SALES AND MARKETING Department MATHEMATICS. 3rd Semester. Probability distributions. Tutorials and exercises

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1 SALES AND MARKETING Department MATHEMATICS 3rd Semester Probability distributions Tutorials and exercises Online document : on section DUT Maths S3. IUT de Saint-Etienne Département TC J.F.Ferraris Math S3 ProbDist TEx Rev2017 page 1 / 14

2 Exercise 1. (Tutorial for lesson page 6) 8 different letters have to be drawn from our alphabet. The success letters are vowels. The random variable X is the number of successes among the 8 drawn letters. a. Justify that the distribution of X is Hypergeometric and give its parameters. b. Calculate p(x = 0), p(x = 3), p(x = 8). c. Calculate the expectation and the standard deviation of X, then comment the expectation. d. Build a bar-chart of this probability distribution. Exercise 2. A supermarket sells 24 fruit species, counting 8 bio label. A blind control consists in choosing 10 fruits of different species. The variable X gives the number of bio species among these Give, with explanations, the probability law of X. 2. Calculate its expected value and standard deviation. 3. What is the probability that less than two bio species would be chosen? Exercise 3. (Tutorial for lesson page 7) A wheel (roulette) is divided into 14 same-sized sectors. 5 sectors are white and the others are red. After spinning the wheel, the success is : "it stops on a white sector". The random variable X gives, after 8 successive attempts, the total number of successes. a. Explain why the law of X is binomial and give its parameters. b. Calculate p(x = 2). On your calculator, list the probabilities of each possible value of X. c. Calculate the expected value and the standard deviation of X, then comment these values. d. Graph (sticks) these results. Exercise 4. A car driver meets five signal lights on his way. They share the same duration of red and green lightening : 40 seconds green and 20 seconds red. These lights are not synchronized, so that the color of one light is independent of the color of another one. 1. At the moment you come to the first light, what s the probability it would be green? 2. What s the probability that the lights would all have been green on crossing them? 3. What s the probability that at least two lights would have been red? 4. What s the mean expected number of green lights driving this way? Exercise 5. The germination capacity of a seed is 0.8 (probability to germinate) seeds are sown. Calculate the probabilities of the following events : a. exactly 5 seeds will germinate. b. At least 7 seeds will germinate. 2. When a seed has germinated, the probability that a slug eats the young plant is 0.4. a. Calculate the probability that a seed will finally become a grown plant. b. How many seeds must be sown to get more than 99% chances of getting at least one grown plant? Exercise 6. According to a survey, 80% of the customers of a product A are satisfied by its purchase. Choosing randomly 10 customers of this product, what s the probability that a. they re all satisfied? b. 80% of them are satisfied? c. at least 80% are satisfied? Exercise 7. 6% of French people are clients of the mobile phoning operator Yellow. A survey consists in asking to 50 randomly chosen French people which is their mobile phoning operator. The variable X gives the number of Yellow clients among these 50 people. 1. a. Justify and give the probability law of X. b. What are the chances that the population proportion of clients would be the same into the sample? c. What is special with this former probability? IUT de Saint-Etienne Département TC J.F.Ferraris Math S3 ProbDist TEx Rev2017 page 2 / 14

