Question

scatterplots & lines of best fit

1. what is the correlation of the graph below?
2. draw a line of best fit for the points in the scatter graph.

(3,2)(2,3)(4,5)(3,3)

1. what is the correlation?
2. use the line to predict the value of y when x = 12.
3. what could be the y - intercept of the line of best fit?

Explanation:

Step1: Analyze the first scatter - plot

Visually, as x increases, y also tends to increase. So, the correlation is positive.

Step2: Draw the line of best - fit for given points

For points (3,2), (2,3), (4,5), (3,3), we can use the least - squares method conceptually. Plot the points on a graph and draw a line that seems to pass as close as possible to most of the points. In a more formal way, we can calculate the slope $m$ and y - intercept $b$ using formulas $m=\frac{n(\sum xy)-(\sum x)(\sum y)}{n(\sum x^{2})-(\sum x)^{2}}$ and $b = \bar{y}-m\bar{x}$ where $n$ is the number of data points, $\bar{x}$ and $\bar{y}$ are the means of $x$ and $y$ values respectively. But for a quick sketch, we can just eyeball it.

Step3: Determine the correlation for the second set of points

As x increases, y does not show a clear upward or downward trend. So, the correlation is weak or close to 0.

Step4: Predict y value when x = 12

After drawing the line of best - fit, we find the corresponding y - value on the line for x = 12. This requires having the equation of the line (either