Effect of Reducing the Production Defective Rugs - Assignment Example

Statistics Module 6 Discussion Forum: The personnel director of a large rug manufacturing firm suggested that each month any employee who produced no defective rugs would receive a citation. Also there would be small party in the person's honour, and he or she would get a half day off. For all employees there was a 0.05 probability that a rug they produced would be defective, since they were equally skilled and the defects were due largely to factors beyond their control. The firm's president was concerned that if this policy was adopted, the cost might be excessive. Also, the president wondered whether the suggested program would have the desired effect of reducing the production defective rugs. If each employee produced about 20 rugs per month, what statistical technique would you use to help answer these questions To answer these questions, we must create and analyze appropriate statistical model. This model should give formalized and clear description of studied random processes with the most important factors and probabilistic parameters. First of all, it is necessary to note that proposed policy has no any effect upon quality of manufactured rugs. Indeed, a non-zero () probability of producing defective rugs is caused by the defects, which are due to factors beyond employees' control. Therefore, any stimulation of employees (receiving a citation, providing parties in the honour, getting a half day off, etc.) to make no defective rugs is ineffective because of absence of real mechanisms for quality control by employees. Let's disclose this proposition. Let probability of producing defective rug is formed by two additive probabilities, and : . Here, probability is caused by all factors beyond employees' control, e.g. malfunctions of automated equipment facilities. Another probability is caused by factors which can be controlled by employees, e.g. managing errors or keying mistakes when operating non-automatically. It is necessary to note that these groups of factors (controlled and uncontrolled by employees) produce so-called antithetic or mutually exclusive events (Feller). Any incentive of employees is unavailing, when and . Let is a number of employees, is a number of produced rugs per month, is cost of produced rug (in USD), and is a value of firm's outlay for incentive of an employee (in USD). Let's compute financial effect of proposed policy. Probability for an employee's incentive is (Feller), i.e. for and . So, total firm's outlay for incentive of employees will be estimated as per month. For example, USD when USD and employees. In this case, proposed policy has no real mechanisms for quality control because of . Therefore, we have defective rugs per month. For example, with total deficient profits of USD per month when USD. So, in this case we have negative profit USD per month. How to avoid such negative profit Let's suppose that rug manufacturing has real and effective mechanisms for quality control by employees, i.e. (for example, and , so ). Then, let's suppose that stimulation of employees (receiving a citation, providing parties in the honour, getting a half day off, etc.) is effective too. It means that probability of producing defective rug can be reduced up to . In this case, we have and USD; and USD. Here, total deficient profits were reduced from USD to USD per month. Moreover, total negative profit can be reduced from USD to USD per month. It is obvious that proposed policy will be most effective at and . What it means In this ideal case all manufacturing factors are controlled by employees and all employees are fully stimulated by proposed policy. Here, probability of producing defective rug can be reduced up to . Therefore, and USD; and USD. So, total firm's outlay for incentive of employees is estimated as USD per month. However, in this ideal case total deficient profits were entirely eliminated from typical values of USD per month. So, proposed policy is effective in the presence of real mechanisms for quality control by employees. To make our estimations more correct, we need for empirical values of following initial manufacturing parameters: , , and . Moreover, we need for estimations of probabilities and . How to make these estimations First of all, we can determine an averaged monthly number of defective rugs. So, we can estimate probability as (Feller). Then, we can determine an averaged monthly number of defective rugs which were produced under the influence of factors beyond employees' control. So, we can estimate probability as and probability as . It is necessary to note that much more precise statistical model can be based upon personalized data, when all probabilities are estimated for each employee. In this case, we can "differentiate" proposed values of firm's outlay for incentive of an employee. Such approach is most efficient for reducing of total firm's outlay for incentive of employees. Module 7 Discussion Forum: Please visit the following website http://davidmlane.com/hyperstat/z_table.html. What are the graphs tells us Could you use these graphs instead of the Z-table Verify the numbers in this graph with some Z values. What are the limits of values you can input in both fields The function defined by is called the normal density function. Its integral is the normal distribution function (Feller). If the origin and the unit of measurement are changed, then is transformed into . This function is called the normal distribution function with mean and variance (or standard deviation ). The graph of is the symmetric, bell-shaped curve. So, at the website we can observe an interactive JavaScript-realization of normal distribution function with the following key parameters: mean, standard deviation, lower and upper limits of integration. Here, to compute the normal distribution function without usage of the Z-table, we need to perform the following actions: 1) set the mean as 0 and standard deviation as 1 (default values for the normal density function); 2) set the cut-off points as consistent with the definition of the , i.e. below the certain value of ; 3) get value for shaded area, i.e. value for the . For example, we can obtain correct values for the ; ; ; ; ; , etc. It is obvious that we can use any value of for computations because of unconditional convergence of integral in definition of the normal distribution function. Bibliography: Feller, William. An Introduction to Probability Theory and Its Applications. New York: John Wiley & Sons, Inc., 1968. Best regards, Writer 4775. Read More