3 d. What s the probability that none of the 50 people would be a Yellow client? e. What s the probability that there would be at least 4 Yellow clients? 2. In this part, the number of individuals to be called is still unknown. How many people would have to be called, to get more than 99% chances finding at least one Yellow client? Exercise 8. (Tutorial for lesson page 8) The law of the variable X is binomial with parameters n = 50 and p = a. Obtain (calculator's lists) p(x = k) for each integer k from 0 to 7. b. Justify the approximation of this law by a Poisson's one whose parameter has to be given. c. Give, by using Poisson's law table, the probabilities asked above. Compare them to the ones obtained with the binomial law. Exercise 9. The shop HighTech sells computers. The variable number of daily sales is distributed like a Poisson s law whose parameter is 4. Calculate the probability that the next day a. No computer would be sold b. At least one computer would be sold c. Exactly 2 computers would be sold. Exercise 10. On a survey implying a large population, only 2% of its individuals accept to give their name. Given that one of the investigators has to interview 250 people, calculate the probability that a. All these people won t give their name. b. At least 5 people will give their name. Exercise 11. A box contains 250 matches. It has been exposed to moisture, so that 20% of matches won t lighten. Taking at random 10 matches, the variable X gives the number of matches that will lighten. 1. Demonstrate that X can be described by a binomial law and give its parameters and expected value. 2. Calculate the following probabilities : a. No match will lighten b. They will all lighten c. At least 3 won t lighten 3. a. Calculate the same probabilities, this time using a Poisson s law. b. Explain the differences between your answers at questions 2 and 3. Exercise 12. In a large population are met on average 0.4% of blind people. 1. Into a sample of 100 people, what s the probability there s no blind person? at least 2? 2. Answer these questions using the correct Poisson s law (justify its use). IUT de Saint-Etienne Département TC J.F.Ferraris Math S3 ProbDist TEx Rev2017 page 3 / 14

4 Exercise 13. (Tutorial for lesson page 10) 1) Let's consider a statistical data set that shows a symmetric distribution with most of central values, from a large population. e.g. : many objects were manufactured and weighed. Their theoretical mass is 3.8 kg, and the weights of 200 objects are distributed as follows: masse (kg) [3.5 ; 3.7[ [3.7 ; 3.77[ [3.77 ; 3.8[ [3.8 ; 3.83[ [3.83 ; 3.9[ [3.9 ; 4.1[ frequency freq. rate Let's graph the frequency histogram of this series : on abscissas: the variable (mass, kg); on ordinates: frequency concentration (in % of objects per kg) the rectangles areas are in proportion with the frequencies. What's the probability, for an object taken randomly, to weigh less than 3.77 kg? 2) Now, the 200 results can be given more precisely, into a greater number of thinner intervals The new frequency histogram is the following (each point is the middle of the high side of a rectangle): A bell curve is appearing, typical of numerous distributions in many concrete fields (production, economy, biology, ecology, ). Which way could we find, using this histogram, the probability that an object's mass (chosen at random) be less than 3.7 kg? be between 3.7 kg and 3.9 kg?. 3) We could consider weighing far more than 200 pieces, and with far more accurate results. The histogram could contain a large number of rectangles and would become difficult to draw and to read! The only useful graph would contain only a points cloud, that would actually follow a bell-shaped curve, that could be modelled by a function f. In this context, how would we calculate the probabilities asked above? Exercise 14. (Tutorial for lesson page 12) Give or calculate the probabilities asked below, from the standard normal law table and the available transformations formulas. Then, check your results thanks to the tool "normal law" of your calculator. a. p(u < 1) b. p(u < 1.96) c. p(u < 2.58) d. p(u > 1) e. p(u > 1.63) f. p(u > 0.35) g. p(1 < U < 2) h. p(0.42 < U < 1.07) i. p(u < -1) j. p(u < -0.88) k. p(u > -0.5) l. p(u > -2.23) m. p(-1.85 < U < -1.07) n. p(-1.12 < U < 0.6) Exercise 15. (Tutorial for lesson page 12) Calculate the probabilities, using the standard normal law table and the available transformations formulas. Then, check your results thanks to the tool "normal law" of your calculator. 1) law of X : N (50, 10). Calculate p(x < 60), p(x < 43), p(45 < X < 55) 2) law of X : N (3, 0.45). Calculate p(x > 4), p(x < 2;55), p(3.2 < X < 3.7) IUT de Saint-Etienne Département TC J.F.Ferraris Math S3 ProbDist TEx Rev2017 page 4 / 14

