Answers and Solutions


  1. Situation Categorical or Numerical Examples of Data
    Ice cream flavour Categorical Vanilla, chocolate, etc.
    Mosquito count Numerical \(340\), \(1000\), etc.
    Types of homes for sale Categorical Town house, apartment, etc.
    Carbon dioxide level Numerical \(280\) ppm, \(335\) ppm, etc.
  2. Situation Discrete or Continuous Examples of Data
    Shoe size Discrete \(6\), \(8.5\), etc.
    Snow fall Continuous \(15\) cm, \(21\) cm, etc.
    Book sales Discrete \(3500\), \(10~000\), etc.
    Tax rate Continuous \(22\%\), \(7\%\), etc.
    Cholesterol level Continuous \(1.4\), \(3.0\), etc.
     
    1. Racing times from fastest time to slowest time:\[13.4,~13.5,~13.7,~13.7,~13.8,~14.1,~14.3,~14.5,~14.9,~15.6,~15.6,~15.9,~16.0,~16.5,~17.0\]
    2. Since almost all of the data points are distinct, recording the frequency of each individual time will not give us more information than we got from the order in part a).
    3. A circle graph does not make much sense since this is not categorical data. Furthermore, since the times are all unrelated (likely from different racers or different races), it does not make sense to connect them in a line graph.
    1. Answers may vary. One example is shoe size data as long as there are "half sizes." Another example is many rating systems that use ratings like \(8.5\) out of \(10\).
    2. Answers may vary. One example might be height data where the data is only recorded to the nearest centimetre.
    1. The inflation rate grew the most during year \(7\). You can determine this from the bar graph by finding the largest jump in bar height. You can also determine this from the line graph, either by the largest jump in point height or by the steepest line.
    2. Inflation rate is continuous data. Since inflation rates change over time, the line graph is likely the most appropriate of the two graph choices.
  6. Answers will vary. While the weight of a letter is continuous data, you can argue that postage costs are discrete according to the rates given. However, it is now possible to mail a letter without purchasing any stamps. Many post offices will charge you based on a continuous postage rate.
    1. Answers will vary. The following table uses only positive numbers but you may have used some negative numbers as well (e.g., \((-1)+ 8 = 7\)).
      \(x\) \(y\)
      \(1\) \(6\)
      \(1.5\) \(5.5\)
      \(0\) \(7\)
      \(3\) \(4\)
      \(5.2\) \(1.8\)
      \(6\) \(1\)
      \(7\) \(0\)
      \(4.6\) \(2.4\)
      \(2.1\) \(4.9\)
      \(1.7\) \(5.3\)
      Since \(x\) and \(y\)  can be any value on the number line, this is continuous data.
    2. The following graph displays the data points from the table in part a). Your graph may look slightly different depending on the values you chose.
      Since for every value of \(x\) there is a corresponding value of \(y=7-x\), and hence a corresponding point for the graph, it makes sense to draw lines between the points in our graph.
    3. The line graph is not broken as is often the case when plotting data. Since the equation \(x+y=7\) is truly describing a linear relationship between \(x\) and \(y\), it makes sense that this line graph looks like a straight line (and not a path of broken lines). 
    1. \(x\) \(y\)
      (rounded value)
      \(1.4\) \(1\)
      \(1.8\) \(2\)
      \(1.9\) \(2\)
      \(2\) \(2\)
      \(2.2\) \(2\)
      \(2.5\) \(3\)
      \(2.8\) \(3\)
      \(3.1\) \(3\)
      \(3.5\) \(4\)

      The variable \(x\) is continuous and the variable \(y\) (the rounded number) is discrete. We can take \(x\) to be any value on the number line, but the rounded value of \(x\) can only be an integer.
    2. The following graph displays the data points from the table in part a).
      If we attempt to draw a line graph through these points then we get the following graph.
      Notice that the line from point \((1.4,1)\) to point \((1.8,2)\) suggests that there is a gradual change from \(y\)-values of \(1\) to \(y\)-values of \(2\). This might be misleading as this is not how rounding works. There should be an abrupt change at \(x=1.5\) which is not reflected in this line graph.
    3. The variable \(x\) is continuous and can take on any value on the number line.  The following graph plots all of the tenths (i.e., \(0.1,~0.2,~0.3,...\)) from \(x = 0\) to \(x = 5\).
      The rest of the graph to the right of \(x=5\) would follow a similar pattern.
      To represent the points for all \(x\) values (including hundredths, thousandths, and so on) on the graph, we use continuous lines as in the following graph.
      Points are plotted at (0,0), (0.5,1), (1.5,2), (2.5,3), (3.5,4), and (4.5,5). A horizontal line is drawn from each point for 1 unit to denote all the x-values that would be rounded to the y-value.
      The right endpoint of each line segment is drawn to remind us that a number like \(1.5\) is rounded up and not down. The rest of the graph to the right of \(x=5\) would follow a similar pattern.
    4. One way to round negative numbers is as follows: If the digit in the tenths place is \(5\) or lower, round to the nearest integer to the right on the number line. Otherwise, round to the nearest integer to the left on the number line. This would mean, for example, that \(-1.2\) and \(-1.5\) are rounded to \(-1\) (on the right), while \(-1.8\) is rounded to \(-2\) (on the left).
      Using this rounding rule, we extend the graph from part c) to include negative numbers. The following is the graph from \(x = -5\) to \(x = 5\).
      Points are plotted at (-4.5,-4), (-3.5,-3), (-2.5,-2), (-1.5,-1), (-0.5,0), (0.5,1), (1.5,2), (2.5,3), (3.5,4), and (4.5,5). A horizontal line is drawn from each point for 1 unit to denote all the x-values that would be rounded to the y-value.
      The rest of the graph would follow a similar pattern.