Using Excel to Construct Confidence Intervals

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1 OPIM 303 Statistics Ja Stallaert Usig Excel to Costruct Cofidece Itervals This hadout explais how to costruct cofidece itervals i Excel for the followig cases: 1. Cofidece Itervals for the mea of a populatio whe the data is umerical ad the stadard deviatio is kow; 2. Cofidece Itervals for the mea of a populatio whe the data is umerical ad the stadard deviatio is ukow; 3. Cofidece Itervals for the proportio i a populatio whe the data is categorical. The terms used i this hadout are as follows. A 99% Cofidece iterval for a parameter of the populatio (say, the mea) meas that i 99% of the cases (99% of the samples, if you were to repeat the samplig process may times) the true parameter (i our case, the mea) will fall iside this iterval. Remember, it s wrog to say that there s a 99% probability that the iterval cotais the true parameter ;-). A cofidece iterval always has a cofidece level, for example 99%, 95%, etc. We will also eed a umber for α, which ca be iterpreted as the ucofidece level, i.e., α= 100-cofidece level. So, for a 95% cofidece level, we have α=5% or I the remaider, stads for the sample size. I geeral, a cofidece iterval always takes o the followig form: Poit estimate ± margi of error 1. Cofidece Iterval for the Mea of a Populatio I this case the poit estimate (i.e., the best guess for the mea of the populatio) is the sample mea, writte as X ( X bar ). The cofidece iterval is the: X ± Z σ Tutorial: Costructig Cofidece Itervals Usig Excel

2 OPIM 303 Statistics Ja Stallaert So, the margi of error is Z σ, ad the Z stads for the value of the stadard ormal distributio (the oe with mea 0 ad stadard deviatio 1) that has a probability of α o the right (this is called a tail probability of α, drawig a little picture of the 2 2 ormal distributio will make this omeclature clear). Sice we assumed that the populatio stadard deviatio σ is kow, ad we should kow what the size was of the sample we took (otherwise, we re i big trouble), the cofidece iterval ca readily be computed. The easiest way to accomplish this i Excel is to use the followig formula to compute the margi of error: =NORMINV(α/2,0,σ/SQRT()) which will the give you a egative umber 1. Example Suppose that our sample of size 25 yields X =7.3, ad we kow that σ=2 i the populatio. The, a 90% cofidece iterval has α=0.10, ad α/2=0.05. The Excel formula becomes: =NORMINV(0.05,0,2/SQRT(25)) ad Excel returs the value: , so our cofidece iterval becomes: 7.3 ± which is the iterval [6.642,7.958]. So, this iterval is iterpreted as the iterval such that i 90% of the samples (if we were to take a lot of samples), the true mea (writte as µ) lies withi i this iterval, ad remember it s *wrog* to thik of this as there s a 90% probability that the true mea is betwee ad ;-). 1 You ca safely igore the sig. Tutorial: Costructig Cofidece Itervals Usig Excel

3 OPIM 303 Statistics Ja Stallaert 2. Cofidece Iterval for the Mea of a Populatio whe σ is ukow Above we assumed that we did t kow the true mea of the populatio, but we did kow its stadard deviatio. This assumptio is oly valid i *very rare* istaces (read ever ). However, we could get a estimate for σ by usig the stadard deviatio computed from the sample, deoted by s (use the EXCEL STDEV() formula). Sice there s more ucertaity i this case (s will probably ot be exactly equal to the true populatio stadard deviatio σ), the result will be that our cofidece iterval will become wider tha i the first case. This is doe by usig values from the t-distributio rather tha the (stadard) ormal distributio whe computig the margi of error. It should be ituitive that the ucertaity about σ is reduced whe we take a larger sample, i.e., a larger sample will probably give us a better estimate for σ tha a smaller sample. So, the value for t will deped o how big a sample we took. For the same cofidece, we will get smaller values for t whe usig a big sample as compared to a smaller sample. This where the degrees of freedom (writte as d.f.) come i. At this poit, we do ot have to worry too much about the precise meaig of the degrees of freedom other tha that for cofidece itervals, the degrees of freedom is always oe less tha the sample size, or: d.f. = -1 The mathematical formula for the cofidece iterval ow becomes: X t ( 1) ± s The term 1) t ( is called the t-value with a tail probability of α/2 ad (-1) degrees of freedom. Its value ca be computed usig Excel with the followig formula (Note that Excel gives a two-tailed value for t, i.e., we do ot divide α i half): =TINV(α,(-1)) Tutorial: Costructig Cofidece Itervals Usig Excel

4 OPIM 303 Statistics Ja Stallaert To compute the margi of error, we will eed to multiply the above quatity for the t value by s ad divide by the square root of (the sample size). Or, the complete Excel formula to compute the margi of error becomes: =TINV(α,(-1))*s/sqrt() with: α: the ucofidece level : the sample size s: the sample stadard deviatio Example Let s cotiue our example from above, so X =7.3, =25, ad we computed a sample stadard deviatio s=1.98 from our sample 2. The cofidece level was set at 90%, so α=0.10 ad α/2=0.05. I the first step, we compute the t value: =TINV(0.1,24) ad Excel returs the value Multiplyig by s (1.98) ad dividig by the square root of gives us the margi of error of Compare this value with the margi of error of computed uder the assumptio that σ was kow. Also ote that the sample stadard deviatio was smaller tha the true stadard deviatio, but we still eded up with a bigger margi of error (ad hece wider cofidece iterval), because the t value corrected for this approximatio used. Our 90% cofidece iterval ow becomes [6.62, 7.98]. 2 Remember, the Excel fuctio STDEV computes this value for you. Tutorial: Costructig Cofidece Itervals Usig Excel

5 OPIM 303 Statistics Ja Stallaert 3. Cofidece Iterval for the Proportio Sice we kow that the sample proportio (writte as p s ) is ormally distributed aroud p ( 1 p) the true populatio proportio p with a stadard deviatio of ( beig the sample size), the mathematical formula for a 100(1-α) cofidece iterval becomes: p Z s ± ps(1 ps) So, the margi of error is ps (1 ps ) Z with Z the value of the stadard ormal distributio that has a probability of α (a tail probability. See 1.) o the right. The 2 followig Excel formula computes this margi of error: = NORMINV(α/2,0,SQRT(Ps*(1-Ps)/)) where Ps stads for the sample proportio, i.e., the oe computed from sample data. Example Suppose that a sample of 200 registered voters was take. 85 of those respoded that they would vote to re-elect the preset presidet. We wat to costruct a 95% cofidece iterval for the true proportio of registered voters who will vote for the curret presidet. For a 95% cofidece iterval, α=0.05, ad α/2= The value for Ps is 85 =0.425, so the margi of error is give by: 200 = NORMINV(0.025,0,SQRT(0.425*( )/200)) which yields or about 6.85%. So, this poll would be reported i the ewspapers as i a recet [ ] poll, oly 42.5% of the voters would re-elect the curret presidet. The poll had a margi of error of ±6.85%. The actual cofidece iterval the becomes [35.65%, 49.35%], so we ca be fairly cofidet (95% cofidet) that less Tutorial: Costructig Cofidece Itervals Usig Excel

6 OPIM 303 Statistics Ja Stallaert tha half the voters will vote for the preset presidet. As usual, it would be *wrog* to say that there s a 95% probability that the umber of voters who will vote for the presidet is less tha half. ;-) Tutorial: Costructig Cofidece Itervals Usig Excel

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