332 CHAPTER 7 Estimating Parameters and Determining Sample Sizes so in this case there are 9 degrees of freedom.) For the methods of this section, the number of degrees of freedom is the sample size minus 1. Degrees of freedom = n − 1 ■ Finding Critical Value t A,2 A critical value ta>2 can be found using technology or Table A-3. Technology can be used with any number of degrees of freedom, but Table A-3 can be used for select numbers of degrees of freedom only. If using Table A-3 to find a critical value of ta>2, but the table does not include the exact number of degrees of freedom, you could use the closest value, or you could be conservative by using the next lower number of degrees of freedom found in the table, or you could interpolate. ■ The Student t distribution is different for different sample sizes. (See Figure 7-4 for the cases n = 3 and n = 12.) 0 Standard normal distribution Student t distribution with n5 12 Student t distribution with n5 3 FIGURE 7-4 Student t Distributions for n = 3 and n = 12 The Student t distribution has the same general shape and symmetry as the standard normal distribution, but it has the greater variability that is expected with small samples. ■ The Student t distribution has the same general symmetric bell shape as the standard normal distribution, but has more variability (with wider distributions), as we expect with small samples. ■ The Student t distribution has a mean of t = 0 (just as the standard normal distribution has a mean of z = 0). ■ The standard deviation of the Student t distribution varies with the sample size, but it is greater than 1 (unlike the standard normal distribution, which has s = 1). ■ As the sample size n gets larger, the Student t distribution gets closer to the standard normal distribution. Procedure for Constructing a Confidence Interval for M Confidence intervals can be easily constructed with technology or they can be manually constructed by using the following procedure. 1. Verify that the two requirements are satisfied: The sample is a simple random sample and the population is normally distributed or n 7 30. 2. With s unknown (as is usually the case), use n - 1 degrees of freedom and use technology or a t distribution table (such as Table A-3) to find the critical value ta>2 that corresponds to the desired confidence level. Estimating Wildlife Population Sizes The National Forest Management Act protects endangered species, including the northern spotted owl, with the result that the forestry industry was not allowed to cut vast regions of trees in the Pacific Northwest. Biologists and statisticians were asked to analyze the problem, and they concluded that survival rates and population sizes were decreasing for the female owls, known to play an important role in species survival. Biologists and statisticians also studied salmon in the Snake and Columbia rivers in Washington State, and penguins in New Zealand. In the article “Sampling Wildlife Populations” (Chance, Vol. 9, No. 2), authors Bryan Manly and Lyman McDonald comment that in such studies, “biologists gain through the use of modeling skills that are the hallmark of good statistics. Statisticians gain by being introduced to the reality of problems by biologists who know what the crucial issues are.” T F a p g i n
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