Exercices

1. The following set of data gives the length of reign in years of the various Kings and Queens of England since the Battle of Hastings in 1066. If the monarch's reign lasted less than six months, then the number was rounded down to zero. Otherwise, the number was rounded to the nearest year.

   Monarch

   Monarch	Years	Monarch	Years
   William the Conquerer	24	Edward VI	7
   William II	13	Jane	0
   Henry I	34	Mary I	5
   Stephen	18	Elizabeth I	44
   Matilda	1	James I	22
   Henry II	34	Charles I	24
   Richard I	10	Charles II	25
   John	18	James II	3
   Henry III	56	William III	13
   Edward I	35	Mary II	5
   Edward II	20	Anne	12
   Edward III	50	George I	13
   Richard II	22	George II	33
   Henry IV	14	George III	59
   Henry V	9	George IV	10
   Henry VI	39	William IV	7
   Edward IV	22	Victoria	64
   Edward V	0	Edward VII	9
   Richard III	2	George V	26
   Henry VII	23	Edward VII	1
   Henry VIII	38	George VI	15

   1. Present the data in the form of an ordered stem and leaf plot. Answer 1a
   2. Do any outliers exist? If so, give a reason for their presence. Answer 1b
   3. Describe the following features of distribution:
      • the number of peaks
      • the general shape
      • the approximate value of the centre of the distribution
      Answer 1c
   4. Calculate cumulative frequency and cumulative percentage. Answer 1d
   5. Draw the ogive with two different vertical axes, one for cumulative frequency and one for cumulative percentage. Answer 1e
   6. Using the information found earlier in this exercise, answer the following questions:
      • How many monarchs have reigned for less than 10 years?
      • How many monarchs have reigned for 50 years or more?
      Answer 1f
   7. The current English monarch is Queen Elizabeth II. She has ruled since February 6, 1952, and has been excluded from the original data set. Calculate the length of her reign, and briefly comment on it in comparison with the other rulers. Answer 1g
2. Eliza often buys a small bag of french fries at Hungry Stats, the local fast-food stand. She was curious to know whether or not she was getting value for her money and was also interested in finding out whether or not she received the same amount of french fries in each bag. So, for three months, Eliza decided to count and record the fries in each bag. At the end of the third month, she realized that she had bought fries a total of 30 times. The resulting data from her 30 visits are as follows:

   44, 46, 54, 38, 49, 46, 45, 31, 55, 37, 42, 43, 47, 51, 48 40, 59, 35, 47, 21,43, 37, 45, 38, 40, 32, 50, 34, 43, 54

   1. Present the data in an ordered stem and leaf plot. If necessary, split the stems. Answer 2a
   2. Do any outliers exist? If so, give a reason for their presence? Answer 2b
   3. Describe the following features of distribution:
      • the number of peaks
      • the general shape
      • the approximate value of the centre of the distribution
      Answer 2c
   4. Calculate cumulative frequency and cumulative percentage. Answer 2d
   5. Draw the ogive with two different vertical axes, one for cumulative frequency and one for cumulative percentage. Answer 2e
   6. Using the information found earlier in this exercise, answer the following questions:
      • How many bags had fewer than 40 fries?
      • What percentage of bags had 45 or more fries?
      Answer 2f
   7. Hungry Stats decided to create a slogan for their new advertising campaign. Complete the following statement using the information gained from Eliza's calculations.

      "Our bags may be small, but half contain at least___ french fries!"

      Answer 2g
3. Imagine that the following table represents (by age group) the number of unemplyed female job-seekers for full-time work.
   

   Table 1. Number of unemployed female job-seekers, by age group

   Age group *	Number of females
   15 to 24	339
   25 to 34	273
   35 to 44	147
   45 to 54	121
   55 to 64	22
   * Age is collected in completed number of years. Thus, the interval 15 to 24 has an upper endpoint of 25 (refer to Stem and leaf plots in the chapter on Organizing data).

   1. Are the age variables discrete or continuous? Answer 3a
   2. Copy Table 1 into your notebook and calculate the cumulative frequency and cumulative percentage. Answer 3b
   3. Draw the ogive with two different vertical axes, one for cumulative frequency and one for cumulative percentage. Answer 3c
   4. Why are there no data for females younger than 15 years old? Answer 3d
   5. In what age group does the cumulative percentage value 50% lie? Answer 3e
   6. What percentage of unemployed female job-seekers are younger than 25 years old? Answer 3f
   7. What percentage of unemployed female job-seekers are 55 years old or older? Answer 3g
   8. How would the Canadian government use this sort of information? Answer 3h
4. Imagine a survey was conducted in order to determine how long it takes for Statistics Canada employees to travel to work. The results, to the nearest minute, were recorded as follows:

   33, 63, 49, 65, 56, 45, 52, 63, 38, 66, 43, 98, 60, 58, 68, 29, 59, 87, 22, 64, 73, 56, 71, 67, 44, 31, 83, 50, 75, 65, 60, 51, 89, 69, 41, 76,58, 62, 25, 52, 64, 77, 61, 55, 80, 45, 12, 69, 40, 37

   1. Are these variables discrete or continuous? Answer 4a
   2. Present the data in a frequency table, using appropriate intervals; including relative and percentage frequencies. Answer 4b
   3. Draw a histogram to represent the data and plot the frequency polygon. Answer 4c
   4. Prepare an ordered stem and leaf plot for the data. Do any outliers exist? If so, give a reason for their presence. Answer 4d
   5. Describe the following features distribution:
      • the number of peaks
      • the general shape
      • the approximate value of the centre of the distribution
      Answer 4e
   6. Calculate the cumulative frequency and cumulative percentage. Find the endpoints and record the results in a table. Answer 4f
   7. Draw the ogive with two different vertical axes, one for cumulative frequency and one for cumulative percentage. Answer 4g
   8. Using the information found earlier in the exercise, answer the following questions:
      • With regards to how long it takes to travel to work, what was the most common time interval for Statistics Canada employees?
      • What percentage of employees took longer than 90 minutes to travel to work?
      • How many employees took less than 40 minutes to travel to work?
      Answer 4h

Class activity

Survey the teachers in your school to find out how long they have been teaching (to the nearest year).

1. Are the variables discrete or continuous?
2. Present the data in a frequency table, using appropriate intervals, including relative and percentage frequencies.
3. Using the information found earlier in this exercise, answer the following questions:
   1. How many years have the majority of teachers been teaching?
   2. What is the percentage difference between the first and the second most common length of service?
4. Draw a histogram to represent the data and plot the frequency polygon. Prepare an ordered stem and leaf plot for the data.
5. Do any outliers exist? If so, give a reason for their presence.
6. Describe the following features of distribution:
   • the number of peaks
   • the general shape
   • the approximate value of the centre of the distribution
7. Calculate cumulative frequency and cumulative percentage.
8. Draw the ogive with two different vertical axes, one for cumulative frequency and one for cumulative percentage.
9. With the new information, answer the following questions:
   1. How many teachers have taught for more than 10 years?
   2. What percentage of teachers have taught for more than 10 years?
   3. What percentage of teachers have taught for less than 10 years?
   4. What is the number of years below which half the teachers have taught?
10. Present your analysis in a report containing the necessary cumulative frequency and percentage tables and graphs.