HW3_Solutions - Chapter 6 Interval Estimation EXERCISES 6.1...

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Unformatted text preview: Chapter 6 Interval Estimation EXERCISES 6.1 6.1.1. (a) We are 99% confident that the estimate value of the parameter lies in the confidence interval. (b) 99% confidence interval is wider (c) When is known but 2 is unknown we use t-distribution for the sample size n 30. If the distribution is binomial and there are enough number of samples such that np 5 and np( 1 p) 5, then we use normal approximation. (d) More the information higher the confidence interval. So the sample size is inversely proportional to the width of the confidence interval. 6.1.2. For y = U as a pivot p ( a < U < b ) = 0.98 F y (a) = 0.01, F y (b) = 0.99 This implies a n = 0.01, b n = 0.99 a = n 0.01, b = n 0.99 p n 0.01 < U < n 0.99 = 0.98 98% confidence interval for is U n 0.99 , U n 0.01 6.1.3. (a) p 2.81 x / n 2.75 = k p 2.81 n x 2.75 n = k p 2.81 n x 2.75 n + x = k p x 2.75 n x + 2.81 n = k 99 100 CHAPTER 6 Interval Estimation (b) Confidence interval for is given by x 2.75 n , x + 2.81 n (c) Confidence level = k 6.1.4. n = 50 x = 15.65 = 0.59 95% confidence interval for mean is x z 0.025 n , x z 0.025 n = 15.65 1.96 0.59 50 , 15.65 + 1.96 0.59 50 = ( 15.945, 15.80 ) 6.1.5. (a) Here x i N( , 2 ) (n 1 )s 2 2 2 (n 1 ) Pivot = (n 1 )s 2 2 ; where only 2 is unknown. p a < (n 1 )s 2 2 < b = 1 p 2 1 / 2 < (n 1 )s 2 2 < 2 / 2 = 1 p 1 2 1 / 2 < 2 (n 1 )s 2 < 1 2 / 2 = 1 p (n 1 )s 2 2 / 2 < 2 < (n 1 )s 2 2 1 / 2 = 1 So ( 1 ) 100% confidence interval for 2 is given by (n 1 )s 2 2 / 2 < 2 < (n 1 )s 2 2 1 / 2 (b) n = 21, x = 44.3, s = 3.96 = 0.1, 1 = 0.9, / 2 = 0.05 2 1 / 2,20 = 2 0.95,20 = 10.851 2 / 2 = 2 0.05,20 = 31.410 90% confidence interval is given by (n 1 )s 2 2 / 2 < 2 < (n 1 )s 2 2 1 / 2 = 20 ( 3.96 ) 2 31.41 < 2 < 20 ( 3.96 ) 2 10.851 We are 90% confident that 2 lies in the interval (9.985, 28.903). 6.1.9. (a) p a x / n b = 1 p z / 2 < x / n < z / 2 = 1 p z / 2 n < x < z / 2 n = 1 Instructors Solutions Manual 101 p z / 2 n x < < z / 2 n x = 1 p x z / 2 n < < x + z / 2 n = 1 So the confidence interval is x z / 2 n , x + z / 2 n ....
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This note was uploaded on 11/14/2010 for the course AMS 312 taught by Professor Zhu,w during the Spring '08 term at SUNY Stony Brook.

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HW3_Solutions - Chapter 6 Interval Estimation EXERCISES 6.1...

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