Merged V2 AMA1501 Test Sem2 2020 21 PDF

Preview

Summary

it is the paper for this couse...

Description

The Hong Kong Polytechnic University Department of Applied Mathematics AMA1501 Introduction to Statistics for Business/ AMA1502 Introduction to Statistics 2020/21 Semester Two Test This paper has FIVE questions. Attempt ALL questions. Time Allowed: 2 hours 15 minutes (10:00 to 12:15) • Your ”Honour Statement” in Question 1 should be signed. The Question 1 is in the last item. You can prepare it before the Home Test. You must do the question 1, otherwise, your solution paper will not be marked. • Hand write solutions showing all proper steps on A4 paper using black or blue pen. Typing is not allowed. • Each solution to one question must be contained in only one pdf file, i.e., please combine the pictures of your solution to each question (in the correct order) into ONE pdf file. Therefore, for the test , you should upload 5 different pdf files for each of the 5 questions of the Test. It is fine if the solution of one question contains more than one page. Rotate the page if the picture is not in the right direction, when you photograph your solutions. • Name the file as StudentID-Q1 (Q2,. . . ,Q5). • Make sure the uploaded file fulfills all above requirements, otherwise at most 5 marks will be deducted. • Scan or take photo of your solutions page by page, make sure that the solutions are legible. Rotate the page if the picture is not in the right direction. No marks will be given if the marker can not read your handwriting. Please use CamScanner app to turn your picture or scan into a pdf file (please see ”Assignment” instructions on how to use CamScanner). • After uploading, please click back to review your uploaded files. Make sure it is correct, before you leave. • You are strongly recommended to start uploading your solutions onto Blackboard before HKT 10:45, after you finish and check the solutions. YOU WILL BE DEDUCTED MARKS if you submit any of your answers AFTER 12:15pm. • The Blackboard Submission links to the Midterm Test will disappear at EXACTLY 12.45pm • NO EMAIL SUBMISSIONS will be accepted (for any reason)!!!

Students will obtain a mark of 0 if they submit their midterm test answers

by email. • There will a mark deduction of -5 marks for every 5 minute interval after the deadline of 12:15pm on Blackboard, with no exceptions, i.e.,: if you submit after the deadline on Blackboard between 1. 12:15-12:19 (-5 marks); 2. 12:20-12:24 (-10marks); 3. 12:25-12:29 (-15 marks); 4. 12:30-12:34 (-20 marks); 5. 12:35-12:39 (-25 marks); 6. 12:40-12:44 (-30 marks); 7. If you submit after 12:45 on Blackboard or by email - TEST WILL NOT BE MARKED (get 0/100 marks). • During 10:00-12:15, if there is any record show you login the Blackboard or receive questions from other channels, you will be considered attending the Home Test. • In your solutions, you should have each of the following elements: (i) technical terms and variables in problem are well defined; (ii) important steps in the solution are clearly shown and justified; (iii) there is an obvious conclusion summarizing each questions answer. • You may assume 4 digits after the decimal in all your numerical answers. • Formula Sheets, Standardised Normal Distribution table and Student’s t-distribution table are attached.

Do not turn over this page until you are told to do so.

1

1. (a) Please copy the following honour statement (in bold face) as the solution to your Question 1 in your midterm test: ”By submitting the Take Home Midterm Test through the Blackboard system, I affirm on my honour that I am aware of the Regulations on Academic Integrity in Student Handbook and have not given nor received any unauthorized aid to/from any person or persons.” (b) At the end of your honour statement, please write your FULL name, Student ID, and your signature.

FOR QUESTIONS 2,3,4 and 5 BELOW, • the NUMBER χ in the question is your LAST NUMBER OF YOUR STUDENT I.D., • for example, if your student I.D. number is 20123456D, then χ = 6; or • for example, if your student I.D. number is 20123459D, then χ = 9, etc...

2. The Hong Kong Health Authority obtained the following sample time data for completing a vaccination procedure (in minutes): Time (in minutes)

Frequency

1– below 4

2

4– below 7

5

7– below 10

9

10– below 13

5

(a) Compute the mean and standard deviation of the time for completing the vaccination procedure. (6 marks) (b) Compute the median and mode time for completing the vaccination procedure.

(6 marks)

(c) Calculate the coefficient of variation and Pearson’s 2nd coefficient of skewness and comment on the shape of the distribution.

(7 marks)

(d) Construct the 90% confidence interval for the population mean time of completing the vaccination procedure. State any assumption(s) and/or approximation(s).

2

(6 marks)

3. (a)

i. There are χ elderly, 3 teachers, and 5 health care workers waiting in a line to receive their vaccination. How many different arrangements are possible if only the elderly must stand together?

