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# Business Statistics Sample Assignment

SECTION A (COMPULSORY)

Answer ALL questions from this Section.

QUESTION 1 (25 Marks)

XYZ, a retail store, operates in a highly competitive market with many competitors. The manager has been tasked to do some statistical analysis on the monthly expenses over the last five years. Table 1.1 shows the data that was collected.

 Monthly Expenses (S\$) Frequency 0 and less than 150 5 150 and less than 300 8 300 and less than 450 15 450 and less than 600 9 600 and less than 750 17 750 and less than 900 6 Total 60

Table 1.1

(a) For the above grouped data, calculate:

(i) Mean 482.50 (2 marks)

(ii) Median 483.33 (2 marks)

(iii) Mode 663.16 (2 marks)

(iv) Sample variance 48,883.47 (3 marks)

(v) Coefficient of Variation 45.82% (3 marks)

(b) Explain with the aid of a diagram, the skewness of the monthly expenses distribution. Left Skewed (3 marks)

(c) Which measure of central tendency (mean, median or mode) would you choose for the above grouped data? Explain. median (3 marks)

(d) If 80% of the monthly expenses is below \$600, then it indicates that the XYZ has not been overspending. From the above grouped data, would you say that XYZ has been overspending? Show your calculation. XYZ has been overspending (3 marks)

(e) The manager wants to do a competitorsâ€™ analysis from two of their closest competitors.

 Statistical Measure Competitorsâ€™ Monthly Expenses Competitor A Competitor B Mean 430 450 Median 450 430 Mode 420 470 Standard Deviation 200 250

Table 1.2

Using the information in Table 1.2, compare XYZ Company with competitor A and competitor B, comment on the following:

(i) Which company has the most variation? Explain. B (3 marks)

(ii) The manager of XYZ Company made the following statement, â€śCompetitor A spends on the average the most amount of money in the past 5 years in the industry.â€ť Is this an Inferential Statistics or a Descriptive Statistics? (1 mark)

Inferential Statistics

#### QUESTION 2 (25 Marks)

(a) A sample of 500 respondents was surveyed in a large sub-urban area to determine various information concerning consumer behaviour. Table 2.1 shows the information obtained.

 Male Female Enjoys shopping for clothes 136 224 Does not enjoy shopping for clothes 104 36

Table 2.1

(i) Compute the probability that a respondent is a female and enjoys shopping for clothes. 0.448 (2 marks)

(ii) What is the probability that a respondent chosen enjoys shopping for clothing given that the respondent is a female? 0.8615 (3 marks)

(iii) Are the categories of â€śEnjoys shopping for clothingâ€ť and â€śFemaleâ€ť statistically independent events? Explain your answer. No (5 marks)

(b) A building contractor claims that he is able to renovate a five room HDB flat in mean time of 40 days with a standard deviation of 5 days. Assume from past experience, that the days to complete similar flats are normally distributed with mean and standard deviation as estimated above.

(i) What proportion of five room HDB flats will be completed in less than 35 days? 0.1587 (2 marks)

(ii) What proportion of five room HDB flats will be completed in between 28 and 32 days? 0.0466 (3 marks)

(iii) What is the minimum number of days to complete the renovation for the slowest 10% of five room HDB flats? 46.4 days (4 marks)

(c) An advertising executive is studying television viewing habits of married men and women during prime time hours. On the basis of past viewing records, the executive has determined that during prime time, husbands are watching television 60% of the time. It has also been determined that when the husband is watching television, 40% of the time the wife is also watching. When the husband is not watching television, 30% of the time the wife is watching television. Find the probability that

(i) the wife is watching television during prime time hours. 0.36 (2 marks)

(ii) if the wife is watching television, the husband is also watching television.

0.667 (4 marks)

SECTION B

Answer any TWO questions from this Section.

QUESTION 3 (25 Marks)

A manager of a bank claims that 80% of their customers use the ATM machine at least once a month to withdraw cash. The average amount withdrawn by these customers is \$1,000 with a standard deviation of \$10.

The average amount withdrawn by the remaining 20% customers is \$200 with a standard deviation of \$5. The maximum number of customers using the ATM machine was found to be 200. A random sample of 36 customers was selected.

(a) Identify the variable(s) for the above and state the type and level of measurement for the variable(s) identified. (4 marks)

(b) What is the probability that in a random sample of 5 customers, 80% will use the ATM machine at least once a month? 0.4096 (3 marks)

(c) What is the probability that in a random sample of 5 customers, at least 80% will use the ATM machine at least once a month? What assumption is necessary for the distribution? 0.7373 (5 marks)

(d) Calculate the average amount of cash per withdrawal. \$840 (3 marks)

(e) Calculate the probability that the amount withdrawn by a customer is at most \$845. 0.7123 (4 marks)

(f) Calculate the probability that the average amount withdrawn by a customer is at most \$842. 0.9292 (4 marks)

(g) Calculate the probability that the average withdrawn by a customer is at least \$840. 0.5 (2 marks)

QUESTION 4 (25 Marks)

A sales manager wants to find out the relationship between the amount his shop spends on advertisements and the sales revenue. He took note of his spending on advertisements and sales revenue earned for the past 7 months.

