371chapter3f2011

371chapter3f2011 - Chapter 3 Estimation of p 3.1 Point and...

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Chapter 3 Estimation of p 3.1 Point and Interval Estimates of p Suppose that we have Bernoulli Trials (BT). So far, in every example I have told you the (numer- ical) value of p .I ns c i e n c e ,u s u a l l yt h ev a l u eo f p is unknown to the researcher. In such cases, scientists and statisticians use data from the BT to estimate the value of p .N o t et h a tt h ew o r d estimate is a technical term that has a precise deFnition in this course. I don’t particularly like the choice of the word estimate for what we do, but I am not the tsar of the Statistics world! It will be very convenient for your learning if we distinguishbetweentwocreatures . ±irst ,is Nature ,whoknowsevery th ingand ,inpar t icu lar ,knowstheva lueof p .Secondi sthere sea rche r who is ignorant of the value of p . Here is the idea. A researcher plans to observe n BT, but does not know the value of p .A f te r the BT have been observed the researcher will use the information obtained to make a statement about what p might be. After observing the BT, the researcher counts the number of successes, x ,inthe n BT. We deFne ˆ p = x/n ,theproportionofsuccessesinthesample ,tobethe point estimate of p . ±or example, if I observe n =20 BT and count x =13 successes, then my point estimate of p is ˆ p =13 / 20 = 0 . 65 . It is trivially easy to calculate ˆ
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Researcher: Well, I don’t actually know what p is, but I certainly hope that it equals 0.65. As I have stated many times, nobody is better than I at obtaining correct point estimates. Reporter: Granted, but is anybody worse than you at obtainingcorrectpointestimates? Researcher: (Mumbling) Well, no. You see, the problem is thatonlyNatureknowstheactual value of p .Nomereresearcherw i l
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371chapter3f2011 - Chapter 3 Estimation of p 3.1 Point and...

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