1. Application of The Central Limit Theorem for Sums 100 North Main Street is the tallest building in Winston-Salem, NC standing at 460ft tall (5520inches). Use the scenario above to determine the selected probabilities below. You may wish to use the Normal Distribution Calculator hosted by the University of Iowa's Department of Mathematical Sciences. Remember: the formatting of this calculator may vary slightly from what is used in class.

a. Given that the heights of American women follow the distribution N(65,3.5)N(65,3.5), what is the probability of that a random sample of 85 women, stacked head-to-foot, would be at least as tall as 100 North Main Street? P(X5520)= _______? (Include three decimal places.)

b. Determine the z-score of X=5520X=5520 for a sample of 85. z=________? (Include three decimal places.)

2. The lengths of adult males' hands are normally distributed with mean of 192 mm and a standard deviation is 7.5 mm. Suppose that 41 individuals are randomly chosen. Round all answers to 4 where possible.

a. What is the distribution of xx ~ N(____,____)

b. For the group of 41, find the probability that the average hand length is more than 193. ___?

c. Find the third quartile for the average adult male hand length for this sample size.___?

d. For part b), is the assumption that the distribution is normal necessary? No? Yes?

3. The amount of coffee that people drink per day is normally distributed with a mean of 16 ounces and a standard deviation of 6.5 ounces. 32 randomly selected people are surveyed. Round all answers to 4 decimal places where possible.

a. What is the distribution of X?X ~ N(___,___)

b. What is the distribution of xx? xx ~ N(___,__)

c. What is the probability that one randomly selected person drinks between 15.5 and 16.1 ounces of coffee per day?

d. For the 32 people, find the probability that the average coffee consumption is between 15.5 and 16.1 ounces of coffee per day. ?___

e. For part d), is the assumption that the distribution is normal necessary? No? Yes?

f. Find the IQR for the average of 32 coffee drinkers.___? Q1 =___ ounces Q3 = ___ounces IQR: ____ounces

4. The average time to run the 5K fun run is 21 minutes and the standard deviation is 2.5 minutes. 46 runners are randomly selected to run the 5K fun run. Round all answers to 4 decimal places where possible and assume a normal distribution.

a.What is the distribution of XX? XX ~ N(__,__)

b.What is the distribution of xx? xx ~ N(___,___)

c. What is the distribution of xx? xx ~ N(__,__)

d. If one randomly selected runner is timed, find the probability that this runner's time will be between 20.5471 and 20.8471 minutes.

e. For the 46 runners, find the probability that their average time is between 20.5471 and 20.8471 minutes. __?

f. Find the probability that the randomly selected 46 person team will have a total time of more than 979.8 __?

g. For parts e) and f), is the assumption of normal necessary? Yes?No?

h. The top 20% of all 46 person team relay races will compete in the championship round. These are the 20% lowest times. What is the longest total time that a relay team can have and still make it to the championship round? ___minutes?