Problem 1: Exponential Distribution and Poisson Process
This exam was written, with the intention of a student not using their notes or the textbook. A formula sheet is included, that should have any
formula you might need. If you attempt to use your notes or the textbook to look things up, you will likely run out of time. You may use a calculator. Please allocate your time appropriately. You must show your calculations to receive full or partial credit. You may round answers to 4 decimal places. You have 45 minutes to complete the exam, and an additional 15 minutes to submit the exam on Gradescope. Please leave yourself the full 15 minutes to submit your exam (you do not need to include images of the formula sheet).
Late submissions will be enabled within Gradescope (submissions will still be accepted within one hour of the exam submission window closing), but will be significantly penalized. No exams will be accepted after the hour following the exam submission window closing. Good Luck
Problem 1. The time between large wildfires in Santa Barbara County follow an exponential distribution with a mean time of 3 months (or 0.25 years).
a. Find the probability that the time between large wildfires in Santa Barbara County is less than 2 months.
b. Find the probability that the time between large wildfires in Santa Barbara County is between 3 and 5 months.
c. What is the probability that between 3 and 4 large wildfires (inclusive) occur in a given year? (Hint, if the time between wildfires can be modeled by an exponential distribution with mean 1/λ, then the number of events per time period can be modeled with a Poisson distribution with mean λ.)
Problem 2. The 42nd annual UCSB Turkey Trot 10k running race, is run the Saturday before the Thanksgiving holiday. The times for the runners are normally distributed with a mean time of 61 minutes and a standard deviation of 8 minutes. A random sample of 50 finishing times is collected.
a. What is the probability that the sample mean time is less than 54 minutes?
b. What is the probability the sample mean time is between 69 minutes and 73 minutes?
c. Construct a 95% acceptance interval for the sample mean.
Problem 3. The soccer balls used by the UCSB soccer teams have a mean circumference of 27.3 inches. Kyle, the team equipment manager tell us that the circumference follows a normal distribution with a standard deviation of 0.6 inches.
a. Per NCAA rules, regulation soccer balls are required to have a circumference between 27 and 28 inches. What proportion of UCSB’s soccer balls will be allowed to be used in competition?
b. NCAA rules also specify the ball weight be between 14 and 16 ounces (at the start of the game). Kyle tells us that the team’s soccer ball weights also follow a normal distribution with a mean weight of 15.3 ounces and a standard deviation of 0.9 ounces.
What is the probability that a randomly selected ball will not be allowed because it is too heavy?
c. What is the probability that a randomly selected ball will not be allowed because it is too light?
Problem 4. Every day at lunch, the Subway sandwich shop will use between zero and three gallons of guacamole. Historical data tells us that any amount of guacamole, up to the maximum of three gallons, is equally likely.
a. If Subway makes $30 per gallon of guacamole sold, how much money from lunchtime guacamole sales does Subway make on average?
b. What is the probability that Subway makes between $52 and $ 76 dollars in lunchtime guacamole sales on any given day?
c. What is the probability that Subway sells between two and four gallons of guacamole
on a given day?