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# 66 Class Note for STAT 30100 with Professor Gundlach at Purdue

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This 6 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Purdue University taught by a professor in Fall. Since its upload, it has received 16 views.

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Date Created: 02/06/15

Section 52 Using the Normal Distribution with Sample Means Quality Control Learning goals for this chapter De ne sampling variability and understand how to minimize it Understand the difference between parameters and statistics Explain how sampling distributions work De ne the Central Limit Theorem and know how to apply it Distinguish between X and f problems Distinguish between forwards and backwards f problems State examples of processes and control vs capability Read create and interpret process control charts Calculate upper and lower control limits and the center line for control charts Know when a process is out of control In section 13 we learned how to use the normal distribution for X if we knew the p and c5 and if X was Normally distributed Here we will see how to use the normal distribution for sample means You just have to use the right u and c5 and everything else works just like in section 13 Normal distribution for X Review of 13 What is X X is a single score account height or weight or scores etc in general When do we use Normal distribution if X is Normally distributed Mean and standard deviation ofX u and cs told to us in the problem Formula we usefor Z Z H 0 Normal distribution for 2 Section 52 What is 7 I is the sample mean When do we use the Normal distribution le is Normally distributed then I is too However according to the Central Limit Theorem no matter what distribution the population has as long as you draw a SRS simple random sample with a fairly large sample size n then the sample mean is Normally distributed This is a very nice and very big deal Mean and standard deviation of a sample mean 3 H w e uf 0 Formula we usefor Z Z How do you tell these J problems apartfrom section 13 X problems Section 13 X problems would have a sample size n l or would talk about the subj ectsunits in a very general sense Section 52 J problems have a speci c sample size bigger than 1 Example The scores of students on the ACT college entrance examination in 2003 had mean u 208 and cs 48 The distribution of scores is only roughly normal a What is the approximate probability that a single student randomly chosen from all those taking the test scores 23 or higher b Now take an SRS of 50 students who took the test What are the mean and standard deviation of the sample mean score 7C of these 50 students c What is the approximate probability that the mean score 7C of these 50 students is 23 or higher To summarize Chapter 5 1 What is your variable a An individual or all members of a population X b A sample mean TC need speci c sample size n 2 Get the appropriate mean and standard deviation TVpe Mean Standard Deviation Individual or all X X p 6x 6 Sample mean 7c p In 039 3 Set up your inequality for example P f lt 02 4 Work the problem as a normal distribution problem from Section 13 Convert to Z using the appropriate mean and standard deviation use the standard normal table etc Know when it is appropriate to use the Normal approximation Section 13 and Section 52 Examples x or E Forwards or backwards The time that a technician requires to perform preventative maintenance on an air conditioning unit is governed by an exponential distribution with a mean of 60 minutes and a standard deviation of 60 minutes Your company operates 70 of these units What is the probability that their average maintenance time exceeds 50 minutes A laboratory weighs lters from a coal mine to measure the amount of dust in the mine atmosphere Repeated measurements of the weight of dust on the same lter vary normally with standard deviation of 008 mg because the weighing is not perfectly precise The dust on a particular lter actually weighs 123 mg Repeated weighings will then have the Normal distribution with mean 123 mg and standard deviation 008 mg What is the probability that the laboratory reports an average of 3 weighings as 124 mg or higher for this lter The level of nitrogen oxides NOX in the exhaust of a particular car model varies with mean 09 grams per mile and standard deviation 015 gmi A company has 125 cars of this model in its eet What is the level L such that the probability that the sample mean is greater than L is only 001 The NCAA requires Division I athletes to score at least 820 on the combined math and verbal parts of the SAT exam to compete in their rst college year In 2002 the scores of the 13 million students taking the SATs were approximately Normal with mean 1020 and standard deviation 207 What percent of all students had scores less than 820 Stau39sljcal Pmnzss Cum 2me Cum A pm is a enam uraemues um turns inputs mm Uulpuls A pmeess 15 hke a pup llau39nn eunmuung all me Uulpuls um wuuld be pmuueeu by me pmeess xfxlran umEr m as presEnl state The Uulpuls pmuueeu luday uths week are a mlz mm Lhs pupLLlauun The mm ufsvzusuml pmeess eunuul Gm1 nuke a pmeess smme War we and keep n stable un1ess planned ehanges are nnue Au prunesses rave manun smusuml smumymmns me pa am urmnanun remns smme nut that there 15 nu vanahun m me mnable mmsureu lncnnlml Cnnunlelmm are smusum1 mus 39halmumlurapmcess and almuswhmthe pmeess ms bean dlsmrbed su mam is nuW Hut ufcumml Tms is 2 513131ch nd and named me muse unne disturbanne Cnmmn can variau39nn A pmeess Hulls m cmtml un1y ms eummun musemnauun Cummm mmewnahums me mhamlmabxhty unne systEm uue m nnny mall muses thalare alwayspresem Special emevanau39nn When 39henurmal runeuumng unnepmeess 15 disturbed by sume unpredmzble evmu wenal muse maum is added u me eummun muse maum We nupe m be able m mscuverwhal hes behmd special muse vananun and Elmmle that muse m regime me smme runeuunng unnepmeess Cnmml chm d39suxgu39shhenwmn me mnlmnn causevar39avim and me glacial causevar39avim UC u zTni How do you know if a process is out of control One point outside the u i 3 limits n I A run of 9 consecutive points above the center line u or 9 consecutive points below the center line In this class we will focus on sample mean 7c charts but other common control charts include s standard deviation f7 sample proportion and R range Control vs Capability I If a process is in control we know what to expect in the nished product Statistical quality control only pays attention to the internal state of the process I If a process is capable the process can meet or exceed the requirements placed on it by some external source client demands goals of the organization etc What do you need to know about Statistical Quality Control for the homework and exams What does in control mean What is a process What is the difference between control and capability Where would you draw the center line and the upper and lower control limits on a control chart 0 How do you know when a process is out of control Be able to recognize from a control chart I Give an example ofa process

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