Final Review Sheet
1) Suppose it is known that a lifetime of a certain type of transistor is lognormally
distributed with an unknown mean and variance.
a) Suppose we take a random sample of 20 transistors. How can we check that the data
follows a lognormal distribution?
b) Suppose our first five transistors have lifetimes of .3679, 2.718, 148.41, 20.086, and
7.3891 minutes. From these data points, do a hypothesis test to determine whether
the mean = 1. What do you conclude at the .10 level? At the .05 level?
2) Suppose 150 years ago, a nuclear reactor exploded somewhere in Nevada and the local
wildlife suffered huge genetic defects. One type of insect grew a second head that seemed
to provide resistance to pesticides. Specifically, it appeared that the larger the head was
in diameter, the more pounds of pesticides it took to kill the insect.
Scientists wanted to determine this effect using the following regression model
Pounds of pesticides needed to kill insect = B0 + B1*Diameter of head in inches + ε
a) Suppose the scientists took a random sample of 3 insects. Their head sizes were 1, 2,
and 2 inches and the respective amounts of pesticides needed to kill them were 2, 3,
and 4 pounds. Estimate B0 and B1 using the least squares estimator. Find s, sB0,
SB1.
b) Using this estimated line, would it make sense to estimate the number of pounds
needed to kill an insect with a head diameter of 1.5 inches? Of 20 inches? Find a
95% prediction interval for the number of pounds needed to kill an insect of 1.5
inches.
c) Suppose the scientists further sampled 17 more insects (20 total). They found that the
estimate for B1 was 1.5 and the SE for B1 was .6. Do a hypothesis test testing B0 = 0
d) Suppose we plotted the residuals, and noted that the residual plots were shaped like a
U. What does this mean? How can we fix this?
3) Suppose Duracell started making special “super-last” batteries that supposedly lasted
longer than regular batteries. Consumer Reports tested the lifetime in hours of these
batteries against regular batteries. Their data is as follows.
Regular: 5.6, 5.8, 5.0, 6.4, 6.0
Super-last: 6.0, 7.0, 6.2, 5.0, 5.8
a) Give a 95% confidence interval for the difference between the two battery lifetimes
and run a hypothesis test that the difference is 0.
b) Suppose it is known that to determine the lifetime, Consumer Reports tested the
batteries in 5 different machines. Their data is as follows (first row is regular, second
is super-last).
Energizer Bunny
5.8
6.0
Toy Soldier
6.0
6.2
Camera
5.0
5.0
Alarm Clock
6.4
7.0
Game Boy
5.6
5.8
Do part a) with this data.
4) Suppose this season of the Real World has a cast of 7 people, Steve, Mark, Norman,
Chauncey, Diane, Kathryn, and Emily. The first four are male and the last three are
female. Additionally, suppose that on each day, two of those people get into a fight, with
each pair equally as likely to get into a fight.
a) Write the state space of the pairs of people that can get into a fight (You can first
letters instead of full names if you would like).
b) What is the probability that on any given day, two girls get into a fight.
c) What is the probability that in any given week, there are fewer than 2 (not including
2) fights involving two girls.
5) Observers took 21 observations of temperatures (in Kelvin) from outer space. Here are
their results:
1, 2, 5, 10, 2, 20, 12, 4, 6, 4, 1, 11, 12, 21, 8, 4, 9, 38, 7, 10, 9
a) Draw a stem-and-leaf and box-plot diagram for this data.
b) Is it more likely that the data came from an exponential distribution or a normal
distribution? Why?