The Mayer Line

1. Order the coordinates according to the independent variable.

2. Separate the distribution into 2 equal groups, if possible.

3. Calculate the mean points of each group |(P_1| and |P_2).|

4. Find the rule of the regression line that passes through the points |(P_1| and |P_2).|

5. Predict values using the rule of the line.

Following a survey of |16| Quebec families, the total spending on sports and recreation was examined in relation to their household income.

The following table of values shows the data collected. The data was then placed on a Cartesian plane to create a scatter plot.

a) A family has an annual household income of |$250\ 000|. If this family follows the same trend as the other Quebec families surveyed, how much do they budget for sports and recreation?

b) A family spends an average of | \$7500| a year on sports and recreation. What is their annual household income if they are a typical Quebec family?

1. Order the coordinates according to the independent variable.

2. Separate the distribution into 2 equal groups, if possible.

The distribution contains |16| pairs of data. The |8| pairs whose household income ranges from |20\ 000\ \$/\text{year}| to |80\ 000\ \$/\text{year}| make up the 1st group. The other |8| pairs form the 2nd group.

3. Calculate the mean points of each group |(P_1| and |P_2).|

Find the mean of the |x| and |y| values of each group to make 2 points.

4. Find the rule of the regression line that passes through the points |\boldsymbol{P_1}| and |\boldsymbol{P_2}.|

Since this is a straight line, the rule has the form |y=ax+b.| We start by calculating the slope |(a).|
||\begin{align}a&=\dfrac{\overline{y}_2-\overline{y}_1}{\overline{x}_2-\overline{x}_1}\\&=\dfrac{8\ 625-3\ 812.5}{123\ 125-53\ 750}\\&\approx 0.07\end{align}|| Next, we replace |a| by |0.07| and the |x| and |y| variables by the coordinates of one of the 2 points. Then, we isolate |b.| ||\begin{align} y &= ax+b \\ y &= 0.07x+b \\ 8\ 625 &= 0.07(123\ 125)+b \\ 8625 &\approx 8619+b \\ 6 &\approx b \end{align}||So, the rule of the Mayer line is |y=0.07x+6,| where |x| is the household income and |y| is the spending on sports and recreation, both in |\$| per year. We can graph this line.

5. Predict values using the rule of the line.

a) A family has an annual household income of | \$250\ 000|. If this family follows the same trend as the other Quebec families surveyed, how much do they budget for sports and recreation?

We can estimate this family's spending on sports and recreation using the regression line. Since the household income in question | \$250\ 000| is outside the range studied (| \$20\ 000| to | \$150\ 000|), this is an extrapolation.

We replace the |x| variable with |250\ 000| in the regression line rule and complete the calculation. ||\begin{align}y&=0.07x+6\\y&=0.07(250\ 000)+6\\y&=17\ 500+6\\y&=\$17\ 506\ \end{align}||Answer: A household with an annual income of | \$250\ 000| would spend approximately | \$17\ 506| on sports and recreation if it followed the same trend as the other Quebec families surveyed.

b) A family spends an average of | \$7500| a year on sports and recreation. What is their annual household income if they are a typical Quebec family?

We can estimate the annual household income of this family using the regression line. This is an interpolation because the annual budget for recreation and sports |( \$7500)| is within the interval studied |(500| to | \$13\ 000).|

We replace |y| with |7500| and isolate |x.| ||\begin{align} y &= 0.07x+6 \\ 7500 &= 0.07x+6 \\ 7500\boldsymbol{\color{#ec0000}{-6}} &= 0.07x+6 \boldsymbol{\color{#ec0000}{-6}} \\ \dfrac{7494}{\boldsymbol{\color{#ec0000}{0.07}}} &= \dfrac{0.07x}{\boldsymbol{\color{#ec0000}{0.07}}} \\\$ 107\ 057\ &\approx x \end{align}||Answer: If a household spends on average | \$7500| per year on sports and recreation, we can predict that the household income is about | \$107\ 057|.

Note: The same problem was solved in the regression line and median-median line concept sheets. In each case, comparable results were obtained.

When the points need to be ordered

• Points are ordered according to their x-coordinates. The x- and y-coordinates should not each be ordered separately, they must be kept as an ordered pair.

• If 2 points have the same x-coordinate, but different y-coordinates, the one with the smaller y-coordinate is placed first.

Example:

When the points need to be separated into 2 groups

• If the number of points can be divided evenly by 2, the groups are equal.
  For example, 16 = 8 + 8.

• If the number of points cannot be divided evenly by 2, one can choose to ignore the middle pair or to include it in one of the 2 groups, as desired.
  For example, 29 = 15 + 14 or 14 + 15 or 14 + 14 + a data value that is left out.

During a hockey season, the points scored by all players are counted. A player's points include both their assists and the goals they scored. In hockey, up to 2 assists are counted for each goal scored, which corresponds to the last 2 passes made just before a goal.

Here are the numbers of assists and points for 10 regular Boston Bruins forwards during the 2022-2023 NHL season.

Based on this team's data, a player who made 60 assists should have finished the season with how many points?

The Mayer method is based on the calculation of the mean. However, the mean is a measure of central tendency that is very much influenced by distant data, known as outliers. In contrast, the median-median method is not influenced by outliers.

In other words, when there are one or more outliers in a distribution, the predictions made using the Mayer line are less reliable, or less representative of the whole data set, than those made using the median-median line. The latter method should therefore be used in these situations.