332 CHAPTER 6 Confidence Intervals Confidence Intervals for S 2 and S You can use the critical values x 2 R and x 2 L to construct confidence intervals for a population variance and standard deviation. The best point estimate for the variance is s2 and the best point estimate for the standard deviation is s. Because the chi-square distribution is not symmetric, the confidence interval for s 2 cannot be written as s2 { E. You must perform separate calculations for the endpoints of the confidence interval, as shown in the next definition. The c@confidence intervals for the population variance and standard deviation are shown. Confidence Interval for S 2: 1n - 12s2 x 2 R 6 s 2 6 1n - 12s2 x 2 L Confidence Interval for S: B1n - 12s2 x 2 R 6 s 6 B1n - 12s2 x 2 L The probability that the confidence intervals contain s 2 or s is c, assuming that the estimation process is repeated a large number of times. DEFINITION Picturing the World The Florida panther is an endangered subspecies of Puma concolor (also known as puma, cougar, or mountain lion). About 120 to 230 adult panthers are in existence, primarily in southwest Florida.The map below shows occurrence data (yellow circles). Recently, legislation was passed in Florida to improve conservation efforts. Tampa Orlando Miami In a study of six Florida panther litters, it was found that the mean litter size was 2.86 kittens, with a standard deviation of 0.82. (Source: Florida Fish and Wildlife Conservation Commission) Construct a 90% confidence interval for the standard deviation of the litter size for Florida panthers. Assume the litter sizes are normally distributed. Constructing a Confidence Interval for a Variance and Standard Deviation In Words In Symbols 1. Verify that the population has a normal distribution. 2. Identify the sample statistic n d.f. = n - 1 and the degrees of freedom. 3. Find the point estimate s2. s2 = Σ1x - x22 n - 1 4. Find the critical values x 2 R Use Table 6 in Appendix B. and x 2 L that correspond to the given level of confidence c and the degrees of freedom. 5. Find the left and right Left Endpoint Right Endpoint endpoints and form the confidence interval for the population variance. 6. Find the confidence Left Endpoint Right Endpoint interval for the population standard deviation by taking the square root of each endpoint. 1n - 12s2 x 2 R 6 s 2 6 1n - 12s2 x 2 L B1n - 12s2 x 2 R 6 s 6 B1n - 12s2 x 2 L GUIDELINES
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