(3 marks)

ii. A box contains (χ + 5) SinoVac, 5 BioNTech, and 9 AstraZeneca vaccines. If 9 vaccines are picked out at random without replacement, determine the probability that 4 SinoVac, 3 BioNTech and 2 AstraZeneca vaccines are drawn.

(4 marks)

(b) Hong Kong adults were asked whether or not they have ever shopped online during COVID-19 pandemic. The following table gives a two-way classification of the responses. Have Never Shopped

Have Shopped

Male

30

90

Female

20

(60+ χ)

i. If one adult is selected at random, find the probability that this adult is a female or have shopped online.

(4 marks)

ii. Given that the selected adult is a female, what is the probability that she has never shopped online?

(2 marks)

iii. Let M denote the event that the adult is male, and ¯S denote the event that the adult has ¯ independent? Justify your answer. (4 marks) never shopped online. Are events M and S (c) Suppose (5 + χ)% of the residents

have the COVID-19 disease. There is

a diagnostic test to detect the COVID-19 disease, but it is not very accurate. Historical evidence shows that if a person actually has the disease, the probability that the test will indicate the presence of the disease is 0.8. For a person who actually does not have the disease, the probability for the test will indicate the presence of disease is 0.25. If a person is selected randomly from this lockdown building to perform the test, and the test results indicate the disease is present, what is the probability that this person actually does not have the disease?

(8 marks)

4. (a) The height of ”CokaKola” soft drink cans are normally distributed with mean 3.7 centimeters and a standard deviation of 0.3 centimeter. i. Find the probability that a randomly selected CokaKola can has the height below 3.55 centimeter.

(4 marks)

ii. Suppose that a CokaKola can is called ”defective” if it has height below 3.55 centimeter or above 3.95 centimeter. Find the probability that in a sample of 10 randomly selected metal pieces, there are less than 3 defective CokaKola cans.

(6 marks)

iii. Using the normal approximation to the binomial distribution, estimate the probability that out of 110 randomly selected cans of CokaKola, more than 25 of them will have height above 3.95cm?

(5 marks)

(b) Suppose that, on average, the number of car accidents in Hong Kong are 6 per day (which has 24 hours). i. Determine the probability that at least χ accidents will occur in 12 hour period. (4 marks)

3

ii. In a two days, if there were 8 car accidents in Hong Kong, what is the probability that 2 accidents have occurred in the first 12 hours of the two days? (6 marks) 5. (a) The number of shrimps in random sample of 8 plates of Yangzhou Fried Rice at the PolyU Student Canteen are listed below: (χ + 1), 4, 5, 6, 7, 8, 8, 9. Find the 99% confidence interval in the mean number shrimp per plate of Yangzhou Fried rice that the PolyU Student Canteen. State your assumption(s) and/or approximation(s). (6 marks) (b) Ms. Louisa Cheng is considering running in the election for the PolyU Student Union President. Before deciding to run in the election, she decides to conduct a poll of 400 PolyU students to what her chances are that she will win. Out of the 400 PolyU students polled, 250 students said they would vote for her, as opposed to the other candidates. i. Find the 90% confidence interval for the population proportion of PolyU students who will vote for Ms. Louisa Cheng. State any assumption(s) and/or approximation(s). (6 marks) ii. Based on the confidence interval you found in part (a), can you say that Ms. Louisa Cheng has more than a 50% chance to be President of the PolyU Student Union? Please justify your answer.

(2 marks)

(c) Suppose you want to find if there is any diffference between the mean (average) lifetime of two types of batteries: ”Energiser” and ”Durasell” batteries. Suppose your random sample of 130 batteries has an average lifetime of 36.8 days and a random sample of 150 batteries averages 28.7 days. Assume by prior research that the population standard deviations for the lifetime of Energiser and Duracell batteries are 1.9 days and 2.5 days, respectively. Find the 94% confidence interval in the difference between the mean (average) lifetime of the two types of batteries? State any assumption(s) and/or approximation(s). ****END****

4

(8 marks)

Table of the Standardized Normal Distribution P The table gives the probability P = Pr(Z > z )

0

where Z ~ N(0,1).

