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A08ans - EXERCISES IN STATISTICS Series A No 8 1 The value...

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EXERCISES IN STATISTICS Series A, No. 8 1. The value of the mean of a random sample of size 20 from a normal pop- ulation is ¯ x = 81 . 2 Find the 95% confidence interval for the mean of the population on the assumption that the variance is V ( x ) = 80. Answer. We have ¯ x = 81 . 2, V ( x ) = 80 , n = 20 and a confidence level of Q = 0 . 95. The confidence interval is derived from the following probability statement: P ¯ x β σ n μ ¯ x + β σ n = Q. The values of β corresponding to Q = 0 . 95 is β = 1 . 960. Therfore the confind- ence interval is ¯ x ± β σ n = 81 . 2 ± 1 . 96 80 20 = [77 . 28 , 85 . 12] . 2. Let ¯ x be the mean of a random sample of size n from an N ( μ, σ 2 ) popu- lation. What is the probability that the interval (¯ x 2 σ/ n, ¯ x + 2 σ/ n ) includes the point μ ? Answer. We have P ¯ x 2 σ n μ ¯ x + 2 σ n = P 2 z = ¯ x μ σ/ n 2 , where z N (0 , 1) is a standard normal variate. From tables, we find that P ( z [0 , 2]) = 0 . 4772. Therefore P ( 2 z 2) = 0 . 9544. 3. The mean of a random sample of size 17 from a normal population is ¯ x = 4 . 17. Determine the 90 % confidence interval for the population mean when the estimate variance of the population is 5.76. Answer. We are given ¯
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