5 Exercise 16. (Tutorial for lesson page 13) In a land, 30 % of the companies do exports. We decide to choose 80 companies at random and we set X as the number of those doing exports among the 80. 1) What is the probability distribution of X and what are its parameters? 2) Justify that a binomial law can be used instead of the former one. 3) Justify that a normal law can be used; give its parameters. 4) Using that normal law, then checking with the tool "binomial law" of your calculator, give: a. the probability that more than 30 companies do exports among the 80. b. the probability that 30 companies do exports among the 80. Exercise 17. (Tutorial for lesson page 13) A company has put its merchant website online. Tests have shown that a connection problem appears on average once in 500. For its brand image, it considers that a bad week is one with more than 50 connection problems and that during the year no more than 5 bad weeks may occur. 1) Given that weekly user connections are expected for the following weeks, buid the probability distribution of the number X of problems per week. 2) Determine which normal law can be used in that case. 3) What is the probability that in a given week more than 50 problems may occur? 4) What is the probability distribution of the number Y of bad weeks during one year? 5) What is the probability that Y would be more than 5? Exercise 18. It s been stated that the variable X mass (kg) of a newborn baby is distributed by the law N (3.4 ; 0.5). 1. What s the probability that a newborn baby weighs more than 4 kg? 2. What s the probability that a newborn baby weighs less than 3 kg? 3. What s the probability that a newborn baby s weight is between 3 and 4 kg? Exercise 19. A company manufactures beacons (flashing lights) for all types of machines, in large quantities. The probability that a beacon is defective is p = A random sample of 600 beacons is taken from the production. X is the random variable that gives the number of defective beacons among the Show that the random variable X has a binomial distribution whose parameters are to be specified. 2. Show that we can approximate the distribution of X by a normal distribution. 3. Determine µ and σ, mean and standard deviation of the variable X for the normal distribution. 4. Then calculate the probability of having at least 27 defective flashing lights in the draw of 600. Exercise 20. A commercial agent is assigned to telephone solicitations. On average, one in five phone calls leads to appeal an order. We name X the random variable number of orders after 60 calls. 1. a. Give the name and the parameters of the probability distribution of X. b. Justify that the law can be approximated by a normal distribution, give its parameters. c. Calculate p(x > 15), p(x < 10), p(x = 12). 2. Find out the minimum number of phone calls that must be passed by the sales agent so that his chances to get at least 10 orders are more than 95%. Exercise 21. It is assumed that on average you re checked once in 20 in the bus by a controller. Mr Don Wanapay makes 800 trips a year on the line. 1. What is the probability that Mr Don Wanapay would be checked between 30 and 50 times a year? 2. Mr Don Wanapay always travels without a ticket. An annual subscription would be 320 / year. At what height must the company fix the fine, so that at least 99% of cheaters would better take an annual subscription? IUT de Saint-Etienne Département TC J.F.Ferraris Math S3 ProbDist TEx Rev2017 page 5 / 14

6 Exercise 22. (Tutorial for lesson page 14) 1) From a normal population, µ = 120 and σ = 40, are taken every SRS of sizes n = 10 and n = 50. a. What are the distributions of these samples means? b. Graph both distributions, roughly, in the same frame, in order to compare them. c. What is the probability that the mean of a random 10-sized sample would be more than 130? d. Same question for a 50-sized sample. 2) Several years ago, the world counted 3.38 billion women and 3.12 billion men. P is the variable giving the proportion of women in every sample of 100 people. a. Give the probability distribution of P. b. What is the probability that, in a sample, there would be more men than women? Exercise 23. In a production of light bulbs, it is assumed that the lifetime of a bulb is a normal random variable whose mean is 900 hours and standard deviation is 80 hours. Calculate the probability that in a random sample of 100 bulbs (SRS), the average lifetime of bulbs exceeds 910 hours. Exercise 24. A candidate obtained 55% of votes cast in an election. 1. What is the probability that, in a sample of 100 people, his result be less than 50%? Among 2000 people? 2. How many people are required so as the probability his result be less than 50% drops below 1%? Exercise 25. In a region, during the summer period, it is assumed that the number of tourists present in a day follows a normal distribution whose mean is 50,000 and standard deviation is 8, The prefecture considers that the tourism is "manageable" (reception, environment, pollution,...) when more than 70% days count less than 55,000 tourists each. What is the actual situation? 2. Officials want to base their thinking on 10 vacation days periods. a. What is the law of X : "Average daily number of vacationers in a sample of 10 days"? b. What is the probability that, in 10 days, this average daily number be less than 55,000? Exercise 26. A large population took an IQ test. The results are normally distributed with µ = 102 and σ = What s the proportion of people whose IQ is less than 100? 2. We wish to analyze the results of a few samples of this population. For that, let's create groups of 20 individuals selected by SRS, and the average IQ of each group will be calculated. a. Give the parameters of the normal distribution of IQ means of all 20-sized samples. b. What is the probability that a group of 20 people has an average IQ below 100? c. Instead of 20, how many people would have to be chosen for this probability to be less than 5%? 3. Using the answer of question 1, what is the probability that, in a group of 20 people, the proportion (of individuals whose IQ is less than 100) is more than 50%? Exercise 27. An elevator can carry a load of 580 kg. It is assumed that someone's weight is a random variable following a normal distribution N (µ, σ) with µ = 70 kg and σ = 16 kg. What is the maximum number of people you may allow to be together in the elevator if you want the risk of overload does not exceed 0.01? IUT de Saint-Etienne Département TC J.F.Ferraris Math S3 ProbDist TEx Rev2017 page 6 / 14

