Math 280 - Homework 2 The Central Limit Theorem and the Law of Large Numbers Sup
ID: 3059576 • Letter: M
Question
Math 280 - Homework 2 The Central Limit Theorem and the Law of Large Numbers Suppose the heights of adult males in a certain population are normally distributed with mean height 70 inches and standard deviation 3 inches. 2. The mean weight for professors in my department (15 people) is 140 pounds; the mean for the 5 professors in the math department is 135 pounds; and the mean weight of the 20 professors in the chemistry department is 160 pounds. What is the overall mean weight of all these people? 3. The distribution of the length of hospital stay in days on a cardiac unit is often positively skewed due to very long stays by some patients. Based on many years of collected data, the mean (H) and standard deviation (a) of length of stay (measured in days) for cardiac surgery patients are known to be-6, = 10, what is the sampling distribution of the sample mean for length of hospital stay (in days) for a sample of 81 patients (ie. what is the shape of the sampling distribution, its mean and its dispersion?) 4. In our neighborhood a trick-or-treater visits an average of 46 houses with a standard deviation of 8. (a) Assuming that the distribution is approximately normal, what is the probability that your child will visit fewer than 42 houses on Halloween? (b) 16 children were randomly selected. What is the probability that their mean number of visits is between 48 and 55? 5. Based on the service records from the past year, the time in hours that a technician re-quires to complete preventative maintenance on an air conditioner follows a strongly skewed distribution, with most jobs taking just a few minutes but with a long tail; the mean time is one hour and the dispersion,o, is 1 hour. Your company has a contract to service 70 air conditioners in a school, and you budget 1.1 hours per unit. Will this be enough, i.e. what is the probability that the technicians will not complete the work in the allotted time? 6. The length of a pregnancy is normally distributed with mean 40 weeks and standard devi-ation 2.5 weeks. Premature births are defined as births prior to 37 weeks while post-mature birth are those that occur after 42 weeks. (a) what is the probability that the next infant to be born in a particular hospital is not premature? (b) what is the probability that the next infant born at the hospital is not post-nature? (e) what is the probability that the gestational age of the next infant born is either prema-ture or post-mature? (d) What is the probability that the mean gestational age for the next 49 infants born in the hospital is between 37 and 42 weeks?Explanation / Answer
Math 280 - Homework 2 The Central Limit Theorem and the Law of Large Numbers Suppose the heights of adult males in a certain population are normally distributed with mean height 70 inches and standard deviation 3 inches. 2. The mean weight for professors in my department (15 people) is 140 pounds; the mean for the 5 professors in the math department is 135 pounds; and the mean weight of the 20 professors in the chemistry department is 160 pounds. What is the overall mean weight of all these people? 3. The distribution of the length of hospital stay in days on a cardiac unit is often positively skewed due to very long stays by some patients. Based on many years of collected data, the mean (H) and standard deviation (a) of length of stay (measured in days) for cardiac surgery patients are known to be-6, = 10, what is the sampling distribution of the sample mean for length of hospital stay (in days) for a sample of 81 patients (ie. what is the shape of the sampling distribution, its mean and its dispersion?) 4. In our neighborhood a trick-or-treater visits an average of 46 houses with a standard deviation of 8. (a) Assuming that the distribution is approximately normal, what is the probability that your child will visit fewer than 42 houses on Halloween? (b) 16 children were randomly selected. What is the probability that their mean number of visits is between 48 and 55? 5. Based on the service records from the past year, the time in hours that a technician re-quires to complete preventative maintenance on an air conditioner follows a strongly skewed distribution, with most jobs taking just a few minutes but with a long tail; the mean time is one hour and the dispersion,o, is 1 hour. Your company has a contract to service 70 air conditioners in a school, and you budget 1.1 hours per unit. Will this be enough, i.e. what is the probability that the technicians will not complete the work in the allotted time? 6. The length of a pregnancy is normally distributed with mean 40 weeks and standard devi-ation 2.5 weeks. Premature births are defined as births prior to 37 weeks while post-mature birth are those that occur after 42 weeks. (a) what is the probability that the next infant to be born in a particular hospital is not premature? (b) what is the probability that the next infant born at the hospital is not post-nature? (e) what is the probability that the gestational age of the next infant born is either prema-ture or post-mature? (d) What is the probability that the mean gestational age for the next 49 infants born in the hospital is between 37 and 42 weeks?