IUT of Saint-Etienne – Sales and Marketing department
Mr Ferraris Prom 2017-2019 30/03/2018
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 (2 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) Complete this sentence : "A residue is a …?... difference between a point and a line." :
minimum maximum vertical mean
2) If a cloud of points (xi, yi) is elongated and going down, then :
Cov(X, Y) < 0 Cov(X, Y) < 0 Cov(X, Y) > 0 Cov(X, Y) > 0
and r < 0 and r > 0 and r < 0 and r > 0
3) The size of a p% confidence interval automatically increases in case :
y’0 increases y’0 increases y’0 decreases y’0 decreases and p increases and p decreases and p increases and p decreases 4) A linear correlation coefficient located between 0 and 0.5 implies that the points (in the cloud)…
are almost distribu- follow a curve are far away from don't follow a
-ted at random each other straight line
Exercise 2 : Chi-square test (5 points)
A study, conducted on 2000 French people, aims to highlight a link between their annual household income and the kind of place they usually opt for during the summer holidays :
Annual household income (k€) [15 ; 25[ [25 ; 35[ [35 ; 60[ 60 and
more total Place to stay
Campsite 345 428 177 27 977
Hotel 44 131 135 82 392
Flat/house 113 245 156 117 631
total 502 804 468 226 2000
1) By the mean of a chi-square testing, is it possible to claim, with a 1% significance level, that there is a link between the household income and the preferred kind of place for summer holidays ? 4 pts
Full Name : Group : B1
2) Identify the two highest part chi-squares, and then give a concrete explanation of their meaning about
French people's habits. 1 pt
Exercise 3 : (3 points)
The laboratory of an agri-food company producing Bolognese sauce is studying the growth of a bacterial culture in this sauce. From four sauce samples, the evolution in time of the amount of bacteria, at room temperature, is marked, which gave the point cloud below (which shows a trend, but does not follow a particular curve because bacterial growth did not proceed at exactly the same rate in the different samples).
1) If a sanitary standard prohibits the consumption of a product when it contains more than 20000 bacteria per millilitre, what is the maximum recommended duration for a consumer to leave a pot open at room
temperature, safely? 1.5 pt
2) It seems that a linear model is not the best one to fit the observed evolution. Analysts opted for an exponential representation and deduced the following equation:
0.4525
2191e x 7000
y= + where y is the number of bacteria per mL and x is the duration in hours.
a. On the graph above, draw the curve representing this equation. Comment. 1 pt
b. Thanks to this equation, give an estimate of the average number of bacteria per mL that would be reached in case a pot would be opened during 10 hours at room temperature. 0.5 pt
Exercise 4 : (3.5 points)
The opposite table has been built by crossing two variables X and Y. One or more individuals correspond to each pair, inside the table.
Thanks to your calculator, give the means and standard deviations of both variables, and calculate their covariance.
Y
X 1 2 4 7
[0 ; 20[ 4 12 20 7
[20 ; 30[ 6 15 14 2
[30 ; 40[ 13 16 3 1
Exercise 5 : (6.5 points)
In France, the turnover of the e-commerce shows a steady growth for several years.
The table below (source: http://fr.statista.com) gives it, in billion euros, from the year 2008 to 2016:
year 2008 2009 2010 2011 2012 2013 2014 2015 2016
turnover 20 25 31 37.7 45 51.1 56.8 64.9 72
We aim to model this evolution by a simple equation, linear if possible, and then to use this one for prediction.
1) Let's name X the number of years passed since 2000 (e.g.: X = 8 for the year 2008, X = 16 for 2016), and Y the turnover of the e-commerce in France.
a. Calculate the covariance of the pair (X, Y) and then its linear correlation coefficient. Is it relevant to model this evolution in time by the mean of a straight line ? 1.5 pt
b. Thanks to your calculator and without any explanation, give the equation of the straight line (least
square method) that can be obtained from the table above. 1 pt
2) a. Let's assume that the e-commerce turnover is to grow at the same rate in the future. Using this line, give a 99% confidence interval of this turnover in France in the year 2025. 3.5 pts
b. What is approximately the probability that, in 2025, this turnover would exceed € 140 billion? 0.5 pt
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