7 Exercise 28. (Tutorial for lesson page 16) A sample of companies of the same industry provided the following results: turnover (M ) [0 ; 2[ [2 ; 3[ [3 ; 4[ [4 ; 5[ [5 ; 7[ size (# of companies) a. Give point estimates of the mean turnover and its standard deviation in the whole set of companies. b. Give the 95% confidence interval of the mean turnover in this industry. c. Give a point estimate of the proportion of companies whose turnover is more than M 4.5. b. Give the 99% confidence interval of this proportion in this industry. Exercise 29. From a vine, 10 grapes have been taken at random and weighed, which gave the following results in kilograms: 2.4 ; 3.2 ; 3.6 ; 4.1 ; 4.3 ; 4.7 ; 5.3 ; 5.4 ; 6.5 ; Give the mean and standard deviation of a grape s mass in this sample. 2. Give a point estimate of the variance of the grape mass in the whole vine (population). 3. Give a 95% confidence interval of the average mass of grapes in the whole population. 4. Calculate the minimum number of grapes that would have to be analyzed so that this interval be 1 kg wide, assuming that the estimated standard deviation (q.2.) is the real one of the population. Exercise 30. A laboratory wishes to analyze the level of contamination of trees by acid rains, in a given territory. After examination of a 100 trees sample, 8 affected trees have been found. Give an estimate of the proportion π of affected trees in this territory, by a 90% confidence interval. Exercise 31. On managing a grain elevator, we wonder about the safety (minimal) required stock that has 99% chances to satisfy customers at any time. For this, the weekly consumption of grain has been analyzed during a sample of 15 weeks. The following results were obtained: weekly consumption (in tons) number of weeks Give the mean and standard deviation of the consumption in this sample. 2. We set X the weekly consumption of grain at any time, and we assume that its distribution is normal. a. Give the point estimates of µ and σ. b. Using this normal law, calculate the value of X that has 99% chances not being exceeded. 3. a. Using the results of question 2, build a 99% confidence interval of the average weekly consumption. b. What s the probability that this average value would exceed the upper limit of this interval? Exercise 32. A company wants to specialize in the delivery of large packages. Those that have already been carried are considered as a representative sample of all future packages. data set of large packages that have already been carried: volume (m³) 0.2 to to to to 1 # of packages Give the point estimates of the mean and standard deviation of the future packages volume. 2. Give a 99% confidence interval of the average volume of the future packages. 3. In this question, the standard deviation of the population is considered known and its value is the one you found in question 1. We want to use a confidence interval of the average volume, whose size would be 0.05 m 3. What would be the confidence level of such an interval? IUT de Saint-Etienne Département TC J.F.Ferraris Math S3 ProbDist TEx Rev2017 page 7 / 14

