ecostat ps4

1 PROBLEM SET 4 CONTINUOUS PROBABILITY DISTRIBUTION SAMPLING DISTRIBUTION 다음의 두 문제를 풀어서 제출할 것 1 Heights of 10-year-old children are normally distributed with a mean of 52 inches and a standard deviation of 4 inches a Find the probability that one randomly selected 10-year-old child is taller than 54 inches b Find the probability that two randomly selected 10-year-old children are both taller than 54 inches Find the probability that the mean height of two randomly selected 10-year-old children is greater than 54 inches 2 The following data give the number of pets owned for a population of 4 families Family A B C D Number of Pets Owned 2 1 4 3 a Find the sampling distribution of X The sample size is 2 b Use the sampling distribution in Question a and directly recalculate the mean and standard deviation of X Compare the two answers What do you conclude? 아래의 문제는 숙제가 아니므로 답안을 제출할 필요가 없다 Part A Uniform distribution 1 The time it takes a student to finish a chemistry test is uniformly distributed between 50 and 70 minutes a What is the probability density function for this uniform distribution? b Find the probability that a student will take between 40 and 60 minutes to finish the test c Find the probability that a student will take no less than 55 minutes to finish the test d What is the expected amount of time it takes a student to finish the test? e What is the variance for the amount of time it takes a student to finish the test? Part B Normal distribution 1 If Z is a standard normal random variable, find the following probabilities: a P(Z ≤ -177) b P(Z ≥ -196) c P(035 ≤ Z ≤ 085) d P(-288 ≤Z ≤ –215) e P(Z ≤ 145) 2 If X is a normal random variable with a mean of 45 and a standard deviation of 8, find the following probabilities: a P(X ≥ 50) b P(X ≤ 32) c P(37 ≤ X ≤ 48) d P(50 ≤ X ≤ 60) e P(X = 45) 2 3 If Z is a standard normal random variable, find the value z for which: a P(0 ≤ Z ≤ z) = 035 b P(–z ≤ Z ≤ z) = 0142 c P(-z ≤ Z ≤ 0) = 0441 4 The recent average starting salary for new college graduates in computer information systems is 47,500 Assume salaries are normally distributed with a standard deviation of 4,500 a What is the probability of a new graduate receiving a salary between 45,000 and 50,000? b What is the probability of a new graduate getting a starting salary in excess of 55,000? c What percent of starting salaries are no more than 42,250? d What is the cutoff for the bottom 5% of the salaries? e What is the cutoff for the top 3% of the salaries? Part C: Sampling distribution 1 A population has a mean of 150 and a standard deviation of 40 A sample of 100 observations will be selected at random from the population a What is the expected value of the sample mean? b What is the standard deviation of the sample mean? c What is the shape of the sampling distribution of the sample mean 2 Assume the time needed by a worker to perform a maintenance operation is normally distributed with a mean of 70 minutes and a standard deviation of 6 minutes What is the probability that the average time needed by a sample of 5 workers to perform the maintenance in between 63 minutes and 68 minutes? 3 The manager of a restaurant believes that waiters and waitresses who introduced themselves by telling their names will get larger tips than those who don’t In fact, she claims that average tip for former group is 18% who while that of latter is only 15% If tips are normally distributed with a standard deviation of 3%, what is the probability that in a sample of 10 tips recorded from waiters and waitresses who introduced themselves and 10 tips from waiters and waitresses who don’t, the mean of the former will exceed that of the latter? 4 The chairman of a statistics department in a certain college believes that 70% of the department’s graduate assistantships are given to international students A random sample of 50 graduate assistants is taken a Assume that the chairman is correct and p = 070 What is the sampling distribution of the sample proportion pˆ ? Explain b Find the expected value and the standard error of the sampling distribution of pˆ c What is the probability that the sample proportion pˆ will be between 065 and 073? d What is the probability that the sample proportion pˆ will be within ± 05 of the population proportion p?