z

z 0.0 0.1 0.2 0.3 0.4

.00 0.5000 0.4602 0.4207 0.3821 0.3446

.01 0.4960 0.4562 0.4168 0.3783 0.3409

.02 0.4920 0.4522 0.4129 0.3745 0.3372

.03 0.4880 0.4483 0.4090 0.3707 0.3336

.04 0.4840 0.4443 0.4052 0.3669 0.3300

.05 0.4801 0.4404 0.4013 0.3632 0.3264

.06 0.4761 0.4364 0.3974 0.3594 0.3228

.07 0.4721 0.4325 0.3936 0.3557 0.3192

.08 0.4681 0.4286 0.3897 0.3520 0.3156

.09 0.4641 0.4247 0.3859 0.3483 0.3121

0.5 0.6 0.7 0.8 0.9

0.3085 0.2743 0.2420 0.2119 0.1841

0.3050 0.2709 0.2389 0.2090 0.1814

0.3015 0.2676 0.2358 0.2061 0.1788

0.2981 0.2643 0.2327 0.2033 0.1762

0.2946 0.2611 0.2296 0.2005 0.1736

0.2912 0.2578 0.2266 0.1977 0.1711

0.2877 0.2546 0.2236 0.1949 0.1685

0.2843 0.2514 0.2206 0.1922 0.1660

0.2810 0.2483 0.2177 0.1894 0.1635

0.2776 0.2451 0.2148 0.1867 0.1611

1.0 1.1 1.2 1.3 1.4

0.1587 0.1357 0.1151 0.0968 0.0808

0.1562 0.1335 0.1131 0.0951 0.0793

0.1539 0.1314 0.1112 0.0934 0.0778

0.1515 0.1292 0.1093 0.0918 0.0764

0.1492 0.1271 0.1075 0.0901 0.0749

0.1469 0.1251 0.1056 0.0885 0.0735

0.1446 0.1230 0.1038 0.0869 0.0721

0.1423 0.1210 0.1020 0.0853 0.0708

0.1401 0.1190 0.1003 0.0838 0.0694

0.1379 0.1170 0.0985 0.0823 0.0681

1.5 1.6 1.7 1.8 1.9

0.0668 0.0548 0.0446 0.0359 0.0287

0.0655 0.0537 0.0436 0.0351 0.0281

0.0643 0.0526 0.0427 0.0344 0.0274

0.0630 0.0516 0.0418 0.0336 0.0268

0.0618 0.0505 0.0409 0.0329 0.0262

0.0606 0.0495 0.0401 0.0322 0.0256

0.0594 0.0485 0.0392 0.0314 0.0250

0.0582 0.0475 0.0384 0.0307 0.0244

0.0571 0.0465 0.0375 0.0301 0.0239

0.0559 0.0455 0.0367 0.0294 0.0233

2.0 2.1 2.2 2.3 2.4

0.0228 0.0179 0.0139 0.0107 0.00820

0.0222 0.0174 0.0136 0.0104 0.00798

0.0217 0.0170 0.0132 0.0102 0.00776

0.0212 0.0207 0.0166 0.0162 0.0129 0.0126 0.00990 0.00964 0.00755 0.00734

0.0202 0.0197 0.0158 0.0154 0.0122 0.0119 0.00939 0.00914 0.00714 0.00695

0.0192 0.0188 0.0150 0.0146 0.0116 0.0113 0.00889 0.00866 0.00676 0.00657

0.0183 0.0143 0.0110 0.00842 0.00639

2.5 2.6 2.7 2.8 2.9

0.00621 0.00466 0.00347 0.00256 0.00187

0.00604 0.00453 0.00336 0.00248 0.00181

0.00587 0.00440 0.00326 0.00240 0.00175

0.00570 0.00427 0.00317 0.00233 0.00169

0.00554 0.00415 0.00307 0.00226 0.00164

0.00539 0.00402 0.00298 0.00219 0.00159

0.00523 0.00391 0.00289 0.00212 0.00154

0.00508 0.00379 0.00280 0.00205 0.00149

0.00494 0.00368 0.00272 0.00199 0.00144

0.00480 0.00357 0.00264 0.00193 0.00139

3.0 3.1 3.2 3.3 3.4

0.00135 0.00097 0.00069 0.00048 0.00034

0.00131 0.00094 0.00066 0.00047 0.00032

0.00126 0.00090 0.00064 0.00045 0.00031

0.00122 0.00087 0.00062 0.00043 0.00030

0.00118 0.00084 0.00060 0.00042 0.00029

0.00114 0.00082 0.00058 0.00040 0.00028

0.00111 0.00079 0.00056 0.00039 0.00027

0.00107 0.00076 0.00054 0.00038 0.00026

0.00104 0.00074 0.00052 0.00036 0.00025

0.00100 0.00071 0.00050 0.00035 0.00024

3.5 3.6 3.7 3.8 3.9

0.00023 0.00016 0.00011 0.00007 0.00005

0.00022 0.00015 0.00010 0.00007 0.00005

0.00022 0.00015 0.00010 0.00007 0.00004

0.00021 0.00014 0.00010 0.00006 0.00004

0.00020 0.00014 0.00009 0.00006 0.00004

0.00019 0.00013 0.00009 0.00006 0.00004

