To fit a straight line [tex]\(y = a + bx\)[/tex] to the given data and predict [tex]\(Y\)[/tex] when [tex]\(X = 9.6\)[/tex], follow these steps:
1. Identify the Data Points:
The provided data points are:
[tex]\[
X: 2, 5, 6, 8, 9
\][/tex]
[tex]\[
Y: 8, 14, 19, 20, 31
\][/tex]
2. Determine the Coefficients [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
We need to find the coefficients [tex]\(a\)[/tex] (the y-intercept) and [tex]\(b\)[/tex] (the slope) of the line [tex]\(y = a + bx\)[/tex] that best fits the data.
After calculations, the coefficients are found to be:
[tex]\[
a \approx 1.000
\][/tex]
[tex]\[
b \approx 2.900
\][/tex]
3. Form the Equation of the Line:
Using the coefficients [tex]\(a\)[/tex] and [tex]\(b\)[/tex], the equation of the best-fit line is:
[tex]\[
y = 1.000 + 2.900x
\][/tex]
4. Predict [tex]\(Y\)[/tex] for [tex]\(X = 9.6\)[/tex]:
To predict the value of [tex]\(Y\)[/tex] when [tex]\(X = 9.6\)[/tex], substitute [tex]\(X = 9.6\)[/tex] into the equation of the line:
[tex]\[
y = 1.000 + 2.900 \times 9.6
\][/tex]
Performing the calculation:
[tex]\[
y \approx 1.000 + 27.84 = 28.84
\][/tex]
5. Result:
So, the predicted value of [tex]\(Y\)[/tex] when [tex]\(X = 9.6\)[/tex] is:
[tex]\[
Y \approx 28.84
\][/tex]
Therefore, the values of the coefficients are [tex]\(a \approx 1.000\)[/tex] and [tex]\(b \approx 2.900\)[/tex], and the prediction for [tex]\(Y\)[/tex] when [tex]\(X = 9.6\)[/tex] is approximately 28.84.
