1. The following data shows the (hourly) number of sales in an ice-cream shop, recorded during different opening hours:

Transcription

1 Statistics I Exercises for Chapter 2 Academic Year 2016/17 Problems 1. The following data shows the (hourly) number of sales in an ice-cream shop, recorded during different opening hours: a) Find the mean, median, and mode of these observations. b) Find (quasi) variance, (quasi) standard deviation, range, inter-quartile range, coefficient of variation. c) Find the five-number-summary and draw a boxplot for this data set. d) Are there any outliers? Is the distribution asymmetric? (Hint: compare the mean and median.) 2. (June 2012 Exam) We are interested in the temperature of a thermostat of a certain model of a car at 100km/h. A simple random sample of 30 cars is taken and the temperature of the thermostat at 100km/h is measured. The results are as follows (in degrees Celsius): 65,8 69,4 69,4 69,7 71,5 72,2 74,1 75,4 75,8 76,3 77,2 77,6 77,6 77,9 78,3 78,7 78,9 81,2 81,2 81,7 82,3 82,3 82,4 84,5 84,7 85,2 85,4 88,2 102,5 105,5 a) Group the data in intervals of the same length starting with [65, 69), and obtain the absolute and relative frequency distributions of the data. b) What percentage of the observations are between 78 and 81 degrees Celsius? c) Calculate the three sample quartiles. Interpret them. d) Draw a boxplot of the data and identify the outliers (if any). Justify your answer. 3. Consider the following (ranked) 20 observations from a variable U a) Find the absolute frequency distributions of this data set (use class intervals of length 2 starting from 2) and use it to make a histogram. b) Based on the graph, describe the shape of the distribution and comment on the absence/presence of outliers. c) Based on the table, calculate (approximately) the mean and the variance of the data set, using the midpoints of class-intervals. d) Based on the data, calculate the exact mean and the variance. Was the approximation from the previous part fairly good? 4. The following table shows the absolute frequency distribution of the variable height (in meters) of n = 500 students. 1

2 Class Absolute interval frequency [1,59, 1,61) 11 [1,61, 1,63) 71 [1,63, 1,65) 159 [1,65, 1,67) 177 [1,67, 1,69) 67 [1,69, 1,71) 15 a) Draw the histogram of the data using the above table and describe the shape of the distribution. b) How many students are taller than or equal to 1.67m? And what percentage of students are shorter than 1.63m? c) Calculate (approximately) the mean and the variance of this data set. 5. The following table shows the number of university graduates of the academic year according to the CCAA of the University in which they obtained their degree (Source I.N.E., Encuesta de inserción laboral de titulados universitarios 2014). Comunidad Numb. Of univ. graduates Andalucía Aragón 4989 Asturias, Principado de 3947 Balears, Illes 1905 Canarias 4615 Cantabria 1751 Castilla y León Castilla - La Mancha 4924 Cataluña Comunitat Valenciana Extremadura 3767 Galicia Madrid, Comunidad de Murcia, Región de 6771 Navarra, Comunidad Foral de 3162 País Vasco 9744 Rioja, La 1005 In case the variable of interest is the CCAA in which the University degree is obtained: (a) What type of variable is it and what population is it? (b) What is the distribution of frequencies in the table above? (c) What statistical measures could you take for that variable? Make a Pareto chart to decide whether each of the following statements is true or false, all referring to the university graduates of the academic year: (a) Less than 25 % of Universities concentrate more than 60 % of university graduates (b) The median of the distribution is Catalonia (c) 20 % of the Autonomous Communities concentrate the Universities with more than 50 % of university graduates (d) From the Universities of the 35 % of the Autonomous Communities come less than 10 % of university graduates 6. Construct the frequency table from the following barchart of cumulative frequencies. 2

