MATH 314 Homework 05

Due 2025-10-08 by 11:59pm

1. Car batteries have a limited lifespan. Let's consider a particular brand of car batteries with a mean lifespan of 5 years.

   a. Identify the appropriate distribution for this question.

   b. What is the probability that a car battery of this brand will fail within the first 3 years of use?

   c. For how many years will 95% of car batteries last?

   d. If a car owner wants to be 95% confident that their battery will last at least 6 years, what should be the mean lifespan of the battery? This one is tricky.

2. In certain remote habitats, rare species can be difficult to observe. Ecologists often rely on random sampling to estimate population densities and study their behavior. Assume, on average, 4 sightings of a rare bird species are recorded every week in the wild.

   a. Identify the appropriate distribution for this question.

   b. What is the probability that in a given week, there will be exactly 3 sightings of this bird species?

   c. What is the probability that in a given week, there will be 5 or more sightings?

   d. After a reforestation initiative, the number of sightings increases to an average of 7 per week. If an ecologist spends a week observing, what is the probability that they will record at least one sighting of this bird species?

3. A software development team is working on improving the response time of an algorithm. Currently, when given random inputs, the algorithm successfully executes within the desired response time in 92% of cases. Suppose the team runs the algorithm with 100 random inputs as a testing phase.

   a. What is the probability that the algorithm will execute within the desired response time for exactly 95 out of the 100 inputs?

   b. What is the probability that the algorithm will execute within the desired response time for 90 or fewer of the 100 inputs?