# Standard Errors and Confidence Intervals

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2 umber of samples of ay give size. Of course, this is ot at all how thigs are i practice, where parameters are ukow. Moreover, drawig eve a sigle sample from a populatio ivolves a great deal of work ad drawig may samples is out of the questio. However, the followig is merely to demostrate the importat ideas: how the repeated drawig of samples is circumveted i practice is explaied the ext sectio. Suppose that 5 samples, each of size 1 are draw a Normal populatio with mea 18 ad stadard deviatio 4.7 (values that accord closely with those for the heights of five-year-old boys). The mea of each of these samples ca be computed ad a histogram of the resultig 5 meas ca be plotted: see figure 1. Samples of size Sample mea Figure 1 The pricipal thig to ote about figure 1 is that the histogram is much less dispersed tha the correspodig histogram of idividual heights see i figure 3 of Data Descriptio, Populatios ad the Normal Distributio. The data i the latter occupied the rage from 95 to 12 cm whereas the figure above is largely cofied to the iterval 15 to 112 cm. Two other features of figure 1 should also be oted: first the distributio of the sample meas does appear to be cetred o μ = 18 ad secod the distributio appears to have the shape of a Normal distributio. Repeatig this exercise but with 5 samples each of size 1 gives the histogram i figure 2. This histogram is also cetred o 18 cm but is eve less dispersed tha that i figure 1, with all the sample meas beig betwee 17 ad 19 cm ad virtually all of them beig betwee 17.5 ad 18.5 cm. 2

6 estimate has a precisio that is appropriate to the purpose of the ivestigatio, rather tha oe that is arbitrarily high. Collectig data o sufficiet patiets to determie the mea blood pressure of a group to withi 1 mmhg is likely to be a waste of time ad moey. Distributio of sample mea This is a useful place to explai a importat feature of the sample mea. Figures 1 ad 2 show that the meas of samples of Normally distributed variables do themselves have a Normal distributio. It is this fact that makes our defiitio of a cofidece iterval work: oly because the sample mea is Normally distributed ca we assert that 95% of sample meas are with 2 SEs of the populatio mea. However, eve whe the samples are of variables that are ot Normally distributed, the sample meas have a distributio that is ofte very close to a Normal distributio. This is a pheomeo explaied by the Cetral Limit Theorem (CLT) ad is illustrated i figure X (a) origial variable Sample mea Sample mea (b) meas of samples of size 1 (c) meas of samples of size 5 Figure 3 Figure 3 (a) is a histogram of 5 observatio from a populatio that has a skewed distributio. Figure 3 (b) is a histogram of the meas of 5 samples, each of size 1 ad figure 3 (c) is the correspodig histogram for samples of size 5. It is clear that the act of takig a mea of just te observatios has produced a quatity that has a much less skewed distributio ad takig the mea of 5 observatios gives a distributio that appears very close to Normal. A techical descriptio of why this is requires the reader to assimilate the details of the CLT. However, somethig of a heuristic explaatio is as follows. The mea of 6

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