Mathematics Accounting - Math Problem Example

Question 1: 50 marks a) For both SAL04 and SAL 09 separately: i) Construct a histogram from the frequency information above. ii) Construct the cumulative frequency diagram, then use it to estimate the median, and the two quartiles Q1 and Q3 Using the cumulative frequency diagram following are the required statistics for the respective salary ranges: SAL04: Median 17,750 Q1 15,375 Q3 20,125 SAL09: Median 27,500 Q1 22,750 Q3 32,250 iii) Using the frequency table, determine the mean, median, standard deviation and inter-quartile range for the salaries. Following are the required statistics using the frequency table: SAL04: Mean 17,750 Median 18,000 Std Dev 2,958 Inter quartile range 4,750 SAL09: Mean 27,500 Median 28,000 Std Dev 5,916 Inter quartile range 9,500 iv) From the individuals’ data in the file Salary Survey.xls use Excel to evaluate the mean, median, standard deviation and inter-quartile range for the SAL04 and SAL09. We get the following values using Excel:   SAL04 SAL09 Mean 17,904 34,539 Median 18,007 23,625.82 Std Dev 949.9 28,961 Inter quartile range 4,750 5,335 c) The two curves can be shown appearing slowly along the line and then stabilizing. During the initial stage, i.e., at the time of graduation, the rate of salary growth is slow. Rate of growth then increases rapidly 5 years after graduation. After some time the growth rate slows down. Question 2 14 marks A company has a large number of accountants. Records show that 30 are qualified to work with SMEs (small or medium sized enterprises), 25 are qualified to work with large international companies and 28 are qualified to work with Local Authorities. Of the accountants who can work with Local Authorities, 3 can work with international companies and SMEs, 5 can work with international companies but not with SMEs, 9 can work with SMEs but not international companies. Of the accountants who can work with international companies, 6 can work with SMEs but not with Local Authorities i) Represent this information in a diagram (a Venn diagram would be suitable). P(LA)= 28 P(LIC AND SME)lLA)=3 P(LIC OR S ii) How many accountants took part in the survey? P(SME U LIC U LA) = 57 accountants took part in the survey iii) If an accountant is chosen at random what is the probability that he/she can: a) Do all 3 types of work? P(all 3 types of work) = 34/57 = 0.59 b) Do only one of the types of work? P (only one of the types of work) = 12/57 + 11/57 + 11/57 = 0.596 ~ 0.60 iv) If those accountants who are qualified to work with international companies are busy; what proportion of the remaining accountants are qualified to work with both SMEs and Local Authorities? 9/32 or 28% of the remaining accountants are qualified to work with both SMEs and Local Authorities v) What practical use could the company make of this information from these records? (3 marks) This information would prove to be very useful for the company in the times to come. This is because it would have a clear picture as to where could the accountants be placed as per their qualifications. Furthermore, it would also show the company management the area where more accountants are needed given the demand for certain requirements in the industry. Question 3 12 marks Applicants for a certain job are given an aptitude test. Past experience shows that the scores from the test are normally distributed with a mean of 60 and standard deviation of 12 marks. i) What percentage of applicants would be expected to pass the test if the minimum score required was 75? Mean = 60, Std dev = 12, X = 75 X = variance * z + mean z= x-mean/var = 0.104167 or 10% of applicants would be expected to pass the test if the minimum score required was 75 ii) What would the pass mark need to be if the company wanted only 4% of applicants to pass? 0.04 = X - 60/12^2 X = 65.76 Therefore, the pass mark needs to be 66 if the company wanted only 4% of applicants to pass iii) What would be the percentage failing if the standard deviation were 20 points? The percentage failing would be 96% if the standard deviation were 20 points. iv) What practical use could the company make of this information? (3 marks) The company could use normal distribution to help predict and adjust for a wide range of goals by optimizing decision-making by applying and graphically mapping financial data into a distribution set of variables. This collection of scores could also be used to calculate descriptive statistics for those scores including mean, standard deviation and z scores. Mean is the average of all of the scores while standard deviation would tell the company how much scores tend to differ from the mean.  It is a measure of how spread out the scores are. Question 4 12 marks A small coach hire company has 10 coaches that it hires out for local use by the day. The daily demand for coaches follows a Poisson distribution with a mean of 8 per day. i) What is the probability on a particular day that a) The demand is for no coaches P(X) = e-μ μx / x! P(x=0) = (e-8 * 80) / 0! = 0.000335 b) Demand is for 9 or fewer coaches P(x ≤ 9) = 1 – P(x=1) = 1 – (e-8 * 81) / 1! = 1 - 0.002684 = 0.997316 ~ 0.997 c) Demand exceeds the number of coaches? P(x > 10) = 1 – P(x=10) = 1 - (e-8 * 810) / 10! = 1 - 0.09926207 = 0.90073793 ~ 0.90 ii) What practical use could the company make of this information? (3 marks) Poisson distribution expresses the probability of several events occurring in a fixed time if these events occur with a known average rate, and are also independent of the time since the last event. So it tells the company the way to forecast demand and hence, profitability at various levels of demand. Next, it could also be used to determine if it is feasible for it to hire out more than 10 coaches a day or not. Question 5 12 marks A small mobile phone retailer has found that one of their phones has a 12% probability of being faulty and a replacement having to be provided for the customer. They have just received a trial order for 10 of these phones from their biggest customer who will take their business elsewhere if 20% or more items are faulty. i) What is the probability that they will lose their biggest customer? As the retailer has found that one of their phones has a 12% probability of being faulty therefore, there is 88% probability of the phone not being faulty. We would use Binomial Distribution method to solve this problem: P(x) = nCx * p^x * q^(n-x) where n = number of trials = 10 x = number of successes among n trials = 20%*10 = 2 p = probability of success in any one trial = 0.12 q = probability of failure in any one trial (q = 1 - p) = 0.88 nCx = combinations of n items, choose x P(they will lose their biggest customer) = 1 – (10C2 * (0.12^2) * (0.88^(10-2))) = 1 - 0.891 = 0.108682 = 10.8% ~ 11% Therefore, the probability that they will lose their biggest customer is 11% ii) What practical use could the company make of this information? (3 marks) The company can make use of the binomial distribution, as above, to determine whether it will manage to gain the new business or not. The binomial distribution can also be used by the company when it is interested in the occurrence of an event, not in its magnitude, here, the probability of losing their biggest customer. The Binomial distribution makes the assumption that the probability p does not change the more trials are performed. Bibliography Pagano, M., and Gauvreau, K. (2000). Priniciples of Biostatistics, 2nd edition. Pacific Grove, CA: Duxbury Press. Bajpai A.C., Mustoe L.R., and Walker D. Engineering Mathematics. John Willey. 1974. Pages 678 to 683 Rosner, B., 2000. Fundamentals of Biostatistics, 5th edition. Pacific Grove, CA: Duxbury Press. Neumann, P., 1966. Über den Median der Binomial- and Poissonverteilung (in German). Wissenschaftliche Zeitschrift der Technischen Universität Dresden 19: 29–33.  Kaas, R.; Buhrman, J.M., 1980. Mean, Median and Mode in Binomial Distributions. Statistica Neerlandica 34 (1): 13–18.  Morse, Philip., 1969. Thermal Physics. New York: W. A. Benjamin. Box, Hunter and Hunter. Statistics for experimenters. Wiley. p. 53. Read More