 X, Amount spent on Advertisements (S\$â€™000) Y, Sales Revenue (S\$â€™0000) 230 152 143 78 177 90 259 142 374 150 132 79 265 151

Table 4.1

(a) Determine and interpret the regression line. = 39.2543 + 0.3590X (10 marks)

(b) Calculate and interpret the correlation coefficient. 0.8442 (5 marks)

(c) Estimate the amount of sales revenue if his shop spends \$140,050 on advertisements. \$895,322.50 (2 marks)

(d) The sales manager realised that he made a wrong calculation on the sales revenue amount. All sales revenue amount should decrease by \$10,000. Without any further calculations, state any change expected on the regression coefficients. (4 marks)

a will decrease by 1, no change to b

(e) Discuss and comment on the accuracy of the following statements made by the manager after conducting the correlation and regression analysis:

(i) â€śAdvertising causes our sales revenue to increase!â€ť (2 marks)

(ii) â€śOur sales revenue is only affected by the change in advertising costs!â€ť

(2 marks)

See lecture notes

QUESTION 5 (25 Marks)

The Body Mass Index (BMI) is a measure for human body shape based on an individual's weight and height. It is a standardised estimate of an individualâ€™s relative body fat calculated from his or her height and weight. You took the measurement of BMI for 25 men and 25 women from the School of Business and Accountancy and calculated the following statistics in Table 5.1.

 Gender Mean Variance Women 25 36 Men 26 9

Table 5.1

The upper limit for a normal BMI for an adult is 23. You may assume that the level of significance is 5%.

(a) Based on the above information, can you say that either the group of women or group of men have above normal BMIs? Explain. (6 marks)

Female: (22.5232, 27.4768)

Male: ( 24.7616, 27.2384 )

(b) Construct the hypothesis test statements for part (a). Briefly explain the construction in Singapore context. Ho: Âµ â‰¤ 23 grams Ha: Âµ > 23 grams (6 marks)

(c) What is the critical value when testing the hypothesis? 1.711 (3 marks)

(d) What are the limits of the acceptance region when testing the hypothesis? (4 marks)

Female: 27.0532

Male: 27.0266

(e) What is the standardised value of the sample mean for both gender? (2 marks)

Female: 1.667

Male: 5.00

(f) Do the sample results provide sufficient evidence to indicate that the mean BMI for an adult is above normal? You are required to explain for both genders. (4 marks)

