semester, Test 1 length : 2 hours – coefficient 1/2

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2017-2019 – S2 – Mathematics – TEST 1 SOLUTIONS – page 1 / 4

IUT of Saint-Etienne – Sales and Marketing department

Mr Ferraris Prom 2017-2019 30/03/2018

MATHEMATICS – 2

^nd

semester, Test 1 length : 2 hours – coefficient 1/2

SOLUTIONS

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 Null hypothesis (H0): income and place are independent.

Calculated Chi-square:

From the subtotals of the observed values table, theoretical frequencies can be deduced:

245.227 392.754 228.618 110.401 977

98.392 157.584 91.728 44.296 392

158.381 253.662 147.654 71.303 631

502 804 468 226 2000

Then, a comparison between both tables leads to the calculation of part Chi-2s:

40.5936195 3.16299902 11.6544538 63.0042011 30.0683965 4.48464981 20.4132433 32.093002 13.0030443 0.29578827 0.4717496 29.286507

248.531654

This total is our χ²calc. Limit Chi-square: at 1% significance level and with 6 dof, χ²lim = 16.8.

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Comparison and decision: χ²calc is largely bigger than χ²lim . To that extent, the null hypothesis can be rejected (with less than 1% chance to be wrong): there is a link between the income and the kind of place people choose for their summer holidays.

2) Identify the two highest part chi-squares, and then give a concrete explanation of their meaning about

French people's habits. 1 pt

These values are 40.59 and 63. As both belong to the first row of the table, they concern people who prefer campsites. The comparison between observed and theoretical frequencies shows us that in general, unlike the rest of the population, people whose income is low prefer campsites, whereas people whose income is high are not interested by campsites.

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

By making an estimate of the fastest bacterial growth observed (dotted curve), we can say that the threshold of 20,000 bacteria per mL is reached after less than 4 hours. For safety, we recommend an opening not exceeding 3:30 or even 3 hours if we consider that other samples could show even faster growths.

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 See the blue curve. Indeed, it seems that it does fit the mean trend of the cloud.

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

0.4525

2191e ^x 7000 209220

y= + = .

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2017-2019 – S2 – Mathematics – TEST 1 SOLUTIONS – page 3 / 4 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

The values of X, of Y and the frequencies, have to be entered in three matching lists, e.g.:

List1 : X List2 : Y List3 : frequencies

10 1 4

25 1 6

35 1 13

10 2 12

25 2 15

35 2 16

10 4 20

25 4 14

35 4 3

10 7 7

25 7 2

35 7 1

With the proper calculator's settings, the following results are obtained:

( )

cov ,

22.212 ; 10.3466 ; 2.8938 ; 1.7108

6460 22.212 2.8938 7.1102 113

X Y

x y

X Y xy x y

n

σ σ

= = = =

=

∑

− × = − × = −

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 the nits linear correlation coefficient. Is it relevant to model this evolution in time by the mean of a straight line ? 1.5 pt

( )

cov , 5234.7

12 44.833333 43.6333 9

X Y xy x y

=

∑

n − × = − × ≈

.

( )

cov , 43.6333333

0.99897 2.5819889 16.9164614

X Y

r X Y

= σ σ ≈ ≈

× . The linear correlation is excellent: modelling the turnover in time by a straight line appears to be relevant.

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

6.545 33.707

y= x− .

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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 Calculation of y_i′ =6.545x_i−33.707:

18.653 25.198 31.743 38.288 44.833 51.378 57.923 64.468 71.013 Calculation of _i ⁱ

i

z y

= y

′:

1.0722 0.9921 0.9765 0.9846 1.0037 0.9946 0.9806 1.0067 1.0139 Mean and standard deviation of Z: 1.00278 and 0.02717424.

Calculation of the point estimate for 2025 (x0 = 25): y₀′ =6.545x₀−33.707 129.918= . Confidence interval:

( ) ( )

[ ]

0 ; 0

129.918 1.00278 2.58 0.02717424 ; 129.918 1.00278 2.58 0.02717424 121.17 ; 139.39

Z Z

y^′ z u σ y^′ z u σ

 − × + × 

 

= − × + × 

=

b. What is approximately the probability that, in 2025, this turnover would exceed € 140 billion? 0.5 pt Hence, this probability is about 0.5%.

____________________ TEST END ____________________