7-2 Estimating a Population Mean 337 225 0 25 50 75 100 125 150 175 200 225 Cotinine (ng/mL) People not exposed to smoke Smokers People exposed to smoke FIGURE 7-6 Comparing Confidence Intervals intervals that do overlap, so it is possible that they have the same mean cotinine level. It is helpful to compare confidence intervals or their graphs, but such comparisons should not be used for making formal and final conclusions about equality of means. Chapters 9 and 12 introduce better methods for formal comparisons of means. YOUR TURN. Do Exercise 30 “Second-Hand Smoke.” Estimating a Population Mean When S Is Known In the real world of professional statisticians and professional journals and reports, it is extremely rare that we want to estimate an unknown value of a population mean m but we somehow know the value of the population standard deviation s. If we somehow do know the value of s, the confidence interval is constructed using the standard normal distribution instead of the Student t distribution, so the same procedure from “Estimating a Population Mean” provided earlier in this section can be used with this margin of error: E = za>2 # s2 n (used with known s) The requirements for using this margin of error are the same as those listed earlier for the t distribution: The sample should be a simple random sample and either n 7 30 or the sample appears to be from a normally distributed population. Choosing the Correct Distribution When constructing a confidence interval estimate of the population mean m, it is important to use the correct distribution. Table 7-1 summarizes the key points to consider. TABLE 7-1 Choosing the Correct Distribution Conditions Method s not known and normally distributed population or s not known and n 7 30 Use Student t distribution with E = ta>2 s2 n s known and normally distributed population or s known and n 7 30 (In reality, s is rarely known.) Use normal (z) distribution with E = za>2 # s2 n Population is not normally distributed and n … 30. Use the bootstrapping method (Section 7-4) or some other nonparametric method. Determining Sample Size If we want to collect a sample to be used for estimating a population mean m, how many sample values do we need? When determining the sample size needed to estimate a population mean, we must have an estimated or known value of the population standard deviation s, so that we can use Formula 7-4 shown in the Key Elements box on the next page. Go Figure Gazillion: An extremely large number that doesn’t have a specific value. It is now accepted as a valid word.
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