STAT 400 1 Spring 2015 Discussion 11 A random sample of size n 9 from a normal distribution is obtained 4 4 3 7 5 1 4 3 4 7 3 7 3 5 4 6 4 7 Recall from last discussion x 4 3 and s 0 55 a Construct a 95 two sided confidence interval for the overall standard deviation b Construct a 90 one sided confidence interval for that provides an upper bound for c Construct a 95 one sided confidence interval for that provides a lower bound for 2 An examination of the records for a random sample of 16 motor vehicles in a large fleet resulted in the sample mean operating cost of 26 33 cents per mile and the sample standard deviation of 2 80 cents per mile Assume that operating costs are approximately normally distributed a Construct a 95 confidence interval for the mean operating cost b Construct a 90 confidence interval for the variance of the operating costs 3 Suppose the time spent on a particular STAT 400 homework follows a normal distribution with an overall standard deviation of 28 minutes and an unknown mean a Suppose a random sample of 49 students is obtained Find the probability that the average time spent on the homework for students in the sample is within 5 minutes of the overall mean b A sample of 49 students has a sample mean of 234 minutes spent on the homework Construct a 90 confidence interval for the overall mean time spent on the homework c What is the minimum sample size required if we want to estimate the overall mean time spent on the homework to within 5 minutes with 90 confidence 4 An economist states that 10 of Springfield s labor force is unemployed A random sample of 400 people in the labor force is obtained of whom 28 are unemployed a Construct a 95 confidence interval for the unemployment rate in Springfield b What is the minimum sample size required in order to estimate the unemployment rate in Springfield to within 2 with 95 confidence Use the economist s guess c What is the minimum sample size required in order to estimate the unemployment rate in Springfield to within 2 with 95 confidence Assume no information is available 5 Let the joint probability density function for X Y be f x y 128 x 2 e 4 y 0 x y zero otherwise a Set up the integral s for the probability P X Y 0 64 b Find the marginal probability density function of X f X x Include its support c Find the marginal probability density function of Y f Y y Include its support 1 a Construct a 95 two sided confidence interval for the overall standard deviation n 1 s 2 n 1 s 2 2 2 1 2 2 Confidence Interval for 2 0 05 1 2 0 975 2 0 025 number of degrees of freedom n 1 9 1 8 2 17 54 2 2 180 1 2 2 9 1 0 3025 9 1 0 3025 17 54 2 180 Confidence Interval for b 0 13797 1 11009 0 13797 1 11009 0 3714 1 0536 Construct a 90 one sided confidence interval for that provides an upper bound for c 0 n 1 s 2 0 9 1 0 3025 21 12 20 90 3 490 0 0 8327 3 490 Construct a 95 one sided confidence interval for that provides a lower bound for n 1 s 2 2 9 1 0 3025 15 51 2 20 05 15 51 0 395 2 An examination of the records for a random sample of 16 motor vehicles in a large fleet resulted in the sample mean operating cost of 26 33 cents per mile and the sample standard deviation of 2 80 cents per mile Assume that operating costs are approximately normally distributed s 2 80 X 26 33 a n 16 Construct a 95 confidence interval for the mean operating cost t 0 025 2 131 n 1 15 degrees of freedom X t 2 s n 26 33 2 131 2 80 26 33 1 49 16 24 84 27 82 b Construct a 90 confidence interval for the variance of the operating costs n 1 15 degrees of freedom n 1 s 2 n 1 s 2 2 2 1 2 2 20 05 25 00 20 95 7 261 16 1 2 80 2 16 1 2 80 2 25 00 7 261 4 704 16 196 3 Suppose the time spent on a particular STAT 400 homework follows a normal distribution with an overall standard deviation of 28 minutes and an unknown mean a Suppose a random sample of 49 students is obtained Find the probability that the average time spent on the homework for students in the sample is within 5 minutes of the overall mean P 5 X 5 X n 49 large plus the distribution we sample from is normal Z n 5 5 P 5 X 5 P Z 28 28 49 49 P 1 25 Z 1 25 0 8944 0 1056 0 7888 b A sample of 49 students has a sample mean of 234 minutes spent on the homework Construct a 90 confidence interval for the overall mean time spent on the homework is known 0 10 234 1 645 c X z The confidence interval 2 0 05 28 49 2 n z 1 645 2 234 6 58 227 42 240 58 What is the minimum sample size required if we want to estimate the overall mean time spent on the homework to within 5 minutes with 90 confidence 2 z 2 2 1 645 28 n 84 86 5 Round up n 85 4 An economist states that 10 of Springfield s labor force is unemployed A random sample of 400 people in the labor force is obtained of whom 28 are unemployed n 400 a y 28 p y 28 0 07 n 400 Construct a 95 confidence interval for the unemployment rate in Springfield p z The confidence interval 2 95 confidence level 0 07 1 960 b 0 07 0 93 400 0 05 p 1 p n 2 0 025 0 07 0 025 z 1 960 2 0 045 0 095 What is the minimum sample size required in order to estimate the unemployment rate in Springfield to within 2 with 95 confidence Use the economist s guess Use p 0 10 the economist s guess 0 02 2 z 2 1 960 n 2 p 1 p 0 10 0 90 864 36 0 02 Round up c n 865 What is the minimum sample size required in order to estimate the unemployment rate in Springfield to within 2 with 95 confidence Assume no information is available Use p 0 50 since no information is available 2 z 2 1 960 n 2 p 1 p 0 50 0 50 2401 0 02 0 02 5 Let the joint probability density function for X Y be f x y 128 x 2 e 4 …
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