A manufacturer
of car batteries claims that the distribution of the lifespans
of its best battery has mean 54 months and standard deviation 6 months. A
consumer group purchases a random sample of 50 batteries and tests the battery lifespans. The sample mean lifespan is 52 months.
Assuming the
manufacturer’s claim is true, what is the approximate probability of observing
a sample mean of 52 or less? Does this make you doubt their claim? (This
example foreshadows the concept of significance testing.)
An elevator has
a capacity of 30 people. The distribution of the weights of elevator passengers
(and whatever they are carrying) has a mean of 168 pounds and a standard
deviation of 10 pounds. Consider rides when the elevator is full with a random
sample of people. What weight capacity should be listed on the elevator, so
there’s only a 0.01 chance the elevator is overloaded? (Kind of a silly
example, but I wanted to remind you of reverse-lookup problems.)
Example 3 (
In roulette,
the probability of winning with a bet on red is 18/38. Suppose you bet 100
times, all on red. What is the approximate probability that you win more than
60% of the bets?
The ideal size
of a first-year class at a particular college is 150 students. The college,
knowing from past experience that only 30 percent of those accepted for
admission will actually attend, uses a policy of approving the applications of
450 students. Find the approximate probability that more than 150 first-year
students attend the college (assume that students independently decide whether
or not to attend).