# chap6 - Chapter 6 Inference for a Population 6.1 Study...

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Chapter 6 Inference for a Population 6.1 Study Suggestions Let X be a random variable with the binomial sam- pling distribution with parameters n and p . In Chap- ter 5 you learned that X can arise when taking a ran- dom sample from a finite population, or when ob- serving a sequence of Bernoulli trials. In Chapter 5 the focus was on using the values of n and p to com- pute the probability of an event involving X . Chap- ter 6 considers the more realistic situation in which the value of p is unknown. In Chapter 6 you learn how the observed value of X can be used to draw an inference about the unknown value of p . The first example of inference is the point esti- mate. The proportion of successes in the sample , ˆ p = x/n , is the point estimate of the proportion of successes in the population . The main advantage of the point estimate is that it is easy to understand and to interpret, and its main disadvantage is that it is usually incorrect. The confidence interval es- timate sacrifices some of the simplicity of a point es- timate to achieve a high confidence of being correct. (Remember that the terms correct and incorrect do not refer to the estimate being computed without or with error; they refer to whether the stated estimate equals, in the case of a point estimate, or contains, in the case of an interval estimate, the actual value of p .) A confidence interval estimate has two parts to it—the interval and the confidence level. The con- fidence level is selected by the researcher and the in- terval is computed from the data. For example, if a researcher selects 95 percent confidence, then the interval is ˆ p ± 1 . 96 s ˆ p ˆ q n . The “95 percent confidence” in the 95 percent confi- dence interval is interpreted as follows. Before col- lecting data, the probability is 95 percent that the data to be obtained will yield an interval that in- cludes the actual value of p . My students find Ta- ble 6.2 and Figure 6.1 on page 197 of the text to be helpful in their achieving an understanding of what “confidence” means. In particular, note that in a real application in which the value of p is unknown, a researcher will never know whether a confidence in- terval is correct. If you find this uncertainty unac- ceptable, then you will probably never consider con- fidence intervals to be of much practical value. I view a confidence interval as a reasonable middle posi- tion between the extremes of taking a census (which is usually prohibitively expensive and time consum- ing), and collecting no data. In the majority of applications I have seen, the researcher simply wants to estimate the value of p . Occasionally, the researcher will consider one of the possible values of p to be of special interest. In this latter case, the researcher can perform a hypothesis test with the null hypothesis stating that p equals the special value of interest, p 0 . With the solid back- ground of Fisher’s test, my students find the hypoth- esis testing in Chapter 6 to be fairly easy to under-

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chap6 - Chapter 6 Inference for a Population 6.1 Study...

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