S&AS: STAT1603 Introductory Statistics

1. (Total: 6 points)

A mutual fund experienced price fluctuations in the last five years as follows:

5 years ago 4 years ago 3 years ago 2 years ago 1 year ago Now

Fund price $100.000 $95.000 $109.250 $98.325 $117.990 $129.789

Explain how the fund manager may misuse statistics to report a greater average

annual return rate, and illustrate a correct use of statistics in this context.

2. (Total: 10 points)

Comment on each of the following students’ claims.

(a) (2 points) Student A claimed:

If events A and Bc are mutually exclusive, then A and B cannot be

independent.

(b) (1 point) Student B claimed:

Suppose X is a discrete random variable with support {3, 4, 5,...} and

probability mass function p(x) = ke2x where k is a constant. Then the

moment generating function of X is

MX(t) = E(etX) = X1x=3

etxke2x = ketX1x=3

e3x = kete9 1 ! e3 = ket+9

1 ! e3 .

(c) (3 points) Student C claimed:

If Y is a normal random variable, then Y 2 must not be a normal random

variable.

Note: If Z is a standard normal random variable, then Z2 is a chi-squared

random variable.

(d) (2 points) Student D claimed:

If W follows a beta distribution with mean 0.5 and variance 0.05, then

W + 1 also follows a beta distribution.

(e) (2 points) Student E claimed:

Suppose a random variable T has a median m, it is impossible that

Pr(T = m) = 0.

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S&AS: STAT1603 Introductory Statistics

3. (Total: 12 points)

A small plane went down and was missing, and the search was organized into three

regions. Starting with the likeliest, they are listed as follows:

Region Initial chance the plane is there Chance of being overlooked in the search

Mountains 0.5 0.3

Prairie 0.3 0.2

Sea 0.2 0.9

The last column gives the chance that if the plane is there, it will not be found. For

example, if it went down at sea, there is 90% chance it will have disappeared, or

otherwise not be found. Since the pilot is not equipped to long survive a crash in

the mountains, it is particularly important to determine the chance that the plane

went down in the mountains.

(a) (1 point) Before any search is started, what is the chance that the plane is in

the mountains?

(b) (2 points) The initial search was in the mountains, and the plane was not

found. Show that the chance the plane is nevertheless in the mountains is 3

13.

(c) (4 points) Following the previous part, the search was continued over the other

two regions, and unfortunately the plane was not found anywhere. Now what

is the chance that the plane is in the mountains?

(d) (5 points) Describe how and why the chances changed from part (a) to part

(b), and to part (c).

4. (Total: 10 points)

Each time a clerk makes a mistake at work, there is a probability of 0.4 for him to

receive a warning letter from the supervisor. The issuance of each warning letter is

independent. Whenever the clerk receives two warning letters, he will be fired. Let

X be the number of mistakes the clerk makes before he is fired.

(a) (4 points) By recognizing the distribution, write down the mean and variance

of X.

(b) (1 point) What is the probability that the clerk makes exactly 2 mistakes before

he is fired?

(c) (2 points) What is the probability that the clerk makes more than 10 mistakes

before he is fired?

(d) (3 points) What is the probability that the clerk makes less than 20 mistakes

before he is fired?

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S&AS: STAT1603 Introductory Statistics

5. (Total: 10 points)

Let X be a random variable following an exponential distribution where its mean

is exactly 2 times its variance.

(a) (3 points) Find Pr(X > 2|X > 1).

(b) (3 points) Find Pr(X < 3|X < 4).

(c) (4 points) Explain whether X is suitable to model the lifetime of a battery.

6. (Total: 13 points)

(a) Suppose Z is a standard normal random variable and let X = 3Z ! 1.

(i) (2 points) Find Pr(!1  X  1).

(ii) (1 point) What is the probability that X is within one standard deviation

of its mean?

(iii) (4 points) Find the value of Pr {(X ! 1)2 > 1}.

(b) The weights (in grams) of a population of mice fed on a certain diet since birth

are assumed to be normally distributed with mean µ = 100 and standard

deviation ! = 20. A random sample of n = 10 mice is taken from this

population.

(i) (4 points) Find the probability that at least two mice weigh between 75

and 90 grams.

(ii) (2 points) Find the probability that the mean weight of the mice is greater

than 110 grams.

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S&AS: STAT1603 Introductory Statistics

7. (Total: 15 points)

The distribution function of a random variable X with a positive parameter ✓ is

given by:

Pr(X  x)=1 ! exp ✓!x2 2✓◆ , for x > 0.

Let {X1,...,Xn} be a random sample of size n from the above distribution.

(a) (7 points) Find the maximum likelihood estimator of ✓.

(b) (5 points) Find E(X) using a suitable substitution method.

Hint 1: Use a substitution method for the integration by letting u = x2 2✓ .

Hint 2: After the substitution compare the integral with a gamma function.

(c) (1 point) Find the method of moments estimator of ✓.

(d) (2 points) Suppose the researchers measurements were as follows:

4, 4, 4, 5, 5, 5, 7, 7, 7, 8, 10, 12, 12, 12, 13

Estimate the parameter ✓ using the estimators in part (a) and part (c), respectively.

8. (Total: 6 points)

The random variables X1, X2,...,X300 are independent and identically distributed,

each having probability density function:

f(x) =

8<:

k, for 0  x  2;

0, otherwise,

where k is a constant. Let Y = (X1 + X2 + ··· + X300)/300.

(a) (2 points) Find the value of k and state the distribution of Xi? (Give the

values of the parameters.)

(b) (3 points) Find E(Xi) and Var(Xi), for i = 1, 2,..., 300, and thus state the

approximate distribution of Y ? (Give the values of the parameters.)

(c) (1 point) Find Pr(15Y  14).

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S&AS: STAT1603 Introductory Statistics

9. (Total: 9 points)

An economist conducted a survey to study the incomes of fresh graduates of a

certain university in Hong Kong. He randomly selected 12 such graduates and

obtained their monthly salary data (in dollars) as follows:

15,400 16,100 18,100 17,000 16,500 15,800

16,900 17,400 17,800 14,500 15,900 16,600

Assume that the salaries follow a normal distribution with an unknown mean µ

dollars and known standard deviation ! dollars.

(a) (1 point) Calculate the sample mean, x12.

(b) (1 point) If the width of a 95% confidence interval for µ is 1,075, find !, rounded

to the nearest whole number.

(c) (1 point) In order to narrow down the 95% confidence interval for µ, the

economist took another random sample of size 13. The mean of this second

sample, x13, is found to be $16,550. Using the combined information of the

two samples, calculate the sample mean of the combined 25 data, x25.

(d) (5 points) The economist found that last academic year the average salary for

fresh graduates was $16,200. Test, at the 0.05 significance level, whether this

academic year’s average salary for fresh graduates is greater than last year’s

average salary, using all the information provided in part (c). Clearly state the

hypotheses, test statistic, critical value and conclusion.

(e) (1 point) Calculate the p-value of the test.

10. (Total: 9 points)

A random variable X has the density function:

f(x) =

8<:|1 ! x|, for 0  x  2;

0, otherwise.

(a) (1 point) Find the median of X.

(b) (2 points) Find the mean of X.

(c) (6 points) Find F(x), the cumulative distribution function of X.

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