3 (a) What type of variable is it? (b) Use the table to calculate the mean and the standard deviation of the data set. (c) Compute the mode, median and the 20th and 80th percentile. 7. (May 2016 Exam) The following tables contain information about 10 companies of the IBEX 35. In particular, three variables are shown: X 1 = average remuneration of the governing board, X 2 = average remuneration of senior management and X 3 = average expenditure per employee (in millions of euros). Source: El País, 8th May Tabla 1 / Table 1 Empresa / Company X1 X2 X3 Figura 1 / Figure 1 BBVA 0,985 1,144 0,455 ACS 0,667 0,540 0,401 FCC 0,720 0,650 0,323 Inditex 1,270 1,730 0,231 A Acciona 0,463 0,590 0,390 Santander 1,484 2,580 0,586 IAG 1,220 2,440 0,809 Iberdrola 0,920 1,979 0,894 Ferrovial 1,330 1,800 0,391 Telefónica 1,240 1,869 0,491 B Tabla 2 / Table 2 X1 X2 X3 Media / Mean 1,030 0,497 Mediana / Median 1,765 0,428 Desv. típica / Standard dev. 0,333 0,756 0,210 Varianza / Variance 0,572 0,044 Q1 0,770 0,774 0,390 Q3 1,263 1,952 Answer to the following questions: (a) Fill in the gaps in Table 2. (b) Determine the shape of the distribution of X 2. Justify your answer. (c) Which of the three variables is more disperse? Justify your answer. (d) Are there any outliers in X 3? Justify your answer. (e) Match the box-plots A and B of Figure 1 with the corresponding variables (X 1, X 2, X 3 ). Justify your answer. (f) It is known that the correlation between X 1 and X 3 is and, on the other hand, that the covariance between X 2 and X 3 is Is it true that the linear relationship between X 3 and X 1 is stronger than between X 3 and X 2? Justify your answer. (Note: this question is from Chapter 3) 3

4 8. The following two graphs were published by the newspaper El Mundo in its paper edition 1 about the data that had recently been published about the diffusion of the Spanish press (OJD, Oficina de Justificación de la Difusión). (a) Do they seem right to you? Why? (b) Represent the data collected in these graphs and compare the graphs obtained with those that appeared published. 9. The following Excel output contains descriptive measures on the price (in euros) of one kilo of tomatoes from different suppliers during the summer campaign: precio verano Percentiles 25% 1,5725 Media 1, % 1,885 Error típico 0, % 2,25 Mediana 1,885 Moda 1,75 Desviación estándar 0, Varianza de la muestra 0, Curtosis 0, Coeficiente de asimetría 0, Rango 1,8 Mínimo 1,04 Máximo 2,84 Suma 64,55 Cuenta 34 (a) Define the population of interest and the variable. What type of variable is it? (b) If you had to provide a maximum price above which you expect prices to be set at most 25 % of suppliers, what value would you propose? (c) Due to several factors, the sale price is reduced by 1 euro per kilo. Calculate the new average price, median, variance, rank and the first quartile for the new prices. If you had to give the data in dollars, How would you calculate these measures? (use the change 1 euro=1,074 US dollars) 1 de 24 de septiembre de Fuente: blog Malaprensa 4

5 10. This problem was incorrect in the previous version. See the slides of Topic 3 for a correct statement of the problem. 11. Next you can see the results of the Encuesta de Condiciones de Vida (which annually performs the I.N.E.) corresponding to the years 2014 and 2006 on the percentage of households with serious difficulties to makes end meet by autonomous communities. Comunidad Andalucía 24,3 16,8 Aragón 9,8 5,7 Asturias, Principado de 4,6 3,1 Balears, Illes 14,7 9,7 Canarias 19,5 18,4 Cantabria 15,2 9,3 Castilla y León 12,1 8,5 Castilla - La Mancha 15,9 10,7 Cataluña 12,2 11,3 Comunitat Valenciana 18,0 12,3 Extremadura 19,6 8,7 Galicia 20,8 11,9 Madrid, Comunidad de 12,4 8,8 Murcia, Región de 22,7 14,1 Navarra, Comunidad Foral de 4,2 6,6 País Vasco 11,5 5,2 Rioja, La 12,9 6,6 Ceuta 32,9 25,7 Melilla 12,9 15,9 The following tables collect information of the variable percentage of households that have serious difficulties to make ends meet in each of the observed periods: Media 15,5895 Media 11,0158 Error típico 1,5677 Error típico 1,2385 Mediana 14,7 Mediana 9,7 Moda 12,9 Moda 6,6 Desviación estándar 6,8333 Desviación estándar 5,3986 Varianza de la muestra 46,6943 Varianza de la muestra 29,1447 Curtosis 1,1493 Curtosis 1,7247 Coeficiente de asimetría 0,6357 Coeficiente de asimetría 1,1245 Rango 28,7 Rango 22,6 Mínimo 4,2 Mínimo 3,1 Máximo 32,9 Máximo 25,7 Suma 296,2 Suma 209,3 Cuenta 19 Cuenta 19 (a) Represent the box plot for the data of What can you say about the distribution of the percentage of households that have difficulties to makes end meet? Which measure of location is more appropriate to use in this case and why? (b) Are there Autonomous Communities with outlier percentages in the data for 2014? (c) In what year do the data show greater variability? (d) In which of the two periods, 2014 or 2006, the Comunidad de Madrid and the Comunitat Valenciana obtain worse results in relation to the general situation of those years? 5

