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.

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 competitors’ analysis from two of their closest competitors.

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, “Competitor A spends on the average the most amount of money in the past 5 years in the industry.” 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.

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 “Enjoys shopping for clothing” and “Female” 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.

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) “Advertising causes our sales revenue to increase!” (2 marks)

(ii) “Our sales revenue is only affected by the change in advertising costs!”

(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 individual’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.

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: µ ≤ 23 grams Ha: µ > 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)

BMI_W for women is not significantly greater than average

Standard Normal Distribution

Z	.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

Left-tailed test a

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
µ	1.282	1.645	1.960	2.326	2.576	3.291

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