Tutorial Handout Statistics, CM-0128M Descriptive Statistics

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1 Tutorial Handout Statistics, CM-0128M January 18, 2013 Exercise 1. The following figures show the annual salaries in of 20 workers in a small firm. Calculate the arithmetic mean, median and mode salaries. 15,180 19,870 14,375 14,767 15,870 15,180 14,375 36,938 15,180 46,132 15,525 19,600 14,375 23,069 16,767 16,767 17,880 14,375 14,375 14,375 Exercise 2. Ten households, each with a new baby were monitored over a 12 months period to estimate their annual expenditure directly related to the arrival of the baby. The following annual expenditures were estimated. (The monetary values are stated in.) 2,750 3,500 2,580 4,600 2,840 2,930 3,000 2,450 6,900 3,120 a) Find the mean annual expenditure on the new baby. b) Find the median annual expenditure on the new baby. Exercise 3. The following recent data shows the age ranges within which males and females were recorded as having stopped smoking in England. (It is assumed that they were smokers before.) Age Males Females , ,070 35, ,100 32, ,460 38, ,800 21,770 Total 98, ,110 From: Dr. A. Csenki 1 of 5 Typeset with L A TEX

2 a) Find the mean and median ages for stopping smoking for males. b) Find the mean and median ages for stopping smoking for females. Exercise 4. Find the mean, median, range, variance and standard deviation of the following 8 items of raw data Exercise 5. The examination results (as percentages) of two students, Julie and Mahmoud, are shown in the table below. a) Find the respective means. b) Find the respective standard deviations. c) Find the respective coefficients of variation. Julie Mahmoud Maths Physics Chemistry French Spanish Exercise 6. The number of visitors, in thousands, to an amusement park over a summer season were recorded as follows: Number of visitors Frequency (in 1,000) (number of days) (The right-hand boundaries of the above classes are not included.) a) How long was the season? b) Calculate the mean attendance during the summer season. c) Calculate the variance and standard deviation for the attendance data. From: Dr. A. Csenki 2 of 5 Typeset with L A TEX

3 Exercise 7. A new production line had a mean daily rejection rate of 196 units with a standard deviation of units in the first three months of operation. In the next three months the mean daily rejection rate was 94 units with a standard deviation of units. Find the coefficient of variation in each case. Exercise 8. An investment analyst receives the following table of data showing the percentage changes in labor costs of senior managers in a multinational company over a 12 month period. The highest change observed was per cent for a senior manager in one of the plants in Latin America. The mean annual percentage change in labour costs for senior managers in this company is per cent. Find the variance and standard deviation for the percentage change in labour costs of senior managers throughout the multinational company. Percentage change Frequency -5 to under to under to under to under to under to under to under to under to under to under to Exercise 9. A company requires that chilled food cabinets in its supermarkets must maintain an average hourly temperature of 3.75 C ± 0.5 C. The manager at one of the supermarkets suspects that the performance of one of the shop s cabinets fails to meet this standard and therefore decides to monitor its performance hourly over a 30 day period with the following results: Temperature Frequency 0 and under and under and under and under and under and under and under 7 1 From: Dr. A. Csenki 3 of 5 Typeset with L A TEX

4 Find the arithmetic mean hourly temperature and the standard deviation to assess whether the equipment conforms to the company s policy. Exercise 10. The following are the grade point averages of 30 students recently admitted to the graduate program at the university a) Calculate the sample mean. b) Calculate the sample standard deviation. c) Determine the proportion of the data values that lie within x ± 1.5s and compare it with the lower bound given by Chebyshev s inequality. d) Determine the proportion of the data values that lie within x ± 2s and compare it with the lower bound given by Chebyshev s inequality. Exercise 11. The following random sample of annual salaries was recorded. (Units are in thousands of pounds.) a) Calculate the mean and standard deviation. b) Using Chebyshev s inequality, between what two bounds will at least 75% of the data lie? c) Using Chebyshev s inequality, between what two bounds will at least 89% of the data lie? Exercise 12. Show that s 2 = 1 n 1 n (x i x) 2 is equivalent to s 2 = ( n n ) 2 x 2 i 1 n x i n 1 From: Dr. A. Csenki 4 of 5 Typeset with L A TEX

5 Exercise 13. The number of defects in 10 rolls of carpets are a) What are the 75th percentile and the 50th percentile?. b) What is the coefficient of skewness? Exercise 14. Assume that the sum of 10 observations is 10, that the sum of the squares of 10 observations is 20, and that the median is 1.5. What is the coefficient of skewness? Exercise 15. The percentage of unemployed workers in each of 20 randomly selected cities are as follows a) Calculate the 20th percentile. b) Calculate the 60th percentile. c) Calculate the 80th percentile. Exercise 16. From a random sample, the mean is 50 and the sample standard deviation is 5. a) What is the z-score if an observation s value is 40? b) What is the z-score if an observation s value is 65? c) What is the value of an observation with a z-score of 1? d) What is the value of an observation with a z-score of 2.5? Exercise 17. The following is a sample of size 10: a) Find the first, second, and third quartiles. b) What are the largest and smallest values? c) Construct a box plot of the data. From: Dr. A. Csenki 5 of 5 Typeset with L A TEX