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

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1 IUT of Saint-Etienne Sales and Marketing department Mr Ferraris Prom /10/2015 MATHEMATICS 3 rd semester, Test 1 length : 2 hours coefficient 1/3 The graphic calculator is allowed. Any personal sheet is forbidden. Your work will be written inside this document. The presentation and the quality of your writings will be taken into account. Your rounded results will own at least four significant figures. Full name : Group : B2 Exercise 1 : (3 points) In a group of 20 people, 8 need to wear glasses. 10 people are to be selected at random, and of these 10, X is the number of those who need glasses. 1) Build the probability distribution of the variable X. 1.5 pt 2) Calculate the probability that X = 4, the one that X = 0 and the one that X = pt

2 Exercise 2 : (7.5 points) 6% of French cars are more than 10 years old (fictional data). The police organize a random series of 50 road controls and the variable X is the number of «more than 10 year-old cars» in this series. 1) Explain, in details, why the variable X can be modeled by a binomial distribution. 1.5 pt 2) What is the probability that at least two cars out of 50 would be more than 10 years old? 1 pt 3) a. On 50 cars, what is the mean expected number of those that are more than 10 years old? 1 pt b. What is the probability that this expected average number would be obtained? Comment. 1 pt

3 4) a. What Poisson distribution can approximate the binomial distribution used previously? 1 pt b. Answer question 2) with the Poisson distribution. 1 pt c. How much, in %, is the error committed by the Poisson distribution in the previous answer? 1 pt Exercise 3 : (5 points) 1) Let be the variable U distributed by the standard normal law N (0, 1). Give or calculate: a. p(u < 2.33) ; p(u < -1) ; p(-0.3 < U < 0.3). 1.5 pt

4 b. u 0 such that p(u > u 0 ) = 2%. 1 pt 2) Let be the variable X distributed by the normal law N (50, 8). Calculate: a. p(x < 42) ; p(x > 66) ; p(55 < X < 58). 1.5 pt b. x 0 such that p(x < x 0 ) = 10%. 1 pt

5 Exercise 4 : (4.5 points) In a large population, 15% of people have already bought the product A. 1) Let s randomly interview 40 people. a. Justify that the number of people who bought the product A, among the 40, can be distributed by the normal distribution which mean is 6 and which standard deviation is pt b. What is then the probability that less than 10 people out of 40 have purchased product A? 1 pt

6 2) Instead of 40, how many people should we interview so that the previous probability would be less than 5%? 2 pts TEST END

7 IUT TC Form for Semester 3 MATHEMATICS Probability distributions Hypergeometric H(n, a, N) n : number of draws ; a : number of success individuals ; N : population size k : number of successes considered after n draws p ( X k) approximation hypergeom. by binomial : if N 20n Binomial B(n, p) n : number of draws p, q : probability of success, failure k k n k ( X = k) = p C p q E( ) Poisson P(λ) k Ca CN = = C n n k a n N E p a X = n N ( ) X = np ( X k) approximation binomial by Poisson: if n 30 and p < 0.1 and npq < 10 Approximation of a binomial distribution B(n, p) by a normal one N(µ, σ) : if n 30 and npq 5 ; we set µ = np and σ = npq Approximation of a Poisson distribution P(λ) by a normal one N(µ, σ) : if λ 20 ; we set µ = λ et σ = λ V V ( ) ( N ) a a N n X = n 2 N N 1 ( ) X = npq k λ λ = = e E ( X ) = λ ; V ( X ) = λ k! Sampling Means sampling distribution Let be a large population (size = N > 30) on which a variable X is defined, knowing its mean µ and standard deviation σ. is the X list of all the the means of all the n-sized samples. σ σ N n The law of X is N µ, on SRS (N 20n), or N in any other situation. n µ, n N 1 Proportions sampling distribution Let be a large population (size = N > 30) on which a characteristic is defined and measured as a proportion π. P is the list of all the the proportions measured in all the n-sized samples. π ( 1 π ) π ( 1 π ) N n The law of P is N π, on SRS, or N π, in any other situation. n n N 1 Estimation n Point estimates of µ, σ, π : ˆ µ = x ˆ σ = s ˆ π = p n 1 Estimation of µ by a confidence interval: σ σ s s σ is known : Iα = x u ; x + u σ is unknown : Iα = x t ; x + t n n n 1 n 1 Estimation of π by a confidence interval: ( 1 ) ( 1 ) p p p p Iα = p u ; p + u n n

8 Tables Poisson law tables probabilities inside: values of p(x = k) for different Poisson parameters λ k λ k λ k

9 Standard normal law N(0, 1) table inside: the probabilities p(u < u) How to get u from x : µ u = x σ p(u < u) u u U