Question

Question 1 Suppose S is the sample variance based on a sample size n from a normal population with unknown mean and variance. Derive a 100(1-a)% a) upper confidence bound for o?. b) lower confidence bound for o?. c) Arik the astronaut decides to prepare for his upcoming trip by experimenting with homemade rockets (using Alka-Seltzer tablets and water to launch them into the air)! He wants to understand the variability of peak height for his rocket design so he constructs 8 rockets, launches each off the top of the LSC and records the peak height. The sample standard deviation of the heights was 3.4 meters. Using this information and your results in the previous parts, provide a 95% upper confidence bound on the true standard deviation of peak heights for Arik's design.

Answer 1

al suppose s² is the sample variance based on sample size n from a normal distribution with Unicrown mean & variance Derive a 100 (1-2)%. @ Upper confidence bound for o2¢ 7 lower confidence bound for r2 * Before we can dessive or develop a confidence Interval for variance (62) we need other distribution ie chi-square distribution.

y? = (halle (1-2)100% of x² values are in interval cn- degree of freedom 212 - 4 x3312 din x 112 x 12 x² are the critical values for the Xidla chi- square test * we get following inequality 1 x 412 < *² < X 2 I

but n2 = (h.1)? Xian < Coletes a M2 dla o2 < (n-1) s2 (ni)s? x 212 confidence interval for r2 A (1-2) X100% is: lower internal bound or level= (n-1) s e 2412 Upper level = (holls? Ki-1/2

© we have: n = sample size = 8, S = simple standaod deviahon = 3.4. a x=95% upper confidence bound on the true standard deviation = I (n-1)52 - upper . confidence bound ch-)) = 7 $? (3.4)2 = 11:56 F1- dl Kidla = x - ogs 2

= xo.ate = 1.6899) -(from chi-square table) ☆ upper confidence bound for standard deviation J +1:56 1.56 -16.91437 U 1.6899 lower confidence bound for standard deviation Kaip 16.0128] = 12 7 856 / 5 2.248) = 2 V 16.0128 that population standard we can say 95% confident deviation is lies within 7 2-248