During a promotion at a restaurant, each customer has a 25% chance of winning a free appetizer.

1.      During a promotion at a restaurant, each customer has a 25% chance of winning a free appetizer. If we observed a sample of 15 customers and just wanted to record whether they won an appetizer or not, find:

a)      The expected number of people to win a free appetizer.                                                                                      

b)     The probability that exactly 5 people win a free appetizer. Show at least two decimal digits in the final answer.                                                                                                                                                                                       

2.       In a large university (45,000 students) it is known that 71% of the students live on campus. 

a)      If we considered samples of 200 students, describe the sampling distribution of the sample proportion of students who live on campus. (Hint - we said there were three things you needed to mention in a full description, and there is check you must perform to be able to describe one of those 3 things!)       [3]

b)     Show the work in finding the probability that in a sample of 200 students at this university more than 75% live on campus.                                                                                                                             [3]

3.    Tires become illegal when their tread depths fall below a certain value. A particular brand and model of tire has lifespans (number of miles that can be driven before they become illegal) that are approximately normal in distribution with mean 34,000 miles and standard deviation 3,800 miles. Use this information (and tables as needed) to answer parts a-c below.

a)      Determine the probability that a single tire, selected at random, has a lifespan of between 30,000 and 40,000 miles.  Show the work.                                                                                                                  [4]

8b) How many miles would a tire of this brand and model have to last in order for it to be in the top 5% of all tire lifespans?                                                                                                                                            [3]

8c) If a sample of 50 of these tires were collected, what is the probability that the sample mean lifespan would be less than 32,500 miles?                                                                                                                                          

4.  For a statistics class the mean midterm test score was 70 with a standard deviation of 12. For a chemistry midterm test (worth more than 100 points) the mean score was 115 with a standard deviation of 21. 

Zeke score 78 on the stats midterm and scored 129 on the chemistry midterm. On which test did he do better relative to his classmates? Support your answer with work.