Question 12.1: Flat-screen TV Sales Study Music Technologies, an electronic......

Step 1: Identify the dependent and independent variables

An essential first step is to correctly identify the independent and dependent variables. A useful rule of thumb is to ask the following question: ‘Which variable is to be estimated?’

The answer will identify the dependent variable, y. In the example:

x = the number of advertisements placed weekly
y = the number of flat-screen TVs sold in the week.

Step 2: Construct a scatter plot between x and y

A scatter plot graphically displays both the nature and strength of the relationship between the independent variable (x) and the dependent variable (y), as is illustrated in Figure 12.1. A visual inspection of the scatter plot will show whether the pattern is linear or not, whether it is direct or inverse, and its strength. Refer to figures 12.3 to 12.7 to interpret the relationship between x and y.

The patterns in figures 12.3 and 12.4 show strong linear relationships between x and y. Estimates of y based on these relationships will be highly reliable. The patterns shown in figures 12.5 and 12.6 are evidence of moderate to weak linear relationships and are of less value for estimation purposes. Finally, the pattern shown in Figure 12.7 is evidence of no statistical relationship between the two numeric measures. In such cases, regression analysis is of little value to estimate y based on x-values. These estimates of y will be unreliable.

In the example, the scatter plot between ads placed and sales recorded is shown in Figure 12.8.

From the scatter plot, there is a moderate positive linear relationship between ads placed and sales of flat-screen TVs. As the number of ads placed increases, there is a moderate increase in flat-screen TV sales.

Step 3: Calculate the linear regression equation

Regression analysis finds the equation that best fits a straight line to the scatter points. A straight-line graph is defined as follows:

\hat{y}=b_0+b_1x        12.1

Where: x = values of the independent variable

\hat{y} = estimated values of the dependent variable

b_0 = y-intercept coefficient (where the regression line cuts the y-axis)

b_1 = slope (gradient) coefficient of the regression line

Regression analysis uses the method of least squares to find the best-fitting straight-line equation. It is a mathematical technique that determines the values b_0 and b_1, such that:

the sum of the squared deviations of the data points from the fitted line is minimised.

A brief explanation of this method is given and illustrated using Figure 12.9.

(a)   First, calculate the deviation (e_i) of each y_i-value from its estimated value y_i.

(b)   Square each deviation to avoid positive and negative deviations cancelling each other out when summed.

(c) Sum the squared deviations to obtain a measure of total squared deviations.

(d) Values for b_0 and b_1 are now found mathematically by minimising the total squared deviations in (c). The calculation is called the method of least squares (MLS).

Without showing the mathematical calculations, the coefficients b_0 and b_1 that result from this method of least squares are given as follows:

b_1=\frac{n\sum{xy} -\sum{x\sum{y} } }{n\sum{x^2}-(\sum{x} )^2 }          12.2

b_0=\frac{\sum{y}-b_1\sum{x} }{n}          12.3

The values of b_0 and b_1 define the best-fitting linear regression line. This means that no other straight-line equation will give a better fit (i.e. a smaller sum of squared deviations) than the regression line. For the example, Table 12.2 shows the calculations of the regression coefficients using Formula 12.2 and Formula 12.3.

From Table 12.2, we have \sum{x}=44,\ \sum{y} =346\ ,\sum{x^2=174}\ ,\sum{xy}=1324\ \text{and}\ n=12. Then:

b_1=\frac{12(1324)-(44)(346)}{12(174)-(44)^2}=\frac{664}{152} = 4.368

b_0=\frac{346-4.368(44)}{12} = 12.817

The simple linear regression equation to estimate flat-screen TV sales is given by:

\hat{y} = 12.817 + 4.368x      for 2 ≤ x ≤ 5

The interval of x-values (i.e. 2 ≤ x ≤ 5) is called the domain of x. It represents the set of x-values that were used to construct the regression line. Thus to produce valid estimates of y, only values of x from within the domain should be substituted in the regression equation.

Step 4: Estimate y-values using the regression equation

The regression equation can now be used to estimate y-values from (known) x-values by substituting a given x-value into the regression equation. In the example, management question 2 asks for a sales estimate of the average number of flat-screen TVs in a week when three advertisements are placed.

Thus substitute x = 3 into the regression equation:

\hat{y} = 12.817 + 4.368(3) = 12.817 + 13.104 = 25.921 = 26 (rounded)

The management of Music Technologies can therefore expect to sell, on average, 26 flatscreen TVs in a week when three newspaper advertisements are placed.