IUT of Saint-Etienne – Sales and Marketing department Mr Ferraris Prom 2012-2014 15/03/2013

MATHEMATICS – 2nd Semester, TEST 1 length : 1 hour 30 min – coefficient 1/3

in english

Graphic calculator is allowed. Form is attached to the subject. Personal sheets are forbidden.

Quality of writing will be taken into account.

Your rounded results will show at least four significant figures.

Write on top : last name, first name, group – S2 Test 1, mathematics, subject 1 - date

Exercise 1 : Quiz (4 points) – fill the correct circles below

1 correct possibility per question ; if wrong, or multiple, or lacking answer : 0

1) The probability that your prediction will be effective is :

� the confidence � the linear corre- � the standard devi- � none of these

level lation coefficient ation of variable Z three proposals

2) A residue is a difference between :

� two points � one point and � one point and the � two regression

in the cloud the mean regression line lines

3) Extracted from a 10 values table, how many moving means are there, after having picked

groups of 4 values ?

� 2 � 2,5 � 6 � 7

4) If a (xi, yi) points cloud is elongated and going down, then :

� Cov(X, Y) < 0 � Cov(X, Y) < 0 � Cov(X, Y) > 0 � Cov(X, Y) > 0

et ρ < 0 et ρ > 0 et ρ < 0 et ρ > 0

Exercise 2 : (8 points)

A survey focused on the comparison between people’s expenditure in high tech furniture and their

income. Each column of the table T below shows, for a given french region, the average monthly

income of working people (X) and the average monthly expenditure (Y) in high tech furniture.

region A B C D E F

income X (€) 1550 1620 1770 1850 1930 2000

expenditure Y (€) 57 61 66 73 76 82

1) a. Compute the covariance, then the linear correlation coefficient, of pair (X, Y).

Interprete both parameters. 1,5 pt

b. Give, asking your calculator, the expression of Y on X regression line. 0,5 pt

c. Determine the 99% confidence interval of the predictable expenditure for people whose

average income would be 2500 €. 2,5 pts

d. What percentage of these people would spend more than 102,5 € ? 1 pt

2) a. Determine the Mayer’s line expression, fitting table T values. 1,5 pt

b. What’s the single prediction of Y given by the Mayer’s line for people whose average income

would be 2500 € ? Compute the difference, by a percentage, between this prediction and the

one used in question 1)c. 1 pt

Name :

Exercise 3 : (8 points)

A supermarket has been selling a product A for many years. During year 2012, its sales price changed a

few times, and the number of sold items has been registered for each period :

price X (€) 40 42 45 37 33 40 42 50

number of sales Y 200 190 158 222 242 195 194 95

1) a. In appendix, plot the points cloud extracted from this table. 1 pt

b. Though the linear correlation coefficient between X and Y is -0.97, we’re going to carry on a

variable change. Comment this decision. 1 pt

2) We set TX

=−

100

60 and our ai mis to use the linear correlation of pair (T, Y).

a. Once values of T listed into your calculator, give the mean and the standard deviation of this new

variable. 1 pt

b. Compute the linear correlation coefficient of pair (T, Y). Comment. 1,5 pt

c. Give, asking your calculator, the expression of Y on T regression line, according to least square

method, and then deduct a relationship between X and Y , that will lead to a curve fitting of the

initial points cloud. 1,5 pt

d. Draw this curve on the appendix’s graph, for x ∈ [25 ; 55]. 1 pt

e. Compute the estimated number of sold items, given that the price were set at 25 €. 1 pt

____________________ END OF SUBJECT ____________________

APPENDIX

250

200

150

100

50

0 25 30 35 40 45 50 55

IUT - TC Mathematics - Form for the first test of semester 2

2-variable statistics :Without contingency

* mean, variant and standard deviation

* covariance and linear correlation coefficient

* parameters in expression of Y on X regression line (least square method)

y = ax + b with and

With contingency* mean and variant

* covariance

* other formulas are identical to those given above

Confidence interval of y for a given value x0 :

where * y' 0 = ax 0 + b * y' i = ax i + b and

* u α = 1.96 (confidence level : 95%) or 2.58 (confidence level : 99%)

FORM

1 1;= == =∑ ∑

n n

i ii i

x yx y

n n( ) ( )

2 2

2 21 1;= == − = −∑ ∑

n n

i ii i

x yV X x V Y y

n n

( ) ( ) ( ) ( );x yX V X Y V Yσ σ σ σ= = = =

( ) 1

., .== −

∑n

i ii

x yCov X Y x y

n

( , )

.x y

Cov X Yρσ σ

=

= −b y ax( )( )

,Cov X Ya

V X=

..11 ; === =∑∑

qp

j ji iji

n yn xx y

N N( ) ( )

22..

12 21 ; === − = −∑∑

qp

j ji iji

n yn xV X x V Y y

N N

( ) 1 1, .= == −∑∑

p q

ij i ji j

n x y

Cov X Y x yN

( ) ( )0 0;α ασ σ′ ′− + z zy z u y z u

=′i

ii

yz

y

IUT Saint-Etienne – Department TC - J.F.Ferraris - Mathematics - S2 - Stat2var - LessonEx - Rev 2012 - page 23 on 23