Decision_of_Uncertainty

Decision of Uncertainty QNT/561 January 18, 2010 Louis Daily Decision of Uncertainty Confidence intervals can be used to make estimations about population parameters since confidence intervals represent an assortment of possible values for the parameter. In statistical inference, one wishes to estimate population parameters using observed sample data. This paper will develop this probability concept to formulate a decision. It will explain research methods and processes for limiting the uncertainty in the decision. Scenario XYZ Company currently has a 10% defective rate for one of its goods. Upper management is interested in improving the production process to shrink this rate. The proposed solution required an investment of $400,000 for retraining of production employees and new equipment. Before approving these changes, management wanted to know if the new process will lessen the defective rate. In this scenario, a random sample from the population run of 400 units was observed, 20 failed to meet specification. This information will help answer management’s question. The question asked by management represents a decision problem in which one has to decide between two possible outcomes. In this scenario, the two possible outcomes are the rate is not lower than 10% and the defective rate is lower than 10%. This process is known as hypothesis testing. Hypothesis testing is the use of statistics to determine the probability that a given hypothesis is true. The usual process of hypothesis testing consists of four steps. These steps are as follows: 1. Identify the null hypothesis H0 and the alternate hypothesis HA. A null hypothesis is a hypothesis set up to be nullified or refuted in order to support an alternative hypothesis, which states there is a difference between the procedures. These hypotheses can be stated in the following terms: a. H0: μ ≥ .10 b. HA: μ < .10 2. When used, the null hypothesis is presumed true until statistical evidence indicates otherwise — that is, when the researcher has a certain degree of confidence, usually 95% to 99%, that the data does not support the null hypothesis. In this situation, it is safe to assume μ = .10. The null hypothesis assumes that any kind of difference or significance seen in a set of data is due to chance. 3. For XYZ, the risk involved is the cost of implementing the new process without reducing the defective rate. If the company decides 5% is a reasonable risk, then the company is willing to incorporate the new process as long as other factors being the same do not exceed the 5% threshold. 4. During the initial production run, there were 20 defects out of the random sample of 400. That is a 5% (20 ÷ 400) defective rate, which gives some validity to the alternative hypothesis. If a 5% sample defective rate would cause XYZ to believe the alternative, then a lower sample defective rate would be even stronger evidence for the alternative. To validate evidence for the alternative hypothesis, one must determine the likelihood of observing a sample defective rate of 5% or less in a random sample of size 400 selected from a population that has a 10% defective rate. Normal distribution P(lower) P(upper) z X mean std.dev .0010 .9990 -3.09 0.05 0.10 0.02 Since the largest z-score is 3.09, then the probability is less than 0.001. There is room for error based on this data, There is 0.001 probability of this happening. The p-value is the probability of obtaining a test statistic at least as extreme as the one that was actually observed, assuming that the null hypothesis is true. The fact that p-values are based on this assumption is crucial to their correct interpretation. The lower the p-value, the less likely the result, assuming the null hypothesis so the more significant the result, in the sense of statistical significance. One often rejects a null hypothesis if the p-value is less than 0.05 or 0.01, corresponding to a 5% or 1% chance respectively of an outcome at least that extreme, given the null hypothesis. 5. The final step in the process is to make a decision. Since the p-value is less than the definition of reasonably small for the chance for the significance level, then the decision is to believe the alternative hypothesis. In this particular scenario, the test statistic does fall in the critical region, therefore, the null hypothesis will be rejected in favor of the alternative hypothesis.