BMIW for women is not significantly greater than average

BMIM for men is significantly greater than average

.00

.01

.02

.03

.04

.05

.06

.07

.08

.09

0.0

.0000

.0040

.0080

.0120

.0160

.0199

.0239

.0279

.0319

.0359

0.1

.0398

.0438

.0478

.0517

.0557

.0596

.0636

.0675

.0714

.0753

0.2

.0793

.0832

.0871

.0910

.0948

.0987

.1026

.1064

.1103

.1141

0.3

.1179

.1217

.1255

.1293

.1331

.1368

.1406

.1443

.1480

.1517

0.4

.1554

.1591

.1628

.1664

.1700

.1736

.1772

.1808

.1844

.1879

0.5

.1915

.1950

.1985

.2019

.2054

.2088

.2123

.2157

.2190

.2224

0.6

.2257

.2291

.2324

.2357

.2389

.2422

.2454

.2486

.2517

.2549

0.7

.2580

.2611

.2642

.2673

.2704

.2734

.2764

.2794

.2823

.2852

0.8

.2881

.2910

.2939

.2967

.2995

.3023

.3051

.3078

.3106

.3133

0.9

.3159

.3186

.3212

.3238

.3264

.3289

.3315

.3340

.3365

.3389

1.0

.3413

.3438

.3461

.3485

.3508

.3531

.3554

.3577

.3599

.3621

1.1

.3643

.3665

.3686

.3708

.3729

.3749

.3770

.3790

.3810

.3830

1.2

.3849

.3869

.3888

.3907

.3925

.3944

.3962

.3980

.3997

.4015

1.3

.4032

.4049

.4066

.4082

.4099

.4115

.4131

.4147

.4162

.4177

1.4

.4192

.4207

.4222

.4236

.4251

.4265

.4279

.4292

.4306

.4319

1.5

.4332

.4345

.4357

.4370

.4382

.4394

.4406

.4418

.4429

.4441

1.6

.4452

.4463

.4474

.4484

.4495

.4505

.4515

.4525

.4535

.4545

1.7

.4554

.4564

.4573

.4582

.4591

.4599

.4608

.4616

.4625

.4633

1.8

.4641

.4649

.4656

.4664

.4671

.4678

.4686

.4693

.4699

.4706

1.9

.4713

.4719

.4726

.4732

.4738

.4744

.4750

.4756

.4761

.4767

2.0

.4772

.4778

.4783

.4788

.4793

.4798

.4803

.4808

.4812

.4817

2.1

.4821

.4826

.4830

.4834

.4838

.4842

.4846

.4850

.4854

.4857

2.2

.4861

.4864

.4868

.4871

.4875

.4878

.4881

.4884

.4887

.4890

2.3

.4893

.4896

.4898

.4901

.4904

.4906

.4909

.4911

.4913

.4916

2.4

.4918

.4920

.4922

.4925

.4927

.4929

.4931

.4932

.4934

.4936

2.5

.4938

.4940

.4941

.4943

.4945

.4946

.4948

.4949

.4951

.4952

2.6

.4953

.4955

.4956

.4957

.4959

.4960

.4961

.4962

.4963

.4964

2.7

.4965

.4966

.4967

.4968

.4969

.4970

.4971

.4972

.4973

.4974

2.8

.4974

.4975

.4976

.4977

.4977

.4978

.4979

.4979

.4980

.4981

2.9

.4981

.4982

.4982

.4983

.4984

.4984

.4985

.4985

.4986

.4986

3.0

.4987

.4987

.4987

.4988

.4988

.4989

.4989

.4989

.4990

.4990

3.1

.4990

.4991

.4991

.4991

.4992

.4992

.4992

.4992

.4993

.4993

3.2

.4993

.4993

.4994

.4994

.4994

.4994

.4994

.4995

.4995

.4995

3.3

.4995

.4995

.4995

.4996

.4996

.4996

.4996

.4996

.4996

.4997

3.4

.4997

.4997

.4997

.4997

.4997

.4997

.4997

.4997

.4997

.4998

3.5

.4998

.4998

.4998

.4998

.4998

.4998

.4998

.4998

.4998

.4998

3.6

.4998

.4998

.4999

.4999

.4999

.4999

.4999

.4999

.4999

.4999

3.7

.4999

.4999

.4999

.4999

.4999

.4999

.4999

.4999

.4999

.4999

3.8

.4999

.4999

.4999

.4999

.4999

.4999

.4999

.4999

.4999

.4999

3.9

.5000

.5000

.5000

.5000

.5000

.5000

.5000

.5000

.5000

.5000

## Right-tailed test a

Student's t Distribution

 1 - a

## Two-tailed test a

 Appendix 3
##### Student's t Distribution
 degrees of freedom Confidence Level ( 1 - a ) 0.80 0.90 0.95 0.98 0.99 0.999 Level of Significance for One-Tailed Test ( a ) 0.10 0.05 0.025 0.01 0.005 0.0005 Level of Significance for Two-Tailed Test ( a ) 0.20 0.10 0.05 0.02 0.01 0.001 1 3.078 6.314 12.706 31.821 63.657 636.619 2 1.886 2.920 4.303 6.965 9.925 31.599 3 1.638 2.353 3.182 4.541 5.841 12.924 4 1.533 2.132 2.776 3.747 4.604 8.610 5 1.476 2.015 2.571 3.365 4.032 6.869 6 1.440 1.943 2.447 3.143 3.707 5.959 7 1.415 1.895 2.365 2.998 3.499 5.408 8 1.397 1.860 2.306 2.896 3.355 5.041 9 1.383 1.833 2.262 2.821 3.250 4.781 10 1.372 1.812 2.228 2.764 3.169 4.587 11 1.363 1.796 2.201 2.718 3.106 4.437 12 1.356 1.782 2.179 2.681 3.055 4.318 13 1.350 1.771 2.160 2.650 3.012 4.221 14 1.345 1.761 2.145 2.624 2.977 4.140 15 1.341 1.753 2.131 2.602 2.947 4.073 16 1.337 1.746 2.210 2.583 2.921 4.015 17 1.333 1.740 2.110 2.567 2.898 3.965 18 1.330 1.734 2.101 2.552 2.878 3.922 19 1.328 1.729 2.093 2.539 2.861 3.883 20 1.325 1.725 2.086 2.528 2.845 3.850 21 1.323 1.721 2.080 2.518 2.831 3.819 22 1.321 1.717 2.074 2.508 2.819 3.792 23 1.319 1.714 2.069 2.500 2.807 3.768 24 1.318 1.711 2.064 2.492 2.797 3.745 25 1.316 1.708 2.060 2.485 2.787 3.725 26 1.315 1.706 2.056 2.479 2.799 3.707 27 1.314 1.703 2.052 2.473 2.771 3.690 28 1.313 1.701 2.048 2.467 2.763 3.674 29 1.311 1.699 2.045 2.462 2.756 3.659 30 1.310 1.697 2.042 2.457 2.750 3.646 40 1.303 1.684 2.021 2.423 2.704 3.551 60 1.296 1.671 2.000 2.390 2.660 3.460 120 1.289 1.658 1.980 2.358 2.617 3.373 Âµ 1.282 1.645 1.960 2.326 2.576 3.291
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