Section 5.2: Using the Normal Distribution with Sample Means, Quality Control

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1 Sectio 5.2: Usig the Normal Distributio with Sample Meas, Quality Cotrol Learig goals for this chapter: Defie samplig variability ad uderstad how to miimize it. Uderstad the differece betwee parameters ad statistics. Eplai how samplig distributios work. Defie the Cetral Limit Theorem ad kow how to apply it. Distiguish betwee X ad problems. Distiguish betwee forwards ad backwards problems. State eamples of processes ad cotrol vs. capability. Read, create, ad iterpret process cotrol charts. Calculate upper ad lower cotrol limits ad the ceter lie for cotrol charts. Kow whe a process is out of cotrol. I sectio 1.3, we leared how to use the ormal distributio for X if we kew the ad ad if X was Normally distributed. Here we will see how to use the ormal distributio for sample meas. You just have to use the right ad, ad everythig else works just like i sectio 1.3. Normal distributio for X (Review of 1.3) What is X? X is a sigle score, accout, height, or weight or scores (etc.) i geeral Whe do we use Normal distributio? if X is Normally distributed Mea ad stadard deviatio of X: ad (told to us i the problem) Formula we use for Z: Z X Normal distributio for X (Sectio 5.2) What is X? X is the sample mea Whe do we use the Normal distributio? If X is Normally distributed, the X is too. However, accordig to the Cetral Limit Theorem, o matter what distributio the populatio has, as log as you draw a SRS (simple radom sample) with a fairly large sample size (), the the sample mea is Normally distributed. (This is a very ice ad very big deal.) 1

2 Mea ad stadard deviatio of a sample mea: Formula we use for Z: Z X How do you tell these X problems apart from sectio 1.3 X problems? Sectio 1.3 X problems would have a sample size = 1 or would talk about the subjects/uits i a very geeral sese. Sectio 5.2 X problems have a specific sample size bigger tha 1. Eample: The scores of studets o the ACT college etrace eamiatio i 2003 had mea = 20.8 ad = 4.8. The distributio of scores is oly roughly ormal. a) What is the approimate probability that a sigle studet radomly chose from all those takig the test scores 23 or higher? b) Now take a SRS of 50 studets who took the test. What are the mea ad stadard deviatio of the sample mea score of these 50 studets? c) What is the approimate probability that the mea score of these 50 studets is 23 or higher? 2

3 To summarize Chapter 5: 1. What is your variable? a) A idividual or all members of a populatio () b) A sample mea ( ), eed specific sample size 2. Get the appropriate mea ad stadard deviatio. Type Mea Stadard Deviatio Idividual or all, = = Sample mea, 3. Set up your iequality, for eample P( < 0.2). 4. Work the problem as a ormal distributio problem from Sectio 1.3. (Covert to Z usig the appropriate mea ad stadard deviatio, use the stadard ormal table, etc.). Kow whe it is appropriate to use the Normal approimatio. 3

4 Sectio 1.3 ad Sectio 5.2 Eamples: or? Forwards or backwards? 1. The time that a techicia requires to perform prevetative maiteace o a aircoditioig uit is govered by a epoetial distributio with a mea of 60 miutes ad a stadard deviatio of 60 miutes. Your compay operates 70 of these uits. What is the probability that their average maiteace time eceeds 50 miutes? 2. A laboratory weighs filters from a coal mie to measure the amout of dust i the mie atmosphere. Repeated measuremets of the weight of dust o the same filter vary ormally with stadard deviatio of 0.08 mg because the weighig is ot perfectly precise. The dust o a particular filter actually weighs 123 mg. Repeated weighigs will the have the Normal distributio with mea 123 mg ad stadard deviatio 0.08 mg. What is the probability that the laboratory reports a average of 3 weighigs as 124 mg or higher for this filter? 3. The level of itroge oides (NOX) i the ehaust of a particular car model varies with mea 0.9 grams per mile ad stadard deviatio 0.15 g/mi. A compay has 125 cars of this model i its fleet. What is the level L such that the probability that the sample mea is greater tha L is oly 0.01? 4. The NCAA requires Divisio I athletes to score at least 820 o the combied math ad verbal parts of the SAT eam to compete i their first college year. I 2002, the scores of the 1.3 millio studets takig the SATs were approimately Normal with mea 1020 ad stadard deviatio 207. What percet of all studets had scores less tha 820? 4

5 Statistical Process Cotrol (Quality Cotrol) A process is a chai of activities that turs iputs ito outputs A process is like a populatio cotaiig all the outputs that would be produced by the process if it ra forever i its preset state. The outputs produced today or this week are a sample from this populatio. The idea of statistical process cotrol Goal: make a process stable over time ad keep it stable uless plaed chages are made. All processes have variatio. Statistical stability meas the patter of variatio remais stable, ot that there is o variatio i the variable measured. (I cotrol) Cotrol charts are statistical tools that moitor a process ad alert us whe the process has bee disturbed so that it is ow out of cotrol. This is a sigal to fid ad correct the cause of the disturbace. Commo cause variatio A process that is i cotrol oly has commo cause variatio. Commo cause variatio is the iheret variability of the system, due to may small causes that are always preset. Special cause variatio Whe the ormal fuctioig of the process is disturbed by some upredictable evet, special cause variatio is added to the commo cause variatio. We hope to be able to discover what lies behid special cause variatio ad elimiate that cause to restore the stable fuctioig of the process. Cotrol charts distiguish betwee the commo cause variatio ad the special cause variatio. UCL 3 LCL 3 5

6 How do you kow if a process is out-of-cotrol? Oe poit outside the 3 limits A ru of 9 cosecutive poits above the ceter lie ( ) or 9 cosecutive poits below the ceter lie. I this class we will focus o sample mea charts, but other commo cotrol charts iclude s (stadard deviatio), ˆp (sample proportio), ad R (rage). Cotrol vs. Capability If a process is i cotrol, we kow what to epect i the fiished product. Statistical quality cotrol oly pays attetio to the iteral state of the process. If a process is capable, the process ca meet or eceed the requiremets placed o it by some eteral source (cliet demads, goals of the orgaizatio, etc.). What do you eed to kow about Statistical Quality Cotrol for the homework ad eams? What does i cotrol mea? What is a process? What is the differece betwee cotrol ad capability? Where would you draw the ceter lie ad the upper ad lower cotrol limits o a cotrol chart? How do you kow whe a process is out of cotrol? Be able to recogize from a cotrol chart. Give a eample of a process. 6