Confidence Intervals for the Mean (s Known) 6.1 298 CHAPTER 6 Confidence Intervals What You Should Learn How to find a point estimate and a margin of error How to construct and interpret confidence intervals for a population mean when s is known How to determine the minimum sample size required when estimating a population mean Estimating Population Parameters Confidence Intervals for a Population Mean Sample Size Number of hours in one day 2 1 1 1 3 1 3 1 2 4 3 3 1 5 5 3 3 3 4 5 2 4 3 3 4 2 2 1 3 5 3 3 2 4 3 2 3 4 5 2 Estimating Population Parameters In this chapter, you will learn an important technique of statistical inference—to use sample statistics to estimate the value of an unknown population parameter. In this section and the next, you will learn how to use sample statistics to make an estimate of the population parameter m when the population standard deviation s is known (this section) or when s is unknown (Section 6.2). To make such an inference, begin by finding a point estimate. A point estimate is a single value estimate for a population parameter. The most unbiased point estimate of the population mean m is the sample mean x. DEFINITION The validity of an estimation method increases when you use a sample statistic that is unbiased and has low variability. A statistic is unbiased if it does not overestimate or underestimate the population parameter. Recall from Chapter 5 that the mean of all possible sample means of the same size equals the population mean. As a result, x is an unbiased estimator of m. When the standard error s 1n of a sample mean decreases by increasing n, it becomes less variable. Finding a Point Estimate A researcher is collecting data about a college athletic conference and its student-athletes. A random sample of 40 student-athletes is selected and their numbers of hours spent on required athletic activities in one week are recorded (see table at left). Find a point estimate for the population mean m, the mean number of hours spent per week on required athletic activities by all student-athletes in the conference. (Adapted from National Collegiate Athletic Association) SOLUTION The sample mean of the data is x = Σx n = 797 40 ≈ 19.9. So, the point estimate for the mean number of hours spent on required athletic activities by all student-athletes in the conference is about 19.9 hours. TRY IT YOURSELF 1 In Example 1, the researcher selects a second random sample of 40 student-athletes and records their numbers of hours spent on required athletic activities in one day (see table at left). Find a point estimate for the population mean m, the mean number of hours spent per day on required athletic activities by all student-athletes in the conference. (Adapted from National Collegiate Athletic Association) Answer: Page A40 EXAMPLE 1 Number of hours in one week 19 18 18 15 21 21 23 20 21 19 16 19 22 15 19 24 20 24 20 17 18 17 19 20 20 20 22 24 22 23 23 21 22 20 17 21 16 18 18 25
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