330 CHAPTER 7 Estimating Parameters and Determining Sample Sizes Estimating a Population Mean It’s rare that we want to estimate the unknown value of a population mean m but we somehow know the value of the population standard deviation s, so we now focus on the realistic situation in which s is not known. Point Estimate As discussed in Section 6-3, the sample mean x is an unbiased estimator of the population mean m. Also, for many populations, sample means tend to vary less than other measures of center. For these reasons, the sample mean x is usually the best point estimate of the population mean m. The sample mean x is the best point estimate of the population mean M. Because even the best point estimate gives us no indication of how accurate it is, we use a confidence interval (or interval estimate), which consists of a range (or an interval) of values instead of just a single value. Confidence Interval The accompanying box includes the key elements for constructing a confidence interval estimate of a population mean m in the common situation where s is not known. Confidence Interval for Estimating a Population Mean with s Not Known Objective Construct a confidence interval used to estimate a population mean. Notation m = population mean n = number of sample values x = sample mean E = margin of error s = sample standard deviation Requirements 1. The sample is a simple random sample. 2. Either or both of these conditions are satisfied: The population is normally distributed or n 7 30. If the second requirement is not satisfied, one alternative is to use the method of bootstrap resampling described in Section 7-4. Confidence Interval Formats: x - E 6 m 6 x + E or x { E or 1x - E, x + E2 • Margin of Error: E = ta>2 # s2 n (Use df = n - 1.) • Confidence Level: The confidence interval is associated with a confidence level, such as 0.95 (or 95%), and a is the complement of the confidence level. For a 0.95 (or 95%) confidence level, a = 0.05. • Critical Value: ta>2 is the critical t value separating an area of a>2 in the right tail of the Student t distribution. • Degrees of Freedom: df = n - 1 is the number of degrees of freedom used when finding the critical value. Round-Off Rule 1. Original Data: When using an original set of data values, round the confidence interval limits to one more decimal place than is used for the original set of data. 2. Summary Statistics: When using the summary statistics of n, x, and s, round the confidence interval limits to the same number of decimal places used for the sample mean. KEY ELEMENTS
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