Department of Economics
Probability, Statistics and Hypothesis Testing Homework
1. The number of peanuts contained in a jar follows the normal
distribution with mean μ and variance
σ2 squared. Quality
control inspections over several periods show that 5% of the jars contain
less then 6.5 ounces of peanuts and 10% contain more than 6.8 ounces.
What is μ?
What is σ2
What percentage of jars contain more than 7
2. Let X ~ N(μ, σ2). A
random sample of three observations was obtained from this population.
Consider the following estimators of μX:
is an unbiased estimator of the population mean.
unbiased estimator of the population mean.
c. The variance of
given by ( /
) σ2 .
of is given
Suppose an underground newspaper is sold on two campuses. A random
selection of weekly sales figures provides the following data:
At the 5% level of significance, test whether the the newspaper sells
more at A than at B.
More papers are sold at A.
Fewer papers are sold at A.
The same number are sold at both
auction you have purchased dice that are purported to have belonged to Bat
Masterson, a notorious gambler of the wild west. Each die
has six faces numbered with spots 1 through 6. The image at left shows
a pair of dice. Such dice are often used in board games like Monopoly. The
dice would be much more valuable if you can show that they are NOT fair.
After all, fair dice are quite commonplace. In craps the shooter loses if he
rolls a 2, 3 or 12 on his initial roll of the dice. The probability distribution
for the number of spots showing on any given throw of the dice is
You roll the dice 900 times and you observe that a 2, 3 or 12 comes up
only 50 times.
a. If the dice are fair, what is the probability of observing a 2,
or 3, or 12 only 50 times in 900 tosses?
b. Using the binomial distribution, construct the Wald statistic to test
whether the dice you bought are fair. W =
have taken a summer job with the University of Delaware Coastal Marine
Mammal Monitoring Project. Each week you patrol about ten miles from
shore from Cape May to Marcus Hook. You record the number of porpoises
that you see during each hour of the trip. At the end of the summer
you report the following data to "Save Our Seas"
a. What is the total number of hours that you observed porpoises?
b. How many porpoises did you see?
c. Forty years ago it was common to see .9 porpoises per hour in the same
waters. What is the average number of porpoises per hour by you?
d. If porpoise sightings follow the Poisson distribution then the mean
should be equal to the variance. Assume that your sample mean is, in
fact, the same as the population mean for whatever probability model
generated the data. Compute the Lagrange Multiplier test statistic for
the hypothesis that there has been no change in the porpoise population off
the Jersey coast over the last forty years. LM =
7. A random variable is normally distributed with mean zero and unknown
variance. A sample of size 19 yields a sample variance of 4.
Compute the likelihood ratio test statistic for the test that the population
variance is 9, against the alternate that it is not. LR =