Download Final test Statistics 2

VOORBLAD VOOR EEN SCHRIFTELIJK TENTAMEN/TOETS
Verplicht gedeelte:
Vaknaam
: Statistics 1 for IBA
Vakcode
: 30J109
Datum tentamen
: 26 maart 2010
Duur tentamen
: 3 uur
Docent
: Gert Nieuwenhuis (tel. 2350 of 0650504640)
Telefoon secretariaat departement: 2430
Studenten worden geacht zich tijdens het tentamen correct te gedragen en de instructies van
examinator en surveillant op te volgen.
Bij constatering van fraude wordt streng opgetreden.
Alle studenten moeten uiteindelijk inleveren:
1) de door hen geproduceerde oplossingen van de opgaven;
2) ook de formulieren met alle vragen.
Vriendelijk verzoek aan de surveillanten: zou u de uitwerkingen van de studenten
per blauwe lijst bij elkaar willen leggen? Zodat onder elke blauwe lijst meteen de
toetsen liggen van de studenten die op deze lijst staan. Wel graag de formulieren met
de opgaven apart op een stapel. Alvast zeer bedankt.
Facultatief gedeelte:
Open boek examen
Schrapkaart
Gebruik eenvoudige calculator
Gebruik grafische calculator
Enkelzijdig afdrukken
Kladpapier beschikbaar
neen
neen
ja
neen
nee
ja
Uitwerkingen dienen op tentamenpapier gemaakt te worden, iedereen dient zijn werk
en de formulieren met de vragen in te leveren.
1
Mid-Exam Statistics 1 for IBA
March 26, 2010, 3 hours
It is NOT allowed to use a graphical, programmable calculator; only a simple pocket
calculator is allowed. Apart from the cover page, this exam contains three pages.
Always explain your answers.
At the end, you have to hand in:
1) the solutions to the six exercises 2) the sheets with the six exercises
On Monday, March 29, these six questions and their solutions will be posted on
Blackboard.
Exercise 1
Last year’s population of the participants of the Mid-Exam of the course Statistics 1
(March 27, 2009) consisted of 464 students. On that population we consider the
continuous variable X = ‘grade (scale 0 – 10) scored for the Mid-Exam’. The dataset
of X is summarized by way of the following classified relative frequency distribution,
with unequal class widths.
[0, 4] (4, 6] (6, 8] (8, 10] total
class
1
rel frequency 0.1272 0.2565 0.3879 0.2284
a. Determine the frequency density at 2.5.
b. Calculate the mean and the variance of this (relative) frequency distribution.
Let F be the cumulative (relative) distribution function of that frequency distribution.
c. Calculate the value of F at the mean you have calculated in b.
Exercise 2
In this exercise we (among others) ask ourselves the question whether Statistics 1
students with higher grades for the team assignment (TA) in general will also score
higher grades for the Mid-Exam. For a random sample of 340 students from previous
years, we get the following summarized results concerning the variables X = ‘grade
(scale 0 – 10) for the TA’ and Y = ‘grade (scale 0 – 10) for the Mid-Exam’:
𝑥̅ = 6.9750;
𝑦̅ = 6.2782;
𝑠𝑋2 = 1.8841;
𝑠𝑌2 = 4.6194
𝑠𝑋,𝑌 = 0.5513
a. Calculate the correlation coefficient of the x- and y-data. What do you think of
the strength of the linear relationship between y and x?
b. What can be concluded from the fact that the variance of the y-data is so much
larger than the variance of the x-data?
c. Write down the regression equation of y on x.
d. One student in the sample scored 7 for the TA and 6.8 for the Mid-Exam.
Calculate the residual of that student.
e. Using Chebyshev’s rule and the above summarized results, find a lower bound
for the percentage of the students in the sample scoring more than 1.5 for the
Mid-Exam.
2
Exercise 3
In a certain random experiment, the events A, B, C, D and E are considered. The
following facts are known:
𝑃(𝐴) = 0.4; 𝑃(𝐵) = 0.3; 𝑃(𝐶) = 0.3; 𝑃(𝐷) = 0.25; 𝑃(𝐸) = 0.45;
A and E are independent; 𝑃(𝐴 ∩ 𝐷) = 0.05; B and E are disjoint;
the events A, B and C constitute a partition of the sample space .
a.
b.
c.
d.
e.
f.
g.
h.
Calculate 𝑃(𝐴 ∪ 𝐶).
Calculate 𝑃(𝐴 ∪ 𝐸).
Calculate 𝑃(𝐵 𝑐 ∪ 𝐸).
