IUT of Saint-Etienne Sales and Marketing department Mr. Ferraris Prom /04/2017

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1 IUT of Saint-Etienne Sales and Marketing department Mr. Ferraris Prom /04/2017 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. Full Name : Group : B1 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 H 0 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 ; ; 4 ; 4 4 ; 4.5 ; 5 ; 5 ; ; 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 Brown Eye colour Green Blue ) 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 S2 Mathematics TEST 1 page 1 / 6

2 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. gross Y year minimum year range wage ( ) X Y X S2 Mathematics TEST 1 page 2 / 6

3 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 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 pt S2 Mathematics TEST 1 page 3 / 6

4 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: vehicle's age (years) : X resale value ( ) : Y 9,200 6,700 5,500 4,300 4,200 3,750 3,400 3,000 1) Here, a linear fitting would be irrelevant; but, a curve fitting would be. Let's perform the variable change: 1 T =, assuming a good linear correlation between T and Y. Give the expression of the Y on T regression X 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 S2 Mathematics TEST 1 page 4 / 6

5 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 S2 Mathematics TEST 1 page 5 / 6

6 IUT TC MATHEMATIQUES FORMULAIRE STATISTIQUES A DEUX VARIABLES S2 Mathematics TEST 1 page 6 / 6

IUT of Saint-Etienne Sales and Marketing department Mr Ferraris Prom /10/2015

IUT of Saint-Etienne Sales and Marketing department Mr Ferraris Prom 2014-2016 22/10/2015 MATHEMATICS 3 rd semester, Test 1 length : 2 hours coefficient 1/3 The graphic calculator is allowed. Any personal