# chapter11 - Chapter 11 Chance Errors in Samples 11.1...

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Chapter 11 Chance Errors in Samples 11.1 Percentages A health study is done on a representative cross section of 4,738 adults. The researcher can sample, say, only 100 of them. Our question is whether the sample is representative? Assume the population had 3,032 (64%) males. It is possible that we might pick a sample and count 63 males in the sample. We might repeat this, say, 250 times; that is, we sample 250 times, samples of 100 individuals and record the sum of males. We could get: By actually counting, we find that only 18 of the 250 samples of 100 in- dividuals had exactly 64 males. This means that getting a sample with 1

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2 CHAPTER 11. CHANCE ERRORS IN SAMPLES the same percentage as the population (64%) is very rare. Below is a his- togram of the number of men from the samples of size 100 persons (i.e. 250 repetitions): Figure 11.1: Sampling distribution of average, n = 100 The proportion of male in the population remains at 64%, but the proportion in any one sample could vary due to chance error. But what if we increase the sample size from 100 to say 400? We get the following distribution: Figure 11.2: Sampling distribution of average, n = 400
11.2. COMPARING THE SE’S OF SUM AND PERCENTAGES 3 Given that the percentage in any sample is equal to the percentage in the population plus some chance error, we find that the bigger the sample, the closer the sample proportion to the population proportion. In other words, with a simple random sample, the expected value for the sample percentage equals the population percentage. However, as we have seen, the percentage of a certain sample need not be exactly equal to the population percentage - it is off because of chance error, which is measured by the standard error (of the percentage). Let’s take an example to concretize ideas. Assume a box model con- taining 1’s and 0’s as follows: Immediately, we realize the SD of the box to be . 64 × . 36 If a sample of 100 random draws with replacement are made, the SE of the sum of draws, i.e. n × SD, is 4.8. To get the SE of percentage, then, is SE = SE of sum n × 100% (11.1) which for our example works out to 4 . 8%. That is, the sampling distribu- tion of percentage of 1’s from a sample has an expected value of 64% and standard error of 4 . 8%.

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