Central Limit Theorem for Sums Suppose random samples of n=36 are collected from an unknown distribution with the properties =54 and =2.

1. Determine the mean of X. _________
2. Determine the standard deviation of X ___________ (Include four decimal places.)

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. (link: Normal Distribution Calculator)

1. Given that the heights of American women follow the distribution 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.)
2. Determine the z-score of X=5520 for a sample of 85. z=___________ (Include three decimal places.)

Suppose that the amount of time that students spend studying in the library in one sitting is normally distributed with mean 48 minutes and standard deviation 17 minutes. A researcher observed 50 students who entered the library to study. Round all answers to 4 decimal places where possible.

1. What is the distribution of X? X ~ N(____,______)
2. What is the distribution of x? x ~ N(______,______)
3. What is the distribution of x? x~ N(______,______)
4. If one randomly selected student is timed, find the probability that this student's time will be between 47 and 50 minutes. ____________
5. For the 50 students, find the probability that their average time studying is between 47 and 50 minutes. ________
6. Find the probability that the randomly selected 50 students will have a total study time more than 2350 minutes. __________
7. For part e) and f), is the assumption of normal necessary? No or Yes
8. The top 10% of the total study time for groups of 50 students will be given a sticker that says "Great dedication". What is the least total time that a group can study and still receive a sticker? __________ minutes