Temple University

Department of Economics

Econometrics 8009

Probability, Statistics and Hypothesis Testing Homework

Name

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 ounces?

2. Let X ~ N(μ, σ2). A random sample of three observations was obtained from this population.  Consider the following estimators of μX:

a.   is an unbiased estimator of the population mean.  True    False

b. is an unbiased estimator of the population mean. True    False

c. The variance of  is given by ( / ) σ2 .

d. The variance of   is given by (/)σ2

3. The average daily output of a certain department within an industrial plant is scheduled to be 85 units.  Twenty five days are selected at random and the output is observed for each day.  From this sample, the average output is calculated to be 81 units and the standard deviation for the sample is found to be 8 units.  Test with 99% confidence whether or not the average output is different from that scheduled.  Check the appropriate box:   Yes, it is different  or   no, t it is not different.

4.  Suppose an underground newspaper is sold on two campuses.  A random selection of weekly sales figures provides the following data:

 Sample Sample Size Mean Standard Deviation Campus A 10 123 15 Campus B 6 108 √185

At the 5% level of significance, test whether the the newspaper sells more at A than at B.

Answer:   More papers are sold at A.   Fewer papers are sold at A.   The same number are sold at both schools.

5. At 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

 Spots 2 3 4 5 6 7 8 9 10 11 12 Probability 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36

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 =

6.  You 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"

 Porpoises sighted in an hour 0 1 2 3 4 5 or more Frequency 144 91 32 11 2 0

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 =