8-5 Resampling: Using Technology for Hypothesis Testing 425 19. Finding Critical Values of X2 For large numbers of degrees of freedom, we can approximate critical values of x2 as follows: x2 = 1 21z + 22k - 122 Here k is the number of degrees of freedom and z is the critical value(s) found from technology or Table A-2. In Exercise 12 “Spoken Words” we have df = 55, so Table A-4 does not list an exact critical value. If we want to approximate a critical value of x2 in the right-tailed hypothesis test with a = 0.01 and a sample size of 56, we let k = 55 with z = 2.33 (or the more accurate value of z = 2.326348 found from technology). Use this approximation to estimate the critical value of x2 for Exercise 12. How close is it to the critical value of x2 = 82.292 obtained by using Statdisk and Minitab? 20.Finding Critical Values of X2 Repeat Exercise 19 using this approximation (with k and z as described in Exercise 19): x2 = k a1 - 2 9k + z A 2 9kb 3 8-4 Beyond the Basics Key Concept The preceding sections of this chapter included three methods for testing claims about a population proportion, population mean and population standard deviation or variance. Those methods have certain requirements that limit the situations in which they can be used. When some of the requirements are not satisfied, we can often use resampling methods that involve the use of technology to “resample” the original sample data many times. Even when requirements are satisfied, resampling methods can be used along with other methods to provide an additional perspective and insight into the data. It is wise to use “holistic statistics” by applying two or more different methods. Resampling methods can serve as a good “second opinion.” Usefulness of Resampling Methods Resampling methods have the following advantages: 1. There is no requirement about the underlying distribution of data. There is no requirement that the data must be from a normally distributed population. 2. There is no requirement of a minimum sample size. Most of the methods used in the preceding sections of this chapter cannot be used with small samples selected from populations not having normal distributions, but resampling methods can be used instead. See the Technology Project near the end of this chapter for a small sample randomly selected from a population with a distribution that is far from normal. 8-5 Resampling: Using Technology for Hypothesis Testing
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