ETF5900 Business Statistics Tutorial 6 Part A (Complete BEFORE tutorial and submit online. Use Excel where necessary and write answers on question sheet) For all your answers, please remember to do the following: 1. Draw curves 2. State the distribution 3. Define the variable A6.1 An automatic machine in a manufacturing process is operating properly if the lengths of an important subcomponent are normally distributed, with mean = 117 cm and standard deviation = 2.1 cm. If the machine is operating correctly: (i) find the probability that one randomly selected unit has a length greater than 120 cm; ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ (ii) find the probability that if three units are randomly selected, their mean length exceeds 120 cm; _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ 1 ETF5900 Business Statistics Tutorial 6 _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ (iii) explain the differences between parts (i) and (ii). _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ Part B: To be completed in class B6.2 (a) Refer to Question A6.1. Draw the probability density function of the length of a subcomponent in A6.1(i) and that of the mean of three subcomponents in A6.1(ii) on a single axis. 2 ETF5900 Business Statistics Tutorial 6 (b) Is the distribution of the sample mean less or more variable than the distribution of the parent population? Explain your answer. __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ B6.3 A statistical analyst who works for a large insurance company is in the process of examining several pension plans. Company records show that the age at which its male clients retire is approximately normally distributed with a mean of 63.7 years and a standard deviation of 3.1 years. (a) Calculate the probability that a randomly selected male client will retire before the age of 65 years. __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ 3 ETF5900 Business Statistics Tutorial 6 (b) If a random sample of 50 male clients is to be selected from the company database, what is the probability that the sample mean will be less than 65 years? ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ (c) Close examination of the ages of recent retirees shows that the assumption of a normal distribution may be false. Which, if either, of your answers above would be changed by this information, and why? ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 4 ETF5900 Business Statistics Tutorial 6 QB6.4 Soft drink bottles are filled so that they contain on average 330 ml of soft drink in each bottle. The standard deviation is 4 ml. Assume that the content of soft drink bottles is normally distributed a) Calculate the probability that a randomly selected bottle will contain less than 325 ml? __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ b) The bottles are sold in 6-packs. What is the probability that in a randomly selected 6-pack the mean amount per bottle is less than 325 ml? __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ 5 ETF5900 Business Statistics Tutorial 6 c) What if the assumption of a normal distribution was incorrect? What will happen to your answers for Parts (a) and (b)? Which, if either, of your answers above would be changed by this information, and why? ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 6