Question

the scatterplot opposite was constructed from the data in the table below.

\

$$\begin{array}{|c|c|c|c|c|c|} \\hline x & 1 & 2 & 3 & 4 & 5 \\\\ \\hline y & 3 & 9 & 19 & 33 & 51 \\\\ \\hline \\end{array}$$

from the scatterplot, the association between \\(y\\) and \\(x\\) is non-linear.

a linearise the scatterplot by applying an \\(x\\)-squared transformation and fit a least squares line to the transformed data.
b give its equation.
c use the equation to predict the value of \\(y\\) when \\(x = 6\\).

Explanation:

🆕 New Concept Discovered: Data Linearisation
Transforming variables to fit a straight line

Step 1: Transform the independent variable

To linearise the data using an \( x^2 \) transformation, we square each of the original \( x \)-values to create a new independent variable, which we will call \( x^2 \).

The original data table is:

\( x \)	1	2	3	4	5

Squaring each \( x \)-value:

\( x^2 \)	1	4	9	16	25

Step 2: Fit a least-squares regression line

We find the equation of the least-squares regression line in the form:
\[ y = a + b(x^2) \]

First, calculate the summary statistics for the transformed variable \( X = x^2 \) and \( y \):

• Number of data points \( n = 5 \)
• Mean of \( X \):

\[ \bar{X} = \frac{1 + 4 + 9 + 16 + 25}{5} = \frac{55}{5} = 11 \]

• Mean of \( y \):

\[ \bar{y} = \frac{3 + 9 + 19 + 33 + 51}{5} = \frac{115}{5} = 23 \]

Next, calculate the slope \( b \):
\[ b = \frac{\sum (X_i - \bar{X})(y_i - \bar{y})}{\sum (X_i - \bar{X})^2} \]

Let's find the individual deviations and products:

\( X_i \)	\( y_i \)	\( X_i - \bar{X} \)	\( y_i - \bar{y} \)	\( (X_i - \bar{X})(y_i - \bar{y}) \)	\( (X_i - \bar{X})^2 \)
4	9	-7	-14	98	49
9	19	-2	-4	8	4
16	33	5	10	50	25
25	51	14	28	392	196
Sum				748	374

Calculate the slope \( b \):
\[ b = \frac{748}{374} = 2 \]

Calculate the intercept \( a \):
\[ a = \bar{y} - b\bar{X} = 23 - 2(11) = 23 - 22 = 1 \]

The equation of the least-squares regression line is:
\[ y = 1 + 2x^2 \]

Step 3: Predict the value of \( y \) when \( x = 6 \)

Substitute \( x = 6 \) into the transformed equation:
\[ y = 1 + 2(6)^2 \]
\[ y = 1 + 2(36) \]
\[ y = 1 + 72 = 73 \]

Answer:

a The transformed data table is:

\( x^2 \)	1	4	9	16	25

b The equation of the least-squares line is:
\[ y = 1 + 2x^2 \]

c When \( x = 6 \):
\[ y = 73 \]