Lebanese University Faculty of Science November 2018 Duration: 2h Partial Exam - S1101 Exercise 1 (12 points) For each of the following 8 questions below, choose only one of the answers: Q1) A distribution of 8 values has a median of 23. If the highest value increases 3 points, the median will become: (a) (b) (c) (d) (e) 23 23.5 25 Cannot be determined without additional information none of the above Q2) The mean and the standard deviation of the sampling distribution are respectively equal to 25 and 0. what must you conclude? (a) (b) (c) (d) (e) Someone has made a mistake All the elements in the sample are null There are no elements in the sample All the elements in the sample are 25 none of the above Q3) The histogram below shows the relative positions of median, mode and mean. What is the proper ordering of these parameters? (a) (b) (c) (d) (e) I I I I I = mean ; II = median ; III = mode =mode ; II = median ; III = mean = median ; II = mean ; III = mode = mode ; II = mean ; III = median = mean ; II = mode ; III = median Q4) Which of the following measures is not a measure of dispersion? (a) (b) (c) (d) (e) The Range Variance Interquartile Range 50th Percentile Standard Deviation 1/3 Q5) In a group of 10 students, the highest grade in statistics is increased by 30 points. What effect will this have on the mean of grades? (a) (b) (c) (d) (e) It will be increased by 10 points It will remain unchanged It will be increased by 3 points It will increase by 30 points None of the above The following information will be used in the next three questions: The distribution of the age of 40 smokers is shown int he following table: Age Density [10,20[ 0.4 [20,30[ d2 [30,x[ 1.5 [x,50[ 0.5 [50,60[ 0.4 Q6) Knowing that the ages of 60% of smokers belong to [30, 50[, then the value of d2 is equal to: (a) (b) (c) (d) (e) 0.1 0.2 0.4 0.8 None of the above Q7) Suppose that d2 = 0.9. If the mode of this sample is equal to 31.5 years, then the value of x is: (a) (b) (c) (d) (e) 34 42 34 54 None of the above Q8) Suppose that d2 = 0.9. The percentage of smoking people whose age is less than 28 years is: (a) (b) (c) (d) (e) 28% 30% 32% 34% None of the above Exercise 2 (18 points) The following table shows the distribution of 20 Lebanese University students according to their study time per day (in minutes), preferred football team and number of times per day they connect to a mobile game. 2/3 Study Time Preferred team 20 23 21 22 30 30.5 30 40 44.5 41 27 28 37.5 38 32.5 32 33 38 31 37 AC Milan AC Milan AC Milan Barcelona Barcelona AC Milan Barcelona Barcelona AC Milan AC Milan AC Milan Barcelona AC Milan AC Milan AC Milan Barcelona AC Milan Barcelona AC Milan Barcelona Number of times they connect 1 1 1 2 2 1 3 4 3 1 5 5 1 2 2 a b c 5 5 1. Precise the population, sample, and the variables and their types. 2. The results for the study time should be summarized in classes of a statistical table: (a) Show that the number of classes and their width are equal to 5. Construct the frequency distribution. (b) Construct the polygon of increasing cumulative frequency for the variable study time. Deduce the median graphically and by calculation. (c) Calculate the study time of individual number 15 (rank =15). 3. We consider that a > 1, b > 3, and c > 3. Find the value of a if the median number of times they connect is equal to 2. 4. In this part we suppose that a=2. Find the values of b and of c if the average number of times they connect to a mobile game is equal to 3 and the third quartile is equal to 4.5. 5. In this part consider that a = 2, b= 10 and c =4. Draw the boxplot for the number of times they connect to a mobile game for students who support AC Milan. 3/3 Lebanese University Faculty of Sciences Midterm Exam November 2018 S1101- Statistic Solution Exercise 1 (12 pts =(1.5*8)) Q1 (a) Q2 (d) Q3 (a) Q4 (d) Q5 (c) Q6 (d) Q7 (a) or (c) Q8 (a) Exercise 2 (18 pts) 1. (4 pts = 0.5*8) Set of Students Population (0.5 pts) 20 Lebanese University Students Sample (0.5 pts) Study Time Preferred Number ot times Variables team they connect (0.5*3 pts) Quantitative – Qualitative Quantitative Nature Continuous – Nominal Discrete (0.5*3 pts) 2. (a) (3 pts = 1+1+1) Let Let number of classes then then the width (or amplitude) of classes then Study Time 20 – 25 25 – 30 30 – 35 35 – 40 40 – 45 Total (b) (4 pts=1+1+1+1) Ni 4 6 13 17 20 1/2 ni 4 2 7 4 3 20 . then Ni 4 6 13 17 20 The median (c) (2 pts) The study time of individual number 15 is then 3. (1 pts) The number of times they connect: 1 1 1 1 1 1 2 2 2 2 3 3 4 5 5 5 5 a b c then We have but we have 4. (2 pts) a=2 then , but or then so that to find then we have . then ( and ) or ( and ). 