Are A and E also disjoint? Explain.
Calculate 𝑃(𝐷 𝑐 |𝐴).
Calculate 𝑃(𝐴 ∪ 𝐵|𝐸).
Calculate 𝑃(𝐷 𝑐 |𝐴𝑐 ).
Are 𝐵 and 𝐶 independent? Explain.
Exercise 4
We reconsider the so-called paradox of Galilei:
“When three fair dice are thrown, then there are six outcomes that all have 10 eyes in
total: 631, 622, 541, 532, 442 and 433. To obtain a total of 9 eyes, there also are six
outcomes: 621, 522, 531, 441, 432 and 333. Still it turns out that the total 10 happens
more frequently than the total 9. How is this possible?”
Galilei solved the problem for three fair dice. However, his wife Galilinda asked
herself the question whether a similar problem comes up when throwing the die of the
swindler three times. Recall that for this die the random variable X = ‘number of eyes
of one roll’ has the following probability density function f :
1
2
3
4
5
6
x
𝒇(𝒙) 1/12 1/6 1/6 1/6 1/6 1/4
You are asked to solve Galilinda’s problem by considering the outcomes of the
random experiment as ordered 3-tuples, thus renaming Galilei’s six “outcomes” 632,
…, 433 with ten eyes and the six “outcomes” 621, …, 333 with nine eyes as six basic
events of ordered outcomes. For instance, Galilei’s “outcome” 442 then becomes the
basic event {(4, 4, 2), (4, 2, 4), (2, 4, 4)}, which consists of three ordered outcomes.
Below, Y denotes the total number of eyes when the die of the swindler is thrown
three times.
a. Are the ordered outcomes of this random experiment (throwing the swindler’s
die three times) equally likely? Formulate all ordered outcomes of the basic
event 622. What is the probability of this basic event?
b. Calculate the probabilities 𝑃(𝑌 = 10) and 𝑃(𝑌 = 9) by first calculating the
probabilities of each of the 12 basic events.
c. Give your comments regarding the problem raised by Galilinda.
3
Exercise 5
The CEO of a company adopts the following discrete probability density function
(pdf) 𝑓 to model the probability distribution of the random variable X = ‘next year’s
profit (in millions of euro)’:
0
1
2
3
x
−2 −1
𝒇(𝒙) 0.08 0.12 0.20 0.30 0.22 0.08
Below, F denotes the cumulative distribution function (cdf) of X.
a.
b.
c.
d.
Calculate the expectation  of X.
Calculate 𝑃(−2 < 𝑋 ≤ 1) and 𝑃(|𝑋 − | ≤ 1.5).
Consider F on the set of all real numbers. Draw the graph of F.
Calculate the variance of X.
The contract of the CEO promises for the year 2012 the bonus 𝑌 = 0.05𝑋 + 0.1 (in
millions of euro), which depends linearly on the profit X in 2011 and might be
negative.
e. Calculate the expectation and the standard deviation of Y.
f. Calculate the probability that Y will be positive.
Exercise 6
The CEO of another company makes use of a continuous pdf 𝑓 to model the
probability distribution of the random variable X = ‘next year’s profit (in millions of
euro)’:
1 1
+ 𝑥
if − 2 < 𝑥 ≤ 0
4 8
1
if 0 < 𝑥 ≤ 2
𝑓(𝑥) = 4
1 1
− 𝑥
if 2 < 𝑥 < 4
2 8
{0
if 𝑥 ≤ −2 or 𝑥 ≥ 4
1
1
So: 𝑓(𝑥) is the straight (increasing) line 𝑓(𝑥) = 4 + 8 𝑥 on the interval (−2, 0], it is
1
1
constantly equal to 0.25 on (0, 2] and it is a straight (decreasing) line 𝑓(𝑥) = 2 − 8 𝑥
on (2, 4). Outside the interval (−2, 4) it is constantly equal to zero: 𝑓(𝑥) = 0.
a.
b.
c.
d.
Draw the graph of f and check whether f indeed is a pdf.
Calculate 𝑃(𝑋 > 2.5).
Calculate the conditional probability 𝑃(𝑋 < 2.5 | 𝑋 > 2).
Use symmetry to determine 𝑃(𝑋 < 1) and 𝐸(𝑋).
The CEO decides to invest in the year 2012 the amount 𝑌 = 0.1𝑋 − 0.05 (in millions
of euro), which depends linearly on the profit X in 2011 and might be negative.
e. Calculate the (15/16)-quantile of Y; that is, calculate the real number b for
which 𝑃(𝑌 ≤ 𝑏) = 15/16.
4