Download Test of Statistics - Prof. M. Romanazzi

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Università di Venezia - Corso di Laurea Economics & Management
Test of Statistics - Prof. M. Romanazzi
17 June, 2010
Full Name
Matricola
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• Total (nominal) score: 30/30 (2 scores for each question).
• Pass score: 18/30.
• Portable computer, computer programs (specifically, R program): allowed; textbook
or class notes: not allowed.
• Detailed solutions to questions must be given on the draft sheet (foglio di brutta copia);
final answers/results must be copied on the exam sheet, on the lines with the small
square.
• Mark below your choice about oral discussion (required when the total score is between
18 and 21). Default: oral discussion, yes.
Oral Discussion Option
NO
YES
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1. Prove that, if A and B are stochastically independent events, then AC and B C are also stochastically
independent. (Note. For any event E, E C is the complement of E.)
2. In a university class there are 30 students, 10 males and 20 females. We choose at random and
without replacement a sample of 6 students. What is the probability that in the sample there is at
least one male?
3. The scores of a test of English Literature vary between 0 and 10 and the probability density function
is
x/50, 0 ≤ x ≤ 10,
fX (x) =
0,
elsewhere.
Compute the shortest interval including 80% of total probability.
4. An investor splits evenly a unit asset into two activities, a risky bond A and a non-risky security
B. The yearly returns XA of A are a normal random variable N (µA = 5%, σA = 3%). The yearly
returns XB of B are constant and equal to 1%. Let X denote the total yearly return of the
investment. Describe the probability distribution of X and compute P (X > 5).
X∼
P (X > 5) =
5. The coordinates X and Y of a point of the cartesian plane are independent uniform R(0, 1) random
variables. Suppose to generate a random sample of 15 points. Consider the events A and B defined
as follows.
A : 5 points are inside the bottom-left square with opposite corners (0, 0), (1/2, 1/2),
B : 5 points are inside the top-right square with opposite corners (1/2, 1/2), (1, 1).
What is the probability of A ∩ B?
6. Compute the expectation of Z = (X − Y )2 , where X and Y are standard normal random variables
and ρX,Y = 1/2.
7. Consider the following computer game. A pointer moves on a ruled line in discrete steps according
to a random variable X. Each second a) it moves a unit step on the left (X = −1) or b) it
moves a unit step on the right (X = 1) or c) it stays at the previous position (X = 0) with equal
probabilities. The starting position is at the origin. Let S300 denote the position of the pointer on
the line after 300 seconds (for example, if the first four moves are x1 = 0, x2 = x3 = 1, x4 = −1,
then s4 = 0 + 1 + 1 − 1 = 1). Compute approximate values of the probabilities of the following
events:
A : S300 = 0, B : S300 < 0, C : |S300 | < 3.
P (A) =
P (B) =
P (C) =
8. A box contains 20 red, 40 white and 40 green balls. We draw 10 balls at random and with
(RED)
satisfies
replacement. What is the probability that the sample proportion of red balls X 10
(RED)
0.2 ≤ X 10
≤ 0.3?
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9. In a university class individuals’ heights (X, cm) have a normal distribution N (µ, σ). Let X 5 denote
the average height of a random sample of n = 5 students and let S52 be the corresponding unbiased
sample variance. Describe the probability distribution of the transformed random variable
Z=
X5 − µ
√ .
S5 / 5
What is the probability of the event |Z| ≤ 2?
Z∼
P (|Z| ≤ 2) < 0.9544997 P (|Z| ≤ 2) = 0.9544997 P (|Z| ≤ 2) > 0.9544997
P (|Z| ≤ 2) =
Other:
10. The stem-and-leaf display shows the CO2 (carbon dioxide) emissions (g/km) of a sample of n = 37
cars (fuel: diesel). Compute median, mean, variance and standard deviation of the sample values.
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14
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n = 37
14|6
Pn is read 146 g/km
xi = 5391
Pi=1
n
2
i=1 xi = 797263
Sample Statistic
Value
Median
Mean
8
3
5777899
000457
26788
0012788999
5779
24
4
Variance
Standard Deviation
11. Consider again the CO2 data. The measured CO2 emissions of a car (similar to the sample units)
are 158 g/km. What is its position within the observed distribution? Choose among the following
answers.
Left tail
Central region
Right tail
Outlier
Other
12. Consider again the CO2 data. Derive the confidence interval for the mean µ of the reference
distribution (confidence level: 95%).
13. We collected the final scores of Mathematics (X) and Statistics (Y ) of n = 40 randomly chosen
students. The table below gives the corresponding sample summaries and the picture on the back
shows the scatter plot. Compute the sample estimates of µX , σX , µY , σY and ρX,Y .
n
40
Pn
i=1
961
x
xi
Pn
i=1
1007
y
yi
Pn
i=1
23397
sX
x2i
Pn
i=1
25539
sY
yi2
Pn
i=1
24332
rX,Y
xi yi
30
4
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28
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24
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22
Statistics
26
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20
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28
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Mathematics
14. Consider again the Maths & Stats data. You are told that Beatrice, a friend of yours, passed both
Maths and Stats and the Maths score was 22. What is your best prediction of her Stats score?
15. Consider again the Maths & Stats data. Let µX and µY denote the populations means of Maths
and Stats, respectively. Consider the statistical hypotheses
H0 : µX = µY ,
H 0 : µX 6= µY .
Suggest a test statistic, describe its distribution under H0 , compute the p-value and interpret the
result.
Test statistic:
Distribution under H0 :
p-value:
Interpretation: