FINAL ASSIGNMENT Code: 1
Lecturer’s Signature
Program: IB, AC, MIS, BDA Course Code: MAT 1004
Course Title: Theory of Probability & Mat. Statistics
Asso. Prof. Nguyễn Hải Thanh Date: 27/12/2021 Time allowed: 3 hours
Department’s Signature Date: 07/01/2022
Starting at 6:30, 07/01/2022
Ending at 9:30, 07/01/2022 Date: 30/12/2021
Instructions to students:
1. At 6:30, 07/01/2022, each student is assigned a final assignment with code number that is
identical to the tens digit of his / her student code (for example: if a student code is 17071365
then the tens digit is 6, and he / she must choose to do .
Final Assignment MAT1004 Code 6)
2. Final assignment consists of 5 problems, students should write the answers by hand to these
problems in 10 pages. Solution of problem 1 is written in pages 1-2, problem 2 in pages 3-4,
problem 3 in pages 5-6, problem 4 in pages 7-8, problem 5 in pages 9-10. Each page is numbered
clearly by hand writing in the right-top corner. Some pages not used are left with blank space.
Students should write clearly his / her full name, student code and course clas s in the first row of
the first page (for example: Nguyễn Văn Thao.17007365.MAT100401).
3. To submit the final assignment, students use CamScanner to scan all the above 10 pages and save
them to a pdf file, starting from page 1, then page 2, 3, 4, 5, 6, 7, 8, 9 and 10.
4. Students should finish submitting the final assignment / the pdf file with the file name: student’s
full name.student code.course class (for example: Nguyen Van Thao.17007365.MAT100401) in
due time through MS Teams Assignment (do not be late in submitting the final assignment in due
time, 9:30, 07/01/2022).
5. Any type of plagiari m and any violation of the above instructions s
can CAUSE ZERO MARK for the final assignment.
Problem 1 (2 points). A successful venture-capital firm notes that it provides financing for
only 30% of the proposals it reviews. This year it reviews 120 proposals.
a/ What is the probability that at least 30 of the 120 proposals submitted will rece ve i financing?
b/ What is the probability that the number of proposals receiving financing will be between 40 and 60?
Problem 2 (2 points). To examine the effectiveness of its five annual advertising promotions,
a mail-order company has sent a questionnair
e to each of its customers, asking how many
of the previous year's promotions prompted orders that otherwise would not have been
made. The following table summarizes the data received, where the random variable X is
the number of promotions indicated in the customers' responses: x 0 1 2 3 4 5 P(X=x) y/100 0.25
1- (y/100+0.25+0.2+0.05+z/100) 0.2 0.05 z/100
In the above table: y, z are the last two digits of your student code (for example: if a student
code is 17071365 then y = 6, z = 5). A previous analysi
s of historical data found that the mean value of orders for promotional
goods is $20, with the company earning a gross profit of 20% on each order. It is assumed
that customer behavior next year will be the same as last year, and the fixed cost of
conducting the five promotions next year is estimated to be $15000, with a variable cost of
$3 per customer for mailing and handling costs.
a/ Find expected value of X, variance of X and the expected gross profit per customer next year.
b/ How large a customer base must the company have to cover the cost of the promotions?
Problem 3 (2 points). In a public opinion survey, 60 out of a sample of 70 high-income
voters and 90 out of a sample of 100 low-income voters supported the introduction of a new national security tax.
a/ Estimate, with 95% confidence level, the true proportion of low-income people who will
vote for the introduction of the tax.
b/ Can we conclude at the 5% level of significance that the proportion of high-income voters
favoring the new security tax is lower than that of low-income voters?
Problem 4 (2 points). A restaurant located in an office building decides to adopt a new
strategy for attracting customers. Every week it advertises in the city newspaper. In the 6
weeks immediately prior to the advertising campaign, the weekly grosse s wer e 350, 320,
307, 398, 420, 335 (in million VND). In the eight weeks after the campaign began, the
weekly grosse was 488, 301, 276 380, 421, 425 (in million VND). s
a/ Test with a = 0.05 to determine whether there is enough evidence to conclude that
expected weekly gross after the advertising campaign differs from 382.
b/ Given that the weekly grosses are normally distributed, can we conclude that the
advertising campaign helped in improving weekly grosse ? s
(Assuming unequal variances and using a = 0.05)
Problem 5 (2 points). The different interest rates charged by some financial institutions may
reflect how stringent their standards are for their loan appraisals: the lower the rate, the
higher the standards, and hence, the lower the default rate. The data below were collected
from a sample of eight financial companies selected at random. Financial company 1 2 3 4 5 6 7 8 Interest rate (in %) x 7.0 6.6 6.0 8.5 8.0 7.5 6.5 7.0
Default rate (in 1000 loans) y 38 40 35 46 48 39 36 37
a/ Find the least square regression line.
b/ Perform the test H0: b1 = 0, HA: b1 ¹ 0 (using α = 0.05) to determine whether there is
enough evidence to infer that a linear dependence exists between i terest n rate (x) and default rate (y).