IUT of Saint-Etienne Sales and Marketing department Mr Ferraris Prom /12/2015

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1 IUT of Sait-Etiee Sales ad Marketig departmet Mr Ferraris Prom /12/2015 MATHEMATICS 3 rd semester, Test 2 legth : 2 hours coefficiet 2/3 The graphic calculator is allowed. Ay persoal sheet is forbidde. Your work will be writte iside this documet. The presetatio ad the quality of your writigs will be take ito accout. Your rouded results will ow at least four sigificat figures. Full ame : Group : B2 Exercise 1 : (4 poits) A compay produces idustrially cream pots. We deote by X the radom variable takig values for the mass i grams of a pot draw at radom from the productio. Assume that X is distributed accordig to a ormal distributio of mea 201 grams ad stadard deviatio 2 g. 1) What is the probability that a pot take at radom have a mass greater tha 200 g? 1.5 pt 2) This compay has to deliver to a cliet boxes cotaiig 25 pots each. This cliet will accept a box oly if its total mass of cream is at least 5 kg - i other words, ad this is the most importat for us: oly if the average mass of a pot is at least 200 g i this sample of 25 pots. a. What is the probability a box be rejected by the cliet? 1.5 pt S3 Mathematics TEST 2 page 1 o 8

2 b. Deduce the umber of rejected boxes, from a delivery of 500 boxes. 1 pt Exercise 2 : (4 poits) A vegetable packig compay istalled a colorimetric detector o its tomatoes packagig lie. This detector rejects tomatoes that appear too gree. 1) Let us cosider a delivery i which the proportio π of gree tomatoes is We deote P the radom variable givig the proportio of gree tomatoes i a simple radom sample of 250 tomatoes. a. What is the probability distributio of the variable P? 1 pt b. Calculate the probability that more tha 10% of the tomatoes be too gree i such a sample. 1 pt 2) Let s cosider aother delivery (aother populatio of tomatoes), ito which the proportio π is ot kow. I a sample of 2000 tomatoes (SRS), 240 appear to be too gree. Give a estimate of π by a 95% cofidece iterval. 2 pts S3 Mathematics TEST 2 page 2 o 8

3 Exercise 3 : (5 poits) A factory produces ad packages sugar cae i packs whose omial et weight is 1000 g. It is assumed that the et mass i grams of a pack is distributed accordig to a ormal distributio whose mea ad stadard deviatio are ukow. Let's take a simple radom sample of 50 packs ad weigh them oe by oe. We obtai the followig results: Mass i grams Number of packs ) Give, with five sigificat figures, the mea mass ad its stadard deviatio, from this sample. 1 pt 2) Give a poit estimate of the stadard deviatio of the masses i the populatio. 1 pt 3) Give a estimatio of the average mass i the populatio, by a 99% cofidece iterval. 1.5 pt 4) What should be the size of a sample such that we would obtai a 99% cofidece iterval whose size (fr : amplitude) be equal to 0.5 g? (we will assume here that the stadard deviatio of the populatio is kow ad that its value is the oe that has bee estimated i questio 2) 1.5 pt S3 Mathematics TEST 2 page 3 o 8

4 Exercise 4 : (3.5 poits) from a tutorial proposed by the Uiversity of Metz, L2 psychology: Psychology has made the demostratio of a amazig pheomeo called "mere exposure effect", or MEE (fr.: effet de simple expositio): subjects submitted to a stimulus evetually "love" this oe. For istace, some sogs that are ot our favorite at all may please us if we liste to them regularly. This effect has bee iterestig advertisers for years, who foud i it a reaso to hope that multiple ad cotiuous displays of the logo of a brad evetually improve its popularity. It is ideed ofte the case, but i large doses there are also surprises. To test this theory, three colors deoted A, B, C have bee chose by researchers. It is kow that i the geeral populatio, if you ask subjects to choose their favorite color amog the three, the respose distributio is uiform, that is to say: 1/3 of people choose each color. A sample of people were exposed to color A durig 3 hours. The, subjects were asked about their favorite color. The results were: color A B C frequecies Treat these results by a statistical test ad coclude S3 Mathematics TEST 2 page 4 o 8

5 Exercise 5 : (3.5 poits) The Pressagrume brad holds 30% market share (fr.: parts de marché) o sales of orage juice i a hypermarket. After a advertisig campaig for the brad, a survey showed that o 500 people who bought orage juice, iterviewed at radom at the exit of the store, 170 of them bought the Pressagrume brad. Ca we cosider that the campaig icreased sales of the brad? We will use a test with a 5% sigificace level. TEST END S3 Mathematics TEST 2 page 5 o 8

6 IUT TC Form for Semester 3 MATHEMATICS Samplig Meas samplig distributio Let be a large populatio (size = N > 30) o which a variable X is defied, kowig its mea µ ad stadard deviatio σ. is the list of all Xthe the meas of all the -sized samples. σ σ N The law of X is N µ, (SRS or N 20), or N i ay other situatio. µ, N 1 Proportios samplig distributio Let be a large populatio (size = N > 30) o which a characteristic is defied ad measured as a proportio π. P is the list of all the the proportios measured i all the -sized samples. π ( 1 π ) π The law of P is N ( 1 π ) N π, (SRS or N 20), or N π, (other). N 1 Estimatio Poit estimates of µ, σ, π : ˆ µ = x ˆ σ = s ˆ π = p 1 Estimatio of µ by a cofidece iterval: σ σ s s σ is kow : Iα = x u ; x + u σ is ukow : Iα = x t ; x + t 1 1 Estimatio of π by a cofidece iterval: ( 1 ) ( 1 ) p p p p Iα = p u ; p + u Statistical testig Adequacy Chi-square test Let be a list of observed values obs, to be compared to a list of theoretical values th. 2 2 ( obsi thi ) The calculated Chi-square of the experimet is χ, sum of the partial Chi-square. calc = th Coformace testig of a mea A mea µ 0 has to be tested for the populatio, by a comparizo with a mea x got from a sample. X µ 0 X µ 0 σ is kow : U = σ is ukow : T = σ S 1 Coformace testig of a proportio A proportio π 0 has to be tested for a populatio, by a comparizo with a proportio p got from a sample. U = π i= 1 P π ( 1 π ) i S3 Mathematics TEST 2 page 6 o 8

7 Stadard ormal law N(0, 1) table iside: the probabilities p(u < u) How to get u from x : µ u = x σ p(u < u) u U u S3 Mathematics TEST 2 page 7 o 8

8 Studet law table iside: the values of t such that p(-t < T < t) = p p -t t T 1 - p 1 - p dof dof χ² law table Iside: the values χ² lim such that p(χ² < χ² lim ) = p p χ² 1 - p 1 - p 1 - p 1 - p dof 1% 2% 5% 10% dof 1% 2% 5% 10% dof 1% 2% 5% 10% dof 1% 2% 5% 10% χ² lim S3 Mathematics TEST 2 page 8 o 8

Expectation and Variance of a random variable

Expectation and Variance of a random variable Chapter 11 Expectatio ad Variace of a radom variable The aim of this lecture is to defie ad itroduce mathematical Expectatio ad variace of a fuctio of discrete & cotiuous radom variables ad the distributio