Exercises

1. Consider the following data set:\[5,~12,~16,~18,~21,~22,~22,~25,~26,~27,~30,~30,~36,~38,~39\]Sort the data as indicated.

   1. Stem	Leaf
      \(0\)
      \(1\)
      \(2\)
      \(3\)

   2. Range	Frequency
      Less than \(20\)
      \(20\) or greater

   3. Range	Frequency
      \(0\) – \(8\)
      \(8\) – \(16\)
      \(16\) – \(24\)
      \(24\) – \(32\)
      \(32\) – \(40\)

      Note: If a number is on the boundary of two intervals, then count the number in the higher interval. For example, the number \(16\) will be counted in the interval \(16\) – \(24\), not \(8\) – \(16\).

2. In each part below, a frequency table is given that has missing entries. Fill in the table so that all intervals are in increasing order and are the same size, then create a data set that would produce this frequency table. Be as creative as you can with the numbers that you choose for your data sets.

   1. Range	Frequency
      \(0\) – \(7\)	\(3\)
      	\(1\)
      	\(0\)
      	\(5\)

   2. Range	Frequency
      	\(0\)
      	\(1\)
      \(12\) – \(15\)	\(2\)
      	\(3\)
      	\(4\)

   3. Range	Frequency
      	\(2\)
      	\(2\)
      \(213.5\) – \(215\)	\(2\)

3. Given the smallest and largest values in a data set, determine a way to organize the data into the indicated number of intervals of equal size.

   1. Smallest value: \(3\)
      Largest value: \(29\)
      Number of intervals: \(6\) 
   2. Smallest value: \(18.5\)
      Largest value: \(85.0\)
      Number of intervals: \(4\)
   3. Smallest value: \(-3.4\)
      Largest value: \(34.5\)
      Number of intervals: \(5\) 
4. The weight of an egg is continuous data. Cartons of \(12\) eggs are weighed and classified as follows:

   Classification	Minimum Weight (g)
   Small	\(504\) (\(42\) per egg)
   Medium	\(588\) (\(49\) per egg)
   Large	\(672\) (\(56\) per egg)
   Extra Large	\(756\) (\(63\) per egg)
   Jumbo	\(840\) (\(70\) per egg)

   1. Reclassify the egg cartons into only three categories.
   2. Is your reclassification likely to be good for the consumers or good for the stores? Explain.
5. The following data shows the mass (in grams) of \(25\) Florida oranges randomly selected from the field.\[116.1,~114.5,~93.2,~112.7,~104.0,~91.9,~105.3,~106.3,~103.5,~91.2,~95.0,~120.0,~105.4,\\ 103.5,~105.1,~106.8,~113.5,~109.0,~92.9,~109.2,~104.1,~114.6,~116.6,~109.0,~112.5\]
   1. Display the data using a stem-and-leaf plot. 
   2. What is the range of the data (the smallest value subtracted from the largest value)?
   3. Display the data in a frequency table with five equally-sized intervals.
6. Senior Care recently completed a survey on the residents of a residence and recorded the residents' ages in a tablular way.

   \(97\)	\(92\)	\(78\)	\(80\)	\(76\)	\(94\)	\(78\)	\(94\)	\(80\)	\(69\)
   \(91\)	\(99\)	\(79\)	\(66\)	\(70\)	\(102\)	\(96\)	\(99\)	\(70\)	\(67\)
   \(100\)	\(96\)	\(90\)	\(92\)	\(100\)	\(102\)	\(91\)	\(66\)	\(85\)	\(82\)
   \(80\)	\(82\)	\(93\)	\(69\)	\(96\)	\(102\)	\(95\)	\(72\)	\(83\)	\(89\)

   1. When you are collecting data on ages, are you collecting continuous or discrete data? Explain.
   2. Display the data in a frequency table with two equally-sized intervals.
   3. Display the data in a frequency table with five equally-sized intervals.
   4. Compare how the data is displayed in the two frequency tables. What does each tell you about the distribution of the ages in the residence?
7. When collecting and organizing data, it is important to think about what types of numbers you are collecting and in which range these numbers will likely fall. Let's consider collecting data of the heights of every student in a school. 
   1. What type of data are you collecting? What range of data are you expecting to get?
   2. When organizing this data into intervals, does it make sense to start your first interval at \(0\)? Where should you end your last interval? Explain.
   3. How will your choices from part b) affect the visual display of the data in a frequency table?
8. There is always some choice involved when determining how to organize numerical data. How you choose to organize data can influence the conclusions that readers might draw from the data. To help with these choices, mathematicians have developed many different methods for systematically choosing appropriate intervals for organizing given data. In this question, we will explore one simple method to organize the following data set:

   \(51.0\)	\(29.0\)	\(26.5\)	\(37.0\)	\(40.0\)	\(31.5\)	\(38.0\)
   \(22.0\)	\(38.5\)	\(38.0\)	\(18.0\)	\(11.0\)	\(35.0\)	\(33.5\)
   \(45.0\)	\(30.5\)	\(24.0\)	\(26.0\)	\(20.0\)	\(35.0\)	\(20.0\)
   \(5.0\)	\(9.0\)	\(15.5\)	\(23.0\)	\(21.0\)	\(22.0\)	\(30.0\)
   \(35.0\)	\(25.0\)	\(24.0\)	\(30.0\)	\(19.5\)	\(32.0\)	\(14.5\)

   1. Let \(N\) be the number of pieces of data in the data set. Calculate the square root of \(N\), to the nearest integer. This is the number of intervals that you will use to organize your data.
   2. Calculate the range of the data and divide the range by the number of intervals you determined from part a). This number will be the approximate length of the intervals you will use to organize your data. For convenience, you may want to choose an integer length for your intervals.
   3. Choose a convenient place to start your first interval. You must choose a starting place that is smaller than the minimum number in the set, and so that your intervals cover the entire range of the data set. 
   4. Organize the data into a frequency table using the decisions that you have made in this question. Did this method produce a useful organization of the data? Why or why not?
   5. Try this method with different sets of data (continuous and discrete data, large and small data values, many or few values, small range and large range of values, etc.). Does this method often produce a useful way of organizing data? Are there particular data sets that are not well organized by this method?