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Confidence Intervals for

Confidence Intervals for. Proportions. Chapter 8, Section 3. Statistical Methods II. QM 3620. Something Changed. Quantitative variables vs. Qualitative variables Sometimes we need to make an estimation about a qualitative variable

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Confidence Intervals for

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  1. ConfidenceIntervalsfor Proportions Chapter8, Section3 StatisticalMethodsII QM3620

  2. SomethingChanged • Quantitative variables vs. Qualitative variables • Sometimesweneedtomakeanestimationaboutaqualitativevariable • Whether or not a customer is satisfied with your company’s services is a qualitative variable. They could answer “Yes” or “No”, or the answer could be “Very Satisfied” “Somewhat Satisfied”, etc. • For accountants, whether or not an account balance is correct is qualitative. The actual amount it is off would be quantitative, but the status would be qualitative.

  3. SomethingChanged • Infact, itislikelythatmoreofthevariablesyouencounter arequalitativethanquantitative (gender, ethnicity, incomelevel, etc.) • But income is quantitative, right? That is a number, not a category • Yes, but we can make a quantitative variable qualitative by reducing the scale level of the data • We could convert a quantitative variable to a qualitative variable by putting the variable in categories. Income between $30K and $40K, etc.

  4. ThePopulationBox Suppose that this box is full of our customers. Let’s say that the ones shaded in red are satisfied with our service. We could simply attempt to talk to ALL of these customers, but that could be difficult and expensive. Or we could take a sample and use it to infer about all the members of the population “box”, whether we actually talk to them or not.

  5. TheSample So,insteadoftalkingtoallofthecustomers,wetakeasamplefromthe“population box”toestimatetheproportionofredindividuals(thosewhoaresatisfied withourservice). But…remembertheproblemswithsamples.Eachtimewetakea sample,wearelikelytoendupwithadifferentproportionsof“red”individuals.We willneedtotakethissamplevariationintoaccount.

  6. SamplingDistribution Weknowthatifwetakeasample,theproportionof“red”individualswillvaryfromsampletosample. Howmuchthesesample proportionsvaryismeasuredbytheirstandarderror. We refertothepopulationproportionasandtheproportioninasampleasp. p=.48 p=.57 p=.49 p=.55 =? p=.52

  7. Is it Magic? So, our purpose for this module is to take ONLY ONE SAMPLE from a closed box and us it to estimate the proportion of, in this case, “red” individuals in the box. How do we do that? Another confidence interval!!

  8. AnotherConfidenceInterval Confidence intervals can be used to estimate pretty much any population parameter. Just keep in mind that a confidence interval is used to estimate a number in a larger group by only looking at the limited information provided by a sample of that group. The only extension we are doing in this module is the application of a confidence interval to a population proportion. This allows us to analyze qualitative values (like gender) versus quantitative variables (like age). So what is different?

  9. AnotherConfidenceInterval Well, the calculations are different because you now have to count the observations rather than average them. The multiple in the confidence interval formula is now based on the z-score rather than the t-value. Read the discussion of interval estimation for proportions and their calculation (pages 358-359). The best part about confidence intervals for proportions is that we need such little information. All we need is the sample size (easy, just count), the proportion that fell into the category we are interested in (almost as easy, just count and divide by the sample size), and …

  10. TheEquation Thegeneralformulaforallconfidenceintervalsis: Point Estimate   StandardErrorof PointEstimate Multiple Theversionforaconfidenceintervalforapopulationproportionis: Sample Proportion StandardErrorof SampleProportion   z-value The calculation to the right hand side of the plus-minus sign (±) is still the margin of error, but in this case it is the margin of error for an estimate of a population proportion.

  11. The Equation The standard error of the sample proportion can again be estimated directly from the sample you took. Sample proportions near 50% create the widest confidence intervals because half belong to one group and half belong to the other group. If the sample is 99% one group and 1% the other group, there will be a lot less variation among the sample members. Like all confidence intervals, more information (i.e. larger sample sizes) lead to smaller intervals. The multiple only handles the confidence level. Greater confidence requires a larger multiple … and subsequently a wider interval.

  12. TheMultiple Nowbasedonaz-score.Thegoodnewsisthatyouhaveseenz-scoresbeforeandshould havesomefamiliarity. Thebadnewsisthatthefunctiontolookupz-scoresinExcelrequires some practice. Idoexpectyoutobeabletofindz-scoresusingExcel, notatableinthebook (anddon’tthinkIcan’taskyouforonethatisnotinthetable.)

