Descriptive statistics
The following figures show the annual salaries in ?s of a population of 20 workers in a small firm
14375 15180 15180 17880 19870 46132 14375 15525 14375 15767
16767 19600 14375 15870 14375 15180 23069 14375 36938 16767.
(a) Calculate the mean, median, and mode of salaries.
(b) Construct a vertical-bar histogram to illustrate the data.
The histogram shall contain:
–Four equal-width classes
–Axis titles, data labels, and class limits below each bar.
–Frequency table used to produce the histogram.
–Indication of the measures of central tendency. Use arrows.
(c) Measures of dispersion: Calculate the variance and the standard deviation of salaries.
(d) Suppose that during your presentation of this histogram one attendant asserts that all
salaries are included in an interval defined as one standard deviation above and below the
mean salary, i.e. ? }?. Are you agree with this person? If not, how many sigmas around
the mean value will cover all the salaries?

Exercise 2.Probabilities.Panel of jurors.
A panel of jurors consists of 16 persons who have had no education beyond high school and 10 who
have had some college education. If a lawyer randomly chooses two of them to ask some questions,
what is the probability that neither of them will have any college education?

Exercise 3.Probabilities. Tropical disease
Suppose that the probability is 0.4 that a rare tropical disease is diagnosed correctly and, if
diagnosed correctly, the probability is 0.6 that the patient will be cured. What is the probability that
a person who has the disease will be diagnosed correctly and cured?

Exercise 4.Probabilities.Driver’s license.
In a Western community of the US, the probability of passing the road test for a driver’s license on
the first try is 0.78. After that, the probability of passing becomes 0.65 regardless of how many
times a person has failed. What is the probability of getting one’s license on the fourth try?

Exercise 5.Probabilities.Fire-fighting equipment.
Suppose that a consumer research organization has studied the service under warranty provided by
the 200 dealers of fire-fighting equipment in Shanghai, and that their findings are summarized in
the following table:
Good service Poor service
Name-brand dealer 55 25

Off-brand dealer 60 60
Table of two hundred dealers of fire-fighting equipment.
If one of these dealers is randomly selected, use the table to find the probability of event choosing
(a) a name-brand dealer;
(b) a dealer who provides good service under warranty;
(c) a name-brand dealer who provides good service under warranty;
(d) a name-brand dealer who provides good service under warranty relative to
Name-brand dealers.

Exercise 6.Probability distribution – Financial application.
Over the last 10 years, a company’s annual earnings increased from the previous year seven times
and decreased from the previous year three times. You decide to model the number of earnings
increases over the next decade as a binomial random variable.
(a) What is your estimate of the probability of success p, defined as an increase in annual
earnings?
For parts (b), (c) and (d) of this problem, take the estimated probability as the actual probability
for the next decade (number of trials is ten).
(b) What is your estimate that earnings will increase in exactly 5 of the next years?
(c) Calculate the expected number (mean of the probability distribution) of yearly earnings
increases over the next 10 years.
(d) Calculate the variance and standard deviation of the number of yearly earnings increases
over the next 10 years.

Exercise 7.Probability distribution – Industry.
A tyre company has invented a revolutionary new product. In order to overcome consumer
resistance a mileage guarantee is offered with the tyre. Road tests suggest that the mean life of the
tyre is 42000 miles, with a standard deviation of 4000 miles. The tests suggest that tyre life is
normally distributed.
(a) What percentage of tyres will last for more than 45000 miles?
(b) What should the guaranteed mileage be if the firm wishes to replace no more than 4 per cent
oftyres?
Exercise 8.Probability distribution – Coffee industry.
At StarCoffee the actual amount of instant coffee that a filling machine deposits into a “6-ounce1”
jars varies from jar to jar and may be looked upon as a random variable having a normal
distribution with a standard deviation of 0.04 ounce. (a) If only 2.5% of the jars are to contain less
than 6 ounces of coffee, what must be the mean fill of these jars? (b) StarCoffee has been informed
by a custom official that hey have to pay a penalty for jars containing less than 6 ounces. The
penalty rate is $100/kg and it concerns 1000 kg of instant coffee. How much will pay StarCoffee?
(c) Finally, the company implements adjustments to the machine in order to reduce the proportion
of jars containing less than 6 ounces of coffee to 0.13%. The machine keeps the mean calculated in
(a). What is the new standard deviation of the normal distribution?

Exercise 9.Sampling and sample size.
When we take a random sample from an ‘infinite’ population, what happens to the standard error of
the mean when we use the mean of the sample to estimate the mean of the population, if the sample
size is decreased from 12250 to 250? Give a quantitative answer.

Exercise 10.Confidence interval – Financial application.
You manage a U.S. core equity portfolio that is sector-neutral to the S&P 500 (its industry sector
weights approximately match the S&P 500’s). Taking a weighted average of the projected mean
returns on the holdings, you forecast a portfolio return of 12 percent. You estimate a standard
deviation of annual return of 22 percent, close to the long-run figure for the S&P 500. For the yearahead
return on the portfolio, you are asked to do the following:
(a) Calculate and interpret a one standard deviation interval for portfolio return
(b) Calculate and interpret a 95 percent confidence interval, with a normality assumption for
returns.
(c) Calculate and interpret a 99% percent confidence interval, with a normality assumption for
returns.
Normal distributions can be considered an approximate model for returns. The sample mean and
the sample standard deviation are point estimates (one single sample is available).

Exercise 11.Probability distribution – Equity Portfolio.
In reference to Exercise 10, you want to estimate the following probabilities, assuming that a
normal distribution describes returns.
(a) What is the probability that portfolio return will exceed 20 percent?
(b) What is the probability that portfolio return will be between 12 percent and 20 percent?
That is, what is P(12% =portfolio return =20%)?
1 Ounce: it’s a unit of weight of one sixteenth of a pound avoirdupois (approximately 28 grams).

Exercise 12.Probability distribution – Risk Management.
Suppose that the mean change in portfolio value over a 7-day horizon is $10 millions, and it follows
a normal distribution. And the standard deviation of changes in portfolio value over this time
horizon is $18.237 millions. Calculate the 98% 7-day normal VaR by using the normal Value at
Risk methodology.

 
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