0.00019 0.00013 0.00008 0.00006 0.00004

0.00018 0.00012 0.00008 0.00005 0.00004

0.00017 0.00012 0.00008 0.00005 0.00003

0.00017 0.00011 0.00008 0.00005 0.00003

Table of the Student's t-distribution The table gives the values of tα ;ν where

α

Pr(Tν > t α; ν ) = α , with ν degrees of freedom

tα ; ν

0.1

0.05

0.025

0.01

0.005

0.001

0.0005

1 2 3 4 5

3.078 1.886 1.638 1.533 1.476

6.314 2.920 2.353 2.132 2.015

12.076 4.303 3.182 2.776 2.571

31.821 6.965 4.541 3.747 3.365

63.657 9.925 5.841 4.604 4.032

318.310 22.326 10.213 7.173 5.893

636.620 31.598 12.924 8.610 6.869

6 7 8 9 10

1.440 1.415 1.397 1.383 1.372

1.943 1.895 1.860 1.833 1.812

2.447 2.365 2.306 2.262 2.228

3.143 2.998 2.896 2.821 2.764

3.707 3.499 3.355 3.250 3.169

5.208 4.785 4.501 4.297 4.144

5.959 5.408 5.041 4.781 4.587

11 12 13 14 15

1.363 1.356 1.350 1.345 1.341

1.796 1.782 1.771 1.761 1.753

2.201 2.179 2.160 2.145 2.131

2.718 2.681 2.650 2.624 2.602

3.106 3.055 3.012 2.977 2.947

4.025 3.930 3.852 3.787 3.733

4.437 4.318 4.221 4.140 4.073

16 17 18 19 20

1.337 1.333 1.330 1.328 1.325

1.746 1.740 1.734 1.729 1.725

2.120 2.110 2.101 2.093 2.086

2.583 2.567 2.552 2.539 2.528

2.921 2.898 2.878 2.861 2.845

3.686 3.646 3.610 3.579 3.552

4.015 3.965 3.922 3.883 3.850

21 22 23 24 25

1.323 1.321 1.319 1.318 1.316

1.721 1.717 1.714 1.711 1.708

2.080 2.074 2.069 2.064 2.060

2.518 2.508 2.500 2.492 2.485

2.831 2.819 2.807 2.797 2.787

3.527 3.505 3.485 3.467 3.450

3.819 3.792 3.767 3.745 3.725

26 27 28 29 30

1.315 1.314 1.313 1.311 1.310

1.706 1.703 1.701 1.699 1.697

2.056 2.052 2.048 2.045 2.042

2.479 2.473 2.467 2.462 2.457

2.779 2.771 2.763 2.756 2.750

3.435 3.421 3.408 3.396 3.385

3.707 3.690 3.674 3.659 3.646

40 60 120 ∞

1.303 1.296 1.289 1.282

1.684 1.671 1.658 1.645

2.021 2.000 1.980 1.960

2.423 2.390 2.358 2.326

2.704 2.660 2.617 2.576

3.307 3.232 3.160 3.090

3.551 3.460 3.373 3.291

α ν

Formula sheet 1. Sample Statistics: Ungrouped data

Grouped data

Σx n

Σf x Σf

Arithmetic Mean

Standard Deviation

s

(x − x) Σ ¯ 2 n−1

=

s

Σ2 − (Σ x x)2 /n n−1

s

v u (Σ f x)2 2 uΣ tf x − f Σ f (x − x Σ ¯ )2 = f −1 Σ f −1 Σ

2. Probability Distributions: (a) Binomial P (r) = n Cr pr (1 − p)n−r (b) Poisson P (r) =

e−λ λr r!

3. Standard Errors: σ (a) Mean √ n (b) Proportion

r

p(1 − p) n

(c) Difference between means

s

σ12 σ22 + n1 n2

(d) Difference between proportions

r

p1 (1 − p1 ) p2 (1 − p2 ) + n2 n1

4. Test Statistics: (a) Z = Z=

(b) t = t=

x¯ − µ √ σ/ n

(one sample)

(¯ x1 − x¯2 ) − (µ1 − µ2 ) s σ12 σ22 + n1 n2 x¯ − µ √ s/ n

(two samples)

(one sample)

(¯ x1 − x¯2 ) − (µ1 − µ2 ) r 1 1 sp + n1 n2

(two samples) where sp2 =

5

(n1 − 1)s12 + (n2 − 1)s22 n1 + n2 − 2

(c) χ2 =

X (O − E)2 E

5. Correlation and Regression: (a) Product moment correlation coefficient r = q

nΣxy − ΣxΣy   nΣx2 − (Σx)2 nΣy 2 − (Σy)2

(b) Spearman’s rank correlation coefficient Rs = 1 −

6Σd2 n(n2 − 1)

(c) Least squares regression line y = a + bx b=

nΣxy − ΣxΣy nΣx2 − (Σx)2

6

a=

Σy bΣx − n n...