8 Exercise 33. (Tutorial for lesson page 17) A die has been rolled 120 times. The results are gathered in the table below. Considering this sample of results, can we say that this die is a fake one, at a 2% significance level? result observed # of throws Exercise 34. An experiment consists in trying something three times, with 1/3 chances of success each time. X is the random variable number of successes at the end of the experiment. 1) Prove that p(x = 0) = 8/27, p(x = 1) = 12/27, p(x = 2) = 6/27 and p(x = 3) = 1/27. 2) Now, let's imagine that 54 people performed this experiment. a. Complete the following table: number of successes per individual total observed number of individuals theoretical number of individuals 54 b. By a χ² test, at a 5% significance level, decide whether the observed results are in adequacy with the expected theoretical ones. Exercise 35. There has been reported, for five French groups in the same industry, the annual budget for promotion on the Internet compared to the global annual budget for promotion: group A B C D E Internet budget (k ) overall budget (k ) Part 1 1. Determine the sample's proportion of groups whose Internet budget exceeds 10% of the overall one. 2. a. Determine the 95% confidence interval of the proportion that could be observed in all French groups in this industry. b. This industry actually consists of 58 groups in France. What is the minimal number, that can be assumed with a confidence level of 80%, of groups whose internet budget exceeds 10% of their overall budget? Part 2 Perform a Chi-squared test to tell, with a significance level of 5%, if the observed data set of five companies is in adequacy with the following assertion: in France, Internet budget is worth 10% of overall budget. Exercise 36. A study was conducted in a sample of 50 plastics companies, getting their 2016 net income (variable R): net income R (M ) [-1 ; 1[ [1 ; 1,5[ [1,5 ; 2[ [2 ; 3[ [3 ; 5[ # of companies Part 1 1) Give the income s mean and standard deviation of this sample. 2) Give a 99% confidence interval of the average net income in the whole large population of plastics companies (you may notice that the population s standard deviation is unknown). Part 2 Our aim here is to make an assertion about the possibility that the net incomes distribution are in adequacy with the normal law N (2, 0.9). Let's name X the variable of this law. 1) a. Calculate : p(-1 < X < 1) ; p(1 < X < 1,5) ; p(1,5 < X < 2) ; p(2 < X < 3) ; p(3 < X < 5). b. Explain why, in adequacy with this normal law, and then in accordance with the five probabilities you just calculated, a theoretical sample of 50 plastics companies would give the following table: IUT de Saint-Etienne Département TC J.F.Ferraris Math S3 ProbDist TEx Rev2017 page 8 / 14

9 net income R (M ) [-1 ; 1[ [1 ; 1,5[ [1,5 ; 2[ [2 ; 3[ [3 ; 5[ # of companies ) a. Then, perform an adequacy χ² test between this normal law and reality, at a 5% significance level. b. Give detailed explanations of this significance level. Exercise 37. The study of 320 families with 5 children has given the distribution of the following table. 5 boys 4 boys 3 boys 2 boys 1 boy 0 boy children 0 girl 1 girl 2 girls 3 girls 4 girls 5 girls # families Are these results compatible with the hypothesis that the births of a boy and a girl are equally likely events? Exercise 38. (Tutorial for lesson page 18) A greengrocer wishes to buy vegetables from a new supplier. The latter claims that his beans measure 10 cm on average. If this value is plausible, or if the estimate is even higher, then the greengrocer will decide to choose this supplier. Of course he won t, in case a sample gives a too low average size. The greengrocer fixed its risk level to 5%. Let X be the random variable "size of a bean (cm)", distributed by N (µ, 2.3). After having taken a sample of n = 25 beans, the calculated average size was x = 9.5 cm. Will he buy the beans here? Exercise 39. (Tutorial for lesson page 18) A career should produce 300 tons of ore, on daily average, not more, not less. It is assumed that the daily mass of produced ore is normally distributed. Produced daily quantities have been examined for 10 days (see the following results, in tons): Can we consider that the all-days average production is 300 tons, at a 5% significance level? Exercise 40. A manufacturer claims that the strings he produces have a 300 kg average tensile strength with a standard deviation of 30 kg. It is assumed that the variable strength of a string is normally distributed. Strengh tests made on 10 strings revealed the following breakdown tensions: Can we consider, thanks to this sample, that the average tensile strength of the whole production is equal to 300 kg? (significance level: 10%) Exercise 41. On 1000 French baccalaureate candidates chosen at random, 675 were successful. Test at a 10% significance level the claim of the minister that the success rate in France is 70%. Exercise 42. In several countries, the weather forecast is given as a probability. Forecasting "the probability of rain tomorrow is 0.4" was made 50 times during the past year and it appeared that the rain actually came 26 times the day after. Test the accuracy of the prediction, with a 5% α-level. IUT de Saint-Etienne Département TC J.F.Ferraris Math S3 ProbDist TEx Rev2017 page 9 / 14

