398 CHAPTER 8 Hypothesis Testing Improving the Exact Method A criticism of the exact method is that it is too conservative in the sense that the actual probability of a type I error is always less than or equal to a, and it could be much lower than a. With the exact method, the actual probability of a type I error is less than or equal to A, which is the desired probability of a type I error. A simple continuity correction improves the conservative behavior of the exact method with an adjustment to the P-value that is obtained by subtracting from it the value that is one-half the binomial probability at the boundary, as shown below. (See Exercise 33 “Exact Method.”) This method is easy to apply if technology is available for finding binomial probabilities. Simple Continuity Correction to the Exact Method Left-tailed test: P@value = P1x or fewer2 - 1 2 P1exactly x2 Right-tailed test: P@value = P1x or more2 - 1 2 P1exactly x2 Two-tailed test: P-value = twice the smaller of the preceding left-tailed and right-tailed values The above “simple continuity correction” is described in “Modifying the Exact Test for a Binomial Proportion and Comparisons with Other Approaches,” by Alan Huston, Journal of Applied Statistics, Vol. 33, No. 7. For another improvement that uses weighted tail areas based on a measure of skewness, see the preceding article by Alan Huston. Hypothesis Test: Proportion Access tech supplements, videos, and data sets at www.TriolaStats.com TECH CENTER Statdisk 1. Click Analysis in the top menu. 2. Select Hypothesis Testing from the dropdown menu and select Proportion One Sample from the submenu. 3. Under Alternative Hypothesis select the format used for the alternative hypothesis, enter significance level, claimed proportion (from null hypothesis), sample size, and number of successes. 4. Click Evaluate. StatCrunch 1. Click Stat in the top menu. 2. Select Proportion Stats from the dropdown menu, then select One Sample—With Summary from the submenu. 3. Enter the number of successes and number of observations (n). 4. Select Hypothesis test for p and for H0 enter the claimed value of the population proportion (from the null hypothesis). For HA select the format used for the alternative hypothesis. 5. Click Compute! Minitab 1. Click Stat in the top menu. 2. Select Basic Statistics from the dropdown menu and select 1 Proportion from the submenu. 3. Select Summarized data from the dropdown menu and enter number of events and number of trials (n). 4. Check the Perform hypothesis test box and enter the proportion used in the null hypothesis. 5. Click the Options button and enter the confidence level. (Enter 95.0 for a significance level of 0.05.) For Alternative Hypothesis select the format used for the alternative hypothesis. 6. For Method select Normal approximation to use the same method in this Section and click OK twice. Im se eq Consequence of a False Positive One week before his scheduled release from prison, Michael Kearney was informed that he tested positive for use of an illegal painkiller, and he was then sentenced to 120 days in solitary confinement. Seven months later, it was found that the test result was a false positive. Michael Kearney spent those 120 days in solitary confinement knowing that he had not used any drugs. O b s r p K in
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