SAS: STAT1603 Introductory Statistics
1. Total: 6 points
A mutual fund experienced price fluctuations in the last five years as follows:
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:
b 1 point Student B claimed:
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
If events A and Bc are mutually exclusive, then A and B cannot be independent.
Suppose X is a discrete random variable with support 3, 4, 5, . . . and probability mass function px ke2x where k is a constant. Then the moment generating function of X is
t X X1 t x 2 x MXtEe e ke ke
x3
t X1 3 x k e t e 9 k e t 9
x3
e 1e3 1e3.
c 3 points Student C claimed:
Note: If Z is a standard normal random variable, then Z2 is a chisquared random variable.
d 2 points Student D claimed:
e 2 points Student E claimed:
If Y is a normal random variable, then Y 2 must not be a normal random variable.
If W follows a beta distribution with mean 0.5 and variance 0.05, then W 1 also follows a beta distribution.
Suppose a random variable T has a median m, it is impossible that PrT m0.
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SAS: 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:
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?
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
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?
P. 3 of 6
SAS: 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 PrX 2X 1. b 3 points Find PrX 3X 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 Pr1 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 12 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 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.
P. 4 of 6
SAS: STAT1603 Introductory Statistics
7. Total: 15 points
The distribution function of a random variable X with a positive parameter is
given by:
PrXx1expx2, forx0. 2
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 EX using a suitable substitution method.
x2 Hint 1: Use a substitution method for the integration by letting u 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:
fx8:k, for0x2;
0, otherwise,
where k is a constant. Let Y X1 X2 X300300.
a 2 points Find the value of k and state the distribution of Xi? Give the values of the parameters.
b 3 points Find EXi and VarXi, for i 1,2,…,300, and thus state the approximate distribution of Y ? Give the values of the parameters.
c 1 point Find Pr15Y 14.
P. 5 of 6
SAS: 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 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 is 1,075, find , rounded
to the nearest whole number.
c 1 point In order to narrow down the 95 confidence interval for , 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 years average salary for fresh graduates is greater than last years 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 pvalue of the test.
10. Total: 9 points
A random variable X has the density function:
fx8:1x, for0x2; 0, otherwise.
a 1 point Find the median of X. b 2 points Find the mean of X.
c 6 points Find Fx, the cumulative distribution function of X.
END OF PAPER
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