If the underlying relationship between an explanatory variable X and a response variable Y is Y = 3 + 2X² which of the following graphical plots of sample X and Y data is most likely to show a linear relationship between the variables passing through the origin of the graph?Y against X²Y against square root of XY against X(Y – 3) against X²
Question
If the underlying relationship between an explanatory variable X and a response variable Y is Y = 3 + 2X² which of the following graphical plots of sample X and Y data is most likely to show a linear relationship between the variables passing through the origin of the graph?Y against X²Y against square root of XY against X(Y – 3) against X²
Solution
The correct answer is "Y against X²".
Here's why:
The given relationship is Y = 3 + 2X². If we plot Y against X², we get Y = 3 + 2(X²), which simplifies to Y = 3 + 2X. This is a linear equation, and the graph of a linear equation is a straight line. However, it does not pass through the origin because the y-intercept is 3, not 0.
The other options do not give a linear relationship:
- Y against square root of X: This would give us a curve, not a straight line, because the square root function is not linear.
- Y against X: This would also give us a curve, because the relationship between Y and X is quadratic, not linear.
- (Y – 3) against X²: This would give us a straight line, but it would not pass through the origin. The equation would be Y - 3 = 2X, which simplifies to Y = 2X + 3. The y-intercept is 3, not 0.
So, none of the options would give a linear relationship that passes through the origin.
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