5. (2pts) a=2, b=10 then the distribution is: 1 1 1 1 1 1 2 2 3 5 5 10 then and and * Number of times 1 1.5 4 5 10 2/2 Supplementary Exercises: Continuous Variable Exercise 1 - We consider the distribution of 200 employees of a bank in Beirut based on their annual income in thousands of $. Annual income [30-50[ [50-70[ [70-90[ [90-110[ [110-130[ [130-150[ Total ni Ni Ni fi 10 10 200 20 30 190 50 80 170 60 140 120 20 160 60 40 200 40 200 0,05 0,05 1 40 20 -58 0,1 0,15 0,95 60 20 -38 0,25 0,4 0,85 80 20 -18 0,3 0,7 0,6 100 20 2 0,1 0,8 0,3 120 20 22 0,2 1 0,2 140 20 42 1 Fi Fi Midpoint : ci Width ai : ci- 1) The population is the employees in a bank in Beirut; the sample is the group of; the variable is the annual income in thousands of $. Its kind is Quantitative continuous. 2) Determine graphically and analytically the mode, the first quartile Q1, the second quartile Q2 and the third quartile Q3. Step 1 – (Same Width) => 60>40>… => Modal Class = [90-110[ Step 2 – Interpretation: The most observed income in employees of bank is 94 thousand of dollars. 3) Draw the cumulative frequency polygon in an ascending and descending direction. Fid Q1, Q2 and Q3. Interpret Me: N/2 = 200/2 = 100 => Median class = [90-110[ Step1 – [90-110[ Setp 2 – Interpretation: 50% of the employees of the bank in Beirut have an income less than 96.67 thousands of dollars and 50% of employees of the bank in Beirut have an n more than 96.67 thousand of dollars. Q1: Step 1 – [70-90[ Step 2 – Interpretation: 25% of the employees of the bank in Beirut have an income less than 78 thousands of dollars and 75% of employees of the bank in Beirut have an n more than 78 thousand of dollars. Q3: Step 1 – [110-130[ Step 2 – Interpretation: 75% of the employees of the bank in Beirut have an income less than 120 thousands of dollars and 25% of employees of the bank in Beirut have an n more than 120 thousand of dollars. 4) Find the Inter-quartile range. Interpret. IQR = Q3-Q1=120-78= 42 Interpretation: 50% of the employees of the bank in Beirut earn between 78 and 120 thousands of dollars. The annual income of 50% of the employees are spread over a range of 42 thousands of $. 5) Determine the income of the employee number 150 We must find P75 Step 1 – [110-130[ Step 2 – 6) Find the proportion of employees in this population whose income exceeds 120 thousand of $. 7) The distribution is homogeneous? Interpretation: the average income of employees of the bank to Beirut is 98 thousands of $ ; Interpretation: The dispersion around the average value of is equal to 28.2 >15% => the distribution is heterogeneous 8) The distribution is symmetric? Interpret Me ≈ M 0 ≈ => the distribution is symmetric ~ 0: Distribution is symmetric Exercise 2- The graph below shows the age (in months) of 100 children treated for respiratory physiotherapy spastic bronchitis at the Hôtel-Dieu in Beirut. 27% 23% 14% 9% 9% 7% 7% 2% 2% 0% 0% 3 8 13 18 23 28 33 38 43 48 0% 53 0% 58 63 68 Âge Age (en (in mois) months) 1) Determine: The Population: children treated for respiratory physiotherapy spastic bronchitis at the Hôtel-Dieu in Beirut. Individual: 1 child The variable: age (in months) Type: Quant. Continuous Age [3-8[ [8-13[ [13-18[ [1823[ [2328[ [2833[ [3338[ [3843[ [4348[ [6368[ T fi ni = fi×N Ni ai Midpoint: Ci 0.27 27 27 5 5.5 0.23 23 50 5 10.5 0.07 7 57 5 15.5 0.09 9 66 5 20.5 0.09 9 75 5 25.5 0.07 7 82 5 30.5 0.14 14 96 5 35.5 0 0 96 5 40.5 0.02 2 98 5 45.5 0.02 2 100 5 65.5 1 100 --------- 2) Find the mode by 2 methods Step 1: Same width => 27>23>… => Modal class = [3-8[ Step 2: 3) Calculate the median (Me) and the first quartile (Q1). Interpret. Median: Step1: [13-18[ Step 2: Interpretation: 50% of children have an age less than 13 months and 50% of children have an age more than 13 months First Quartile Q1: Step 1: [3-8[ Step 2: Interpretation: 25% of children have an age less than 7.7 months and 75% of children have an age more than 7.7 months 4) Calculate the average (mean) age of the children. Interpret. 5) The statistician at the hospital has found that the standard deviation of the variable studied is 4.3. Interpret this value. The dispersion of observations around the average value of is equal to 4.3