circle the final answer. (c) Based on a simple random sample (with replacement) of size 4 selected from this class, find the exact variance of the sample total. Show all the steps. Circle the final answer. (d) According to the graduate school, a student fails a course if a student scores less than 72. Based on a simple random sample (without replacement) of size 4 selected from this class, find the exact variance of the sample proportion of students who failed this course. Show all the steps. Circle the final answer. Hint: 𝑉(𝑝 ) =

 



󰇡



󰇢.

Page 4 of 5

STA304H5

Midterm Exam

Nov. 2, 2021

Problem 4. (7 marks: 1, 3, 3) Let 𝑦 , … , 𝑦 be a simple random sample of size 𝑛 from UTM students. Assume that the population size is 𝑁.

Suppose that 𝑦 equals 1 (if the participant is COVID-19 fully vaccinated) or 0 (if the participant is not COVID-19 fully vaccinated). Let 𝑝 be the population proportion and 𝑞 = 1 − 𝑝. Define 

𝑝 = ∑ 𝑦 and 𝑞 = 1 − 𝑝 . 

Some useful facts: 𝐸(𝑦 ) = 𝑝, 𝑉(𝑦 ) = 𝑝𝑞. You may use any needed fact about expected value/variance from previous courses such as STA256H5 without proof. Also, well-known facts from STA304H5 maybe used directly without proof. Only describe what you are doing. (a) If sampling is with replacement, find 𝑉(𝑝 ). Show all steps. (b) If sampling is with replacement, show that 𝑉(𝑝 ). Show all steps. (c) If sampling is without replacement, find E󰇡





 

is unbiased estimator of

󰇢. Show all steps. Simplify



your answer.

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