  13. FindingazvalueusingExcel ThezDistribution TofindazvalueinExcel,youuse theNORMSINVfunction.The NORMSINVfunctionneedsonly 97.5%total onebitofinformation: 1)Theconfidencelevelyouwant 2.5% 95% 2.5% ThisiswhereitgetsmessyinExcel.Togettheproperzvaluefora confidencelevel, youneedtoincludethelowerpartofthe“unconfidence” regioninthefunction.Forexample,tolookupthezvaluefora95% confidenceinterval, youwouldtakethe95%andaddhalfofthe “unconfidence“of5%or2.5%.ThatmeanstheNORMSINVfunctionlooks up97.5%(or.975)fora95%confidenceinterval.Youonlywantthe middle95%,buttheExcelfunctionwon’tprovideitthatway.Fora99% confidenceinterval,youwouldusedNORMSINV(.995).For90% confidence, youwoulduseNORMSINV(.95). TheNORMSINVfunctionlooksuptheblueareainthegraphabove. +zvalue So,azvaluefora95% confidence(or.95indecimal format)wouldbe: =NORMSINV(.95+(1-.95)/2); or=NORMSINV(.975)

  14. Wehaveeverythingweneed Putting it all together … We have a point estimate … the proportion we calculated from the sample … referred to as p We have the sample size … the number of observations in the sample … which is referred to as n We have the multiple … the z-value from Excel … which is referred to as z That gives us everything we need to calculate a confidence interval for a proportion.

  15. TheCalculation Rememberourconfidenceintervalequationfromanearlierslide, StandardErrorof Sample Proportion z-valueSampleProportion  p1 - p n pz Algebraically,theequationlookslike: pSampleProportion Where p(1p) n zz-value StandardErrorof SampleProportion and

  16. Factors AffectingMarginofError p1-p n z Marginof Error Sampleproportion(p): Samplesize: ConfidenceLevel(CL): asp.50 asn asCL Marginof Error Marginof Error Marginof Error    …andmarginoferrorisdirectlyrelatedtoprecision.A smallermarginoferrorisa morepreciseestimate.

  17. SoWhatwasthePointof All This? Remember, ourwholereasonforthisseriesof thoughtsandcalculationswastohelpusbest estimatetheproportionwithsomecharacteristicin alargegroupbyusingonlythelimitedinformation providedbyasamplefromthatgroup.

  18. UsingStatistics The Point Estimate The point estimate for the population proportion is the sample proportion. Generally, when you want to estimate something in a population, use the same calculation in the sample … generally. The Confidence Level Confidence levels mean the same thing regardless of what you are estimating. A 95% confidence level means that 19 out of 20 times you will have an accurate interval. Using the point estimate alone will pretty much guarantee that you are incorrect. Precision or accuracy – your choice.

  19. UsingStatistics The Standard Error of the Point Estimate The sample proportion, which is our point estimate, is going to change from sample to sample. We are going to take this into account when forming our confidence interval. The Assumptions The sample size is going to need to be large enough to justify our use of the normal distribution to estimate the proportion. This will be true when the number of observations that fall into each group is at least 5. So if you are estimating the proportion of a certain characteristic in a population, make sure that the number of observations in your sample that have the characteristic is at least 5, and the number of observations that does not have the characteristic is at least 5.

  20. ApplicationTime Let’strythisforreal

  21. BusinessApplicationHighlights Businessapplicationonpage360 - QuickLubeoperatesoil-changeoutlets. Acomputerisusedtosendoutremindercardstocustomers3monthsaftera service. Managementisinterestedinthesuccessrateofthisremindersysteminbringing backcustomers. Asampleof100customersisevaluatedandofthe100, 62returnedwithinone monthofreceivingtheremindercard.

  22. ANOTHEREXAMPLE

  23. Business ApplicationHighlights A multinational corporation would like to estimate the proportion of its workers who currently commute to work by using a carpool. The corporation hopes to develop a proposal to encourage more employees to forgo their automobile as means of maker the company “greener” A random sample of 156 employees was taken from company records and the most-often used transportation method to work of these employees was recorded.

  24. Business ApplicationHighlights • The responses were separated into one of three categories: • carpool • the employee’s own vehicle • public transportation • Management has asked for an estimate at the 90% • confidence level

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