CHAPTER 9: ESTIMATING THE VALUE OF A PARAMETER USING CONFIDENCE INTERVALS
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1 CHAPTER 9: ESTIMATING THE VALUE OF A PARAMETER USING CONFIDENCE INTERVALS 1 By: Wadi Dig
2 1. Poit estimate: is the value of a statistic that estimates the value of a parameter. For eample, the sample mea is a poit estimate of the populatio mea,. Suppose: we wat to estimate the average weight for all studets i MTSU for this semester, we could take a radom sample of 100 studets ad fid the average weight of these studets, say, 130 pouds, this kid of estimate is called a poit estimate. Ofte, there is aother questio to be asked, it is, how good is a poit estimate? There is o way of kowig how close a particular poit estimate is to the populatio mea if the populatio is large. For this reaso, statisticias prefer aother type of estimate, called iterval estimate.. Iterval estimate: a iterval for ukow parameter is a iterval or a rage of values used to estimate the parameter with the specific cofidece level of estimate. It is also called cofidece iterval.
3 To make our iterval estimate more reasoable ad cofidet, we usually use a degree of cofidece to describe iterval estimate, like 95%, 98%, 99% The level of cofidece: represets the epected proportio of itervals that will cotai the parameter if a large umber of differet samples is obtaied. The level of cofidece is deoted 1- α*100%. For eample, a 95% level of cofidece α=0.05 implies that if 100 differet cofidece itervals are costructed, each based o a differet sample from the same populatio, the we will epect 95 of the itervals to iclude the parameter ad 5 to ot iclude the parameter. 4. Formula for the cofidece iterval of the populatio mea for a specific α, where is kow ad populatio is ormally distributio or sample sie 30 3
4 E: for 95% cofidece iterval for a populatio mea: α=1-95%=5%; α/=0.05/=0.05, Based o the stadard ormal table α/ α/
5 Cofidece iterval estimates for the populatio mea ca be writte i this form, too. poit estimate margi of error =E is margi of error, also called maimum error of estimate. There are three factors which affect the margi of error. 1 level of cofidece. Note: α=1-level of cofidece. sample sie,. 3 stadard deviatio of the populatio. Note: The value of is called critical value of the distributio ad the et slide shows commo critical values used lot i cofidece itervals. 5
6 The commo critical value for 90%, 95%, 99% cofidece level: Level of cofidece 1-α*100% Area i each tail, α/ Critical vale 90% % % Iterpretatio of a cofidece iterval: A 1-α*100% cofidece iterval idicates that 1-α*100% of all simple radom samples of sie from the populatio whose parameter is ukow will cotai the parameter. 6
7 Commets for cofidece iterval: Lower boud: Upper boud: 7
8 E: a survey of 30 adults foud that the mea age of a perso s primary vehicle is 5.6 years. Assumig the stadard deviatio of the populatio is 0.8 year, fid the 99% cofidece iterval of the populatio mea ad iterpret it. As: based o the 30 adults sample; =30; is the stadard deviatio of populatio. cofidece level is 99% ad α=1-99%=1%,
9 ad the fial aswer is: or Iterpretatio: oe ca be 99% cofidet that the mea age of all primary vehicles is betwee 5. years ad 6.0 years, based o 30 vehicles. Commets: the margi of error i this eample is 0.376, if we icrease sample sie, ad the margi of error will decreasig. If possible, collectig more data to reduce the margi of error. 9
10 5. Calculatig the ecessary sample sie: E a E a * a E * a E * 10
11 E: the college presidet asks the statistics teacher to estimate the average age of the studets at their college. How large a sample is ecessary? The statistics teacher would like to be 99% cofidet that the estimate should be accurate withi 1 year. the stadard deviatio of the age is kow to be 3 years. As: α=1-99%=1%=0.01, α/=0.01/=0.005, Z a/ =.575, E=1 a * E.575* so, roud up to60 The sample sie at least is 60 studets. 11
12 9. WHEN THE POPULATION STANDARD DEVIATION UNKNOWN As we kow, if the stadard deviatio of populatio is kow ad or 1the sample is draw from a ormal distributio sample sie 30 whe paret populatio is ukow. We will use -value based o stadard ormal distributio to costruct cofidece iterval. If the populatio stadard deviatio is ukow ad sample sie <30, the we will use t-value based o studet s t distributio NOT use -value based o stadard ormal distributio. 1
13 9. WHEN THE POPULATION STANDARD Studet s t-distributio: DEVIATION UNKNOWN Suppose that a simple radom sample of sie is take from a populatio. If the populatio from which the sample is draw follows a ormal distributio. The distributio of t s follows a studet s t-distributio with -1 Degree of freedom, where is the sample mea ad s is the sample stadard deviatio. Note: the t-statistic represets the umber of sample stadard errors is from the populatio mea,. 13
14 9. WHEN THE POPULATION STANDARD Properties of the t-distributio: DEVIATION UNKNOWN 1 The t-distributio is differet for differet degree of freedom. The t-distributio is cetered at 0 ad is symmetric about 0. 3 The area uder the curve is 1. The area uder the curve to the right of 0 equals to the area uder the curve to the left of 0, which equals ½= As t icreases without boud, the graph approaches, but ever equals, ero. As t decreases without boud, the graph approaches, but ever equals, ero. 5 The area is the tails of the t-distributio is a little greater tha the area i the tails of the stadard ormal distributio, because we are usig s as a estimate of, thereby itroducig further variability ito the t-statistic. 14
15 9. WHEN THE POPULATION STANDARD DEVIATION UNKNOWN 6 As the sample sie icreases, the desity curve of t gets closer to the stadard ormal desity curve. This result occurs because, as the sample sie icreases, the values of s get closer to the value of, by the law of large umbers. Page 46, figure 8 shows these properties. A. How to fid t-values: Let s go over the eample o page 46. After this eample, you should kow how to fid the t-values. 15
16 9. WHEN THE POPULATION STANDARD DEVIATION UNKNOWN Costructig a 1-α*100% cofidece iterval for, ukow: s t t s lower boud : t s upper boud : t s 16
17 9. WHEN THE POPULATION STANDARD DEVIATION UNKNOWN E: the below data represets a sample of the umber of home fires started by cadles for the past several years, fid the 99% cofidece iterval for the mea umber of home fires started by cadles each year. 5460, 5900, 6090, 6310, 7160, 8440, 9930 Aswer: step 1: fid the sample mea ad stadard deviatio based o the sample Step : fid the t α/ sample sie =7, so degree of freedom is -1=7-1=6 t α/ = s
18 9. WHEN THE POPULATION STANDARD DEVIATION UNKNOWN Step 3: formula used: s s t t How to iterpret this result? 18
19 Summary: 9. WHEN THE POPULATION STANDARD DEVIATION UNKNOWN 1 If populatio stadard deviatio is kow, ad a sample sie 30, usig -iterval. b sample sie <30, but the sample is draw from a ormal populatio, usig -iterval. If the sample ot from ormal populatio, usig o-parametric method. If populatio deviatio ukow, ad a sample sie 30, usig t-iterval. b sample sie <30, if sample is from ormal populatio, the usig t-iterval, if ot from ormal populatio, usig o-parametric method. 19
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