Quantitative Data Analysis and Decision Making - Assignment Example

i. The important quantitative concept The normal distribution concept is the most useful. This concept is found to have a wide application due to the fact that there are many natural processes random variations will be found to be conforming to the normal distribution which is the most commonly observed type of probability distribution. The use of the normal distribution was linked to mathematicians de Morvre and Laplace in 1700’s and in early 1800’s the distribution was used by a German mathematician and physicist Karl Gauss in analysis of astronomical data and this so it being named Gaussian distribution by the scientist community. With the shape of the normal distribution resembling that of a bell, the distribution has been sometimes been referred to as bell curve. One of the most unique thing about normal distribution is the fact that by use of two parameters there is complete specification of the distribution. The two parameters are the mean and standard deviation. Having knowledge the mean and standard deviation translates to knowing as much as what could be known when one has access to every point of the data set. The empirical rule of normal distribution Normal distribution has an empirical rule which is seen as a handy quick estimate of the spread of the data when the mean and standard deviation of the data have been given. According to the empirical rule in a normal distribution 68%, 95% and about 99.7% will be found to fall within 1, 2 and 3 standard deviations of mean respectively. These are approximations which are use for convenience. An example for clarification is where in accordance to normal probability density functions 95% of the data will be found to fall within 1.96 standard deviations of the mean thus using a2 as the number of standard deviations is for convenience purposes. Central limit theorem The central limit theorem (CLT) is important in the application of normal distribution to a population that is not normally distributed. In most of the statistical inferences and estimation techniques there is assumption of normal distribution. But in reality with samples being used in the techniques coming from real world the distribution may not be actually normal. By applying the CLT it is possible to apply the normal distribution theory in inferring about population from a non-normal sampling distribution. This involves working with mean sample data as opposed to the individual values. The theorem states that not putting into consideration the distribution of the population, the distribution of the means in random samples will conform to normal distribution where large sample sizes are used. The standard normal distribution Normal distribution consists of a large number of distributions having different means and standard deviations. As a way of making it possible to tabulate the normal probabilities that have various parameter values normal distributions can be transformed to a standard form. When we have a variable X it is possible to transform it by using the relationship Through this transformation a new variable that has its mean as 0 and a variance of 1 The transformation is expressed in the a theorem that states : if a variable has a normal distribution with a mean  and variance  the transformation  will yield a variable Z that that has a normal distribution with a mean of 0 and variance 1. The transformation results to the basic scales of x values being converted so as to have measurements being made on a standardized scale. The Z can be defined as the number of standard deviations a value is a way from the means. At a value corresponding to the mean the number of times will be 0. Normal distribution and Forecasting of Demand In most of inventory management problems the demand for future planning is seen as a random variable having probability distribution that is unknown. The models that are used in these problems aim at putting to a minimum the sum of expected overage as well as the underage costs. To have a successful inventory management system is highly dependent on having good demand forecast that will provide data for inventory replenishing decisions. Usually the output of the forecast in literature is presented as the forecasted demand quantity, which in reality is an estimation of the expected demand in the planning period. This as a result makes the purpose of forecasting to misunderstand as being to generate this single number, even when the standard deviation of demand is given as an estimate. Most forecasting methods that are commonly used are parametric in nature, with the assumption being that demand is normally distributed, where the distribution is updated through making updates of the parameters of distribution ie the mean and the standard deviation. Moving average and the exponential smoothing are the most commonly applied method in updating of the parameters. In the moving averages method there is use of n most recent observations on demand in coming up with a forecast for the expected demand in the next period. The n is referred to as the order of the moving average which has typical values ranging between 3 and 6 or beyond. Brown (2) is linked to the introduction and popularization of the exponential smoothing method which has been to be very popular in practice. In the method the forecast of expected demand ˆD t+1, for the next time period t + 1, is given by αxt + (1 − α)ˆDt with xt being the observed demand for the current period t and the smoothing constant α taking values in the range of 0 to 1 relative to how much weight is being placed on the demand that is being observed currently. Typical cases have α taking values between 0.1 and 0.4 with an increase in α values being effected when the deviation absolute value observed between forecast and the observed demand goes beyond tolerance times the standard deviation. For expected demand having relatively smooth pattern small α like 0.1 will have been used whereas there is prediction of values that have significantly greater variation when large α in the tune of 0.4 are used, but this are associated with better job in tracking of demand series. By use of larger α values there is