STAT 400 Lecture AL1 Spring 2015 Dalpiaz Answers for 5 5 7 1 Let Z be a N 0 1 standard normal random variable Then X Z 2 has a chi square distribution with 1 degree of freedom 2 MX t E et Z 1 2 e e t z2 1 2 z 2 2 1 2 t since 1 2 t 1 2 e z 2 X has a 2 1 2 t 2 e z2 2 dz dz 1 1 2 t 12 t 1 2 is the p d f of a N 0 1 random variable 1 2t 2 1 distribution Let X and Y be be two independent 2 random variables with m and n degrees of freedom respectively Then W X Y has a chi square distribution with m n degrees of freedom MX t 1 1 2 t m 2 t 1 2 MW t MX t MY t W has a MY t 1 1 2 t m n 2 2 m n distribution 1 1 2 t n 2 t 1 2 t 1 2 1 A manufacturer of TV sets wants to find the average selling price of a particular model A random sample of 25 different stores gives the mean selling price as 342 with a sample standard deviation of 14 Assume the prices are normally distributed Construct a 95 confidence interval for the mean selling price of the TV model is unknown n 25 small The confidence interval X t 2 s n n 1 25 1 24 degrees of freedom 95 confidence level 342 2 064 14 25 0 05 342 5 78 0 025 2 t 24 2 064 2 336 22 347 78 2 The following random sample was obtained from N 2 distribution 16 a 12 18 13 21 15 8 17 Compute the sample mean and the sample standard deviation x 16 12 18 13 21 15 8 17 120 15 x n 8 8 x x2 x x x x x 2 16 12 18 13 21 15 8 17 256 144 324 169 441 225 64 289 16 12 18 13 21 15 8 17 1 3 3 2 6 0 7 2 1 9 9 4 36 0 49 4 0 112 OR 1912 x 2 2 s x 2 n 1 n 1912 120 2 8 7 s s b 16 2 s 2 x x n 1 2 112 16 7 16 4 Construct a 95 confidence interval for Confidence interval x t 2 95 confidence level 0 05 15 2 365 4 8 s n 15 3 3446 0 025 2 n 1 7 degrees of freedom t 0 025 2 365 11 6554 18 3446 b Construct a 90 confidence interval for Confidence interval x t 2 90 confidence level 0 10 15 1 895 c 4 8 s n 0 05 2 15 2 68 t 0 05 1 895 12 32 17 68 Construct a 90 confidence upper bound for s X t n n 1 7 degrees of freedom 4 15 1 415 8 d n 1 7 degrees of freedom t 0 10 1 415 17 Construct a 99 confidence lower bound for s X t n n 1 7 degrees of freedom 4 15 2 998 8 t 0 01 2 998 10 76
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