6 12. (June 2016 Exam) The following tables contain information about 10 companies of the Dow Jones. In particular: X 1 = highest-paid CEO (in million dollars) and X 2 = share price (in dollars). Source: El País, 8th May Tabla 1 / Table 1 Figura 1 / Figure 1 X1 X2 44,91 106,11 27,29 87,81 24,2 53,95 23,79 113,69 23,37 29,97 22,58 158,31 22,03 100,31 21,98 64,21 A 20,01 110,68 19,82 147,6 B C a) Draw the box-plot for X 2 and identify the outliers (if any). Justify your answer. b) Determine if X 1 and X 2 have the same type of asymmetry. Justify your answer. c) Determine which box-plot (A, B, C) corresponds to X 1. Justify your answer. d) If 1 euro = 1.14 dollars, calculate the average salary for those CEOs (in million euros) and the variance. Questions 1. We want to study the time that is used in correcting an error in a computer program. For this purpose, the times used to correct 100 errors are collected. If the histogram shown below is obtained, which of the following statements is true? (a) Mode is greater than the median. (b) The shape of the distribution is flared. (c) The distribution is skew and, therefore, the coefficient of variation must be used to measure the dispersion. (d) More than 50 % of the errors take less time to correct than the average correction time. 2. A college decides to conduct a survey among its recent graduates to obtain information on their annual salaries. 200 recent graduates were randomly selected and questionnaires were sent to their current jobs. However, of those 200, only 86 completed and returned the questionnaire. Suppose 6

7 that the average wage reported was 75, 000. Would it be correct for the university to think that 75, 000 is a good approximation to the average salary level of all its graduates? Why? Fuente: Introducción a la Estadística. Sheldon M. Ross 3. In the blog Malaprensa (December, 29th, 2013) appears the following chart published in El Pais with the score of the Ministers in the surveys. The graph reflect losses of, for example, 214 % in the case of Minister Wert. If these scores are measured on a scale of 0 to 10, does it make any sense? what error is it? 4. Which of the following two job offers would you choose? Offer 1: The average salary in the company is 2000 euros per month. Offer 2: The average salary in the company is 800 euros per month. With what offer would you stay? Do you have enough information to make a good decision? Source: Trivial de la Estadística, actividad organizada para celebrar el Día Mundial de la Estadística ( ) 5. A politician promises that if he/she is elected in the next election everyone will receive a salary higher than average, would you vote for her/him? Source: Trivial de la Estadística, actividad organizada para celebrar el Día Mundial de la Estadística ( ) 6. According to the following histogram on the amount of the monthly telephone bill in the different departments of a company, what could you say about the distribution of the data? 7

8 (a) Its gaussian. (b) We don t have information enough. (c) There is evidence that there could be two kinds of departments. (d) Its symmetrical 7. The following list is ordered from smallest to largest: 2.5, 2.6, 2.6, 3.0, 3.3, Which of the following statements is (are) true? I. The mean is greater than the median II. The mode is 2.6 III. There are no outliers in the data a) I only b) I and II only c) I and III only d) II and III only 8. Which of the following statements is (are) true? I. The median is resistant to extreme values. II. The mean is resistant to extreme values. III. The standard deviation is resistant to extreme values. a) I only b) II only c) III only d) II and III only 8