forecasts that is more responsive with regard to changes in demand process, but resulting to forecasts errors having higher variance. The disadvantage that is associated by both the two is that where there is a definite trend in the demand process which could be falling of growing, the forecast obtained by applying any of the two will lag being the trend. There has been proposition of variations of the exponential smoothing method to track trend that is linear in time but this have not had any popularity. ii. Problem and solution Competition between Mercedes and BMW The rivaltry between BMW and Mercedes-Benz may be traced back to 1959, where there was attempt of the later to take over the former through the Deutsche Bank when it was (BMW) on the verge of bunkrupsy. But through the effort made by the minor shareholders, the dealers and the unionized workforce the take over was prevented the last minute. In the following few years, Quandt family become the majority stakeholder of BMW, but even with new cash pumped in the company, it was until the early 1970s when BMW was able to come up with competitive model range. From then up two now there has been a stiff competition between the two at the marketplace. The scenario has been that the two never aim at gaps in their ‘enemies’ product portfolio but they always take on each other head on! Some of the car models that reflect this rivalry are : 3-series vs. C-class, 1-series vs. A/B-class, 5-series vs. E-class, 6-series vs. SL, X3 vs. GLK, 7-series vs. S-class ,X5 vs. ML, Z4 vs. SLK, , M division vs. AMG, BMW Sauber vs. McLaren-Mercedes, Mini vs. Smart, Rolls-Royce vs. Maybach. There have been attempt by both to go downmarket through joining volume brands but failure was the result for both. While Mercedes has ventured into trucks and buses BMW has not while Mercedes has not ventured into motorcycles as BMW has. It is only till recently that the two rivals have recognized the importance of hybrids. More success has gone to BMW with regards to small-car segment with its pricey and prestigious Mini brand. The opposing Smart brand has only the rear-engine ForTwo, which has been reported to being dynamically flawed. There was quick withdrawal of the roadster/coupe and ForFour as a result of slow sales while the ForMore did not materialize. Some selected reasons that have made BMW have upper hand on their rival are BMW have had more luxurious models than previously available that they have managed to sale successfully They have had a lot of focus on customer quality and environmental responsibility BMW has sales pitch targeting younger market while Mercedes have stuck on mature customers In responding to this challenge from BMW there has been attempt by Mercedes to change their image so as to attract younger buyers and female customers There is need by both company to have a partner so as to help in booting volumes and cutting down costs but the Not Invented Here syndrome has been persistent in both R&D centers and the corporate quarters. This put the archirivals in a strange deadlocked situation even with both being able to see the likely benefits that can be found from direct tie-ups, the two brands which are at a critical point are drifting apart. The interesting this would be to see the executives of the two companies and the Quandt family being able to streamline their efforts maybe with a third party being involved. This is because thriving of the two is what is the most desirable even though car enthusiasts could have benefited from the fifty-year rivalry How does Mercedes A200 compete with BMW 120i? Mercedes group has concern that dealer prices are not consistent. The Average price is $42,900 with a standard deviation of $2,891. Suppose also that Mercedes believes that at $45,000, the A200 is priced out of the BMW 120i market. Average price for a BMW 120i is $41,900 with a standard deviation of $2,367. Assuming that dealer prices follow a Normal distribution. What percentage of dealer prices for the Mercedes A200 are more than $45,000, hence priced out of the BMW 120i market? Dealer prices for A200 are normally distributed with an average price of $42,900 and a standard deviation of $2,891. Assigning X as the dealer price of A200 and X being normally distributed and having mean  and standard deviation  The point of interest is P(X>$45000) The z score is given by  This shows that 23.3% of Mercedes dealers have A200 priced at more than $45000 What percentage of BMW dealers are pricing the 120i at more than the average price of the A200? Assigning X as the dealer price of 120i and X being normally distributed and having mean  and standard deviation  The point of interest is P(X>$42900) The z score is given by  This shows that 33.72% of BMW dealers have 120i priced at more than $42900 which is the average price of A200 What percentage of Mercedes dealers are pricing the A200 at less than the average price for a 120i? Dealer prices for A200 are assumed to be normally distributed with an average price of $42,900 and a standard deviation of $2,891. Assigning X as the dealer price of A200 and X being normally distributed and having mean  and standard deviation  The point of interest is P(X>$41900) The z score is given by  This shows that 36.32% of Mercedes dealers have A200 priced at less than the average price of BMW 120i This shows that 23.3% of Mercedes dealers have A200 priced at more than $45000 From the calculation it has been found that 23.3% of Mercedes dealers have the price of A200 at more than $45000 which is out of the market of BMW 120i. This is substantially a very high percentage. Also having 36.32% of Mercedez dealers selling A200 at a price below the average means that 63.68% of the dealers prices are beyond the average price of 120i. This shows that pricing of A200 is high and this could be affect its market share. There is need for Mercedes to consider reducing their price so as to compete with BMW. iii. Limitation The limitation that is in the calculation comes from the fact that the sample may not be the actual representation of the population. In order to address this it important to ensure that the right procedure is used in picking the sample for it to be a true representation. Read More