IUT of Saint-Etienne – Sales and Marketing department
Mr. Ferraris Prom 2016-2018 14/04/2017
MATHEMATICS – 2
ndsemester, Test 1 length : 2 hours – coefficient 1/2
Graphic calculator is allowed. Any personal sheet is forbidden.
Your work has to be written down inside this document.
The presentation and the quality of your writings will be taken into account.
Your rounded results will show at least four significant figures.
Exercise 1 : MCQ (3 points) – tick the right boxes below
One correct answer only per question - 0 point in case of wrong/missing/multiple answer at a question 1) Fill the following sentence, using the correct item: "In a Chi-square testing, the significance level is the
probability that H0 may be ___[1]___ given that we ___[2]___ it."
[1] right [1] right [1] wrong [1] wrong
[2] reject [2] accept [2] reject [2] accept 2) From the series 3 ; 5 ; 4 ; 6 ; 4 ; 7 , the 2 by 2 moving means are:
4 ; 5 ; 5.5 3 ; 4 ; 4 4 ; 4.5 ; 5 ; 5 ; 5.5 8 ; 9 ; 10 ; 10 ; 11 3) Between a straight line and a point cloud, if all residues are very little, then:
|r| ≈ 1 cov(X,Y) ≈ 0 |r| ≈ 0 cov(X,Y) ≈ 1
Exercise 2 : χ² testing (5.5 points)
In the following table are gathered 418 women, sorted by their hair colour and their eye colour:
Hair colour
Black Brown Red Blond
Eye colour
Brown 82 118 20 35
Green 11 16 11 8
Blue 33 42 18 24
1) By the mean of a Chi-square testing, can we claim, with a 2% significant level, that hair colour and eye
colour are related in the population this sample comes from? 3 pts
Full Name : Group : B1
2) Give a concrete explanation of the significance level. 1 pt
3) On having a closer look on the part chi squares in details, say for which hair colour people are not
distributed by eye colour like the rest of the population. 1.5 pt
Exercise 3 : (6.5 points)
The table below displays the evolution of the French hourly minimum wage for the past 13 years. The corresponding scatter plot is also displayed.
year
year range
gross minimum
wage (€)
X Y
2005 1 8.03
2006 2 8.27
2007 3 8.44
2008 4 8.71
2009 5 8.82
2010 6 8.86
2011 7 9.10
2012 8 9.31
Y
1) a. Give the expression of the Y on X fitting line, according to the least square method. 1 pt
b. Draw this line on the scatter plot above. 1 pt
c. Using this linear fitting, calculate an estimate of the amount of this gross (fr.: "brut") minimum wage in
the year 2025. 1 pt
2) It seems that a linear fitting may not be the best way to model the growth of the minimum wage:
let's perform the variable change T = X .
a. Calculate the covariance of the pair (T, Y) and its linear correlation coefficient. Comment. 2 pts
b. Give the expression of the Y on T fitting line, according to the least square method. 0.5 pt
c. Then, give with this new model an estimate of this minimum wage in the year 2025. 1 pt
Exercise 4 : (5 points)
A car consulting website has identified the resale values of several vehicles of the same model based on their age. The numerical results and the corresponding scatter plot are given below:
1) Here, a linear fitting would be irrelevant; but, a curve fitting would be. Let's perform the variable change:
T 1
= X , assuming a good linear correlation between T and Y. Give the expression of the Y on T regression
line, according to the least square method. 1 pt
2) We would like to have an estimate of the selling price of a 10 year-old vehicle of the same model.
a. Give the point estimate for this price. 1 pt
b. Give its estimate by a 95% confidence interval (first, you will build a confidence interval for T using the rates method, and you will then translate it into an interval for X). 2.5 pts
vehicle's age (years) : X 1 1.5 2 3 3 4 5 7
resale value (€) : Y 9,200 6,700 5,500 4,300 4,200 3,750 3,400 3,000
c. According to this confidence interval, what is the rate of such 10 year-old vehicles whose selling price
would be more than €2,560 ? 0.5 pt
____________________ TEST END ____________________