10 IUT TC Form of the 3rd semester MATHEMATICS Probability distributions Hypergeométric H (n, a, N) k: number of desired successes, after n draws k n k Ca C a N a p( X = k ) = E( X ) = n n V C N Poisson P(λ) n: number of draws; a: number of "success" individuals; N: population's size; approx. hypergeometric by binomial: if N 20n. We set p = a/n if n 30 and p < 0.1 and npq < 10. We set λ = np Approximation of B (n, p) by N (µ, σ) : if n 30 and npq 5 ; we set µ = np and σ = npq Approximation of P (λ) by N (µ, σ) : if λ 20 ; we set µ = λ and σ = Sampling Means sampling distribution Given a big population (size = N), we examine a variable X has a mean µ and a standard deviation σ. We consider the mean of every sample of the same size n > 30. σ σ N n X is distributed by N µ, on SRS (or if N 20n), or N µ, in the other situations. n n N 1 Proportions sampling distribution Given a big population (size = N), we measure a proportion π for some characteristic. We consider the proportion in every sample of the same size n > 30. ( ) π 1 π π 1 π N n P is distributed by N π, on SRS (or if N 20n), or N π, (other). n n N 1 Estimation n Point estimate of µ, σ, π : ( X k ) Estimate of µ by a confidence interval: σ σ σ is known: Iα = x u ; x + u n n σ is unknown: IUT de Saint-Etienne Département TC J.F.Ferraris Math S3 ProbDist TEx Rev2017 page 10 / 14 λ ( ) Estimate of π by a confidence interval: p 1 p p 1 p Iα = p u ; p + u n n Statistical tests Adequacy Chi-square test (comparison to a distribution) n observed values obs, matching n theoretical ( ) 2 n 2 obsi thi values th. The Chi-square of the experiment is the number χ calc =, sum of the part χ². th Conformance test of a mean A mean µ 0 is tested for the population, by a comparison to a mean x got from a sample. σ s σ is known: X N µ 0, σ is unknown: X St µ, n n 0 n 1 with -1 dof Conformance test of a proportion A proportion π 0 is tested for the population, by a comparison to a proportion p got from a sample. N p k λ = = e λ k! ( X ) E ( N ) a a N n = n 2 N N 1 Binomial B (n, p) n: number of draws; p, q: probabilities of success, of failure p( X = k ) = C k p k q n k E( X ) = np V( X ) = npq approx. binomial by Poisson: ( X ) λ ; V( X ) = = λ n ˆ µ = x ˆ σ = s n 1 i= 1 i ˆ π = p s s Iα = x t ; x + t n 1 n 1 ( ) ( ) ( 1 π ) π 0 0 P N π0, n

11 Tables Poisson's law table Table of probabilities: values p(x = k) for several Poisson's distributions λ k λ k λ k IUT de Saint-Etienne Département TC J.F.Ferraris Math S3 ProbDist TEx Rev2017 page 11 / 14

12 Standard normal law table gives probabilities p(u < u) x µ How to get u from x : u = σ p(u < u) u u U IUT de Saint-Etienne Département TC J.F.Ferraris Math S3 ProbDist TEx Rev2017 page 12 / 14

13 Student's law table This table gives the values t such that p(-t < T < t) = p p -t t T α = 1 - p α = 1 - p dof dof IUT de Saint-Etienne Département TC J.F.Ferraris Math S3 ProbDist TEx Rev2017 page 13 / 14

14 χ² law table This table gives the values χ² lim such that p(χ² > χ² lim ) = α α α α α dof 1% 2% 5% 10% dof 1% 2% 5% 10% dof 1% 2% 5% 10% dof 1% 2% 5% 10% IUT de Saint-Etienne Département TC J.F.Ferraris Math S3 ProbDist TEx Rev2017 page 14 / 14