IUT of Saint-Etienne – Sales and Marketing department Mr Ferraris Prom 2019-2021 03/2020 MATHEMATICS – 2

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IUT of Saint-Etienne – Sales and Marketing department

Mr Ferraris Prom 2019-2021 03/2020

MATHEMATICS – 2

^nd

semester, 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: Chi-square test (4 points)

A market study crosses two characteristics of a tool supplier's clientele: the type of clientele (individual, professional) and the category of goods purchased (tools, materials, products, electricity). The following table shows, at the end of the study, the number of customers for each type/category crossover (e.g. 22 professional customers purchased materials).

tools materials products electricity

individual 24 25 15 38 102

professional 14 22 25 17 78

38 47 40 55 180

After conducting a Chi-square test, discuss how much confidence can be placed in the statement "The category of goods purchased depends on the type of clientele".

Full Name : Group : B1

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Exercise 2: linear regression (8.5 points)

A study was carried out to compare the spending habits of individuals on high-tech equipment compared to their income. Each column in the table below represents, in a given French region, the median monthly income of working people (X) and the average monthly expenditure (Y) on high-tech equipment.

region A B C D E F

income X (€) 1550 1620 1770 1850 1930 2000

expenditure Y (€) 48 51 56 60 62 66

1) a. Calculate the covariance, then the linear correlation coefficient, of the pair

(

^{X Y}^,

)

^.

Give a comment for both parameters. 2 pts

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b. Using the calculator, give the Y in X regression line expression. 0.5 pt

c. Determine the 99% confidence interval of the estimated high-tech equipment expenditure for

individuals with a median income of 2500 €. 2.5 pts

d. Of these, what percentage would spend more than €83.1? 1 pt

2) a. Determine Mayer's line expression relative to the table T. 1.5 pt

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b. What point estimate of Y would be given by this line for individuals with a median income of €2500?

What is the percentage difference with the point estimate calculated in question 1)c.? 1 pt

Exercise 3: crossed table (3.5 points)

Mrs Betty Cash, sales representative, analyses her activity and efficiency. She has classified her customers' orders according to two variables: the duration of a visit (X, in minutes) leading to an order, and the number of items ordered (Y) at the end of this visit. The numbers of visits are visible in the intersections of the table opposite.

X : duration of a visit

(minutes)

Y : number of items ordered per visit

1 2 3 4

[0 ; 30[ 3 1 1 0

[30 ; 60[ 2 4 3 1

[60 ; 90[ 0 4 5 3

1) What is the meaning of the frequency "5" in the table? 1 pt

2) Calculate manually the average time spent per order placed. 1.5 pt

3) What process should be followed to estimate the number of items sold during a visit, for a group of visits

lasting two hours? 1 pt

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Exercise 4: variable change (4 points)

The table opposite gives the price list of a private company, in metropolitan France, for the delivery of letters or packages sent, according to their mass.

Our goal is to analyse and model, thanks to this table, the evolution rate of Y according to X; then to propose a suitable price to be applied in the case of packages heavier than those proposed in this table.

X : mass (g) Y : price (€)

[0 ; 20[ 0.72

[20 ; 50[ 1.15

[50 ; 100[ 1.65

[100 ; 250[ 2.50 [250 ; 500[ 3.33 [500 ; 1000[ 4.38 [1000 ; 2000[ 5.50 [2000 ; 3000[ 6.55 1) Graphing this table would make it clear that a linear regression would not be appropriate here. It is

proposed to perform the following variable change: T =X^0.3.

a. After entering the values of T and Y into your calculator, calculate the linear correlation coefficient

between these two variables, and then interpret. 1.5 pt

b. Thanks to your calculator, give the least square line’s expression of Y with respect to T. Deduce a

relationship between Y and X. 1 pt

2) According to this last relationship, what price should be applied for packages weighing 6 kg? 1.5 pt

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IUT TC MATHEMATICS BIVARIATE STATISTICS FORM

χ

² law table

The table gives values χ²lim

such that p(χ² > χ²lim) = α

α α α α

χ²lim

χ

^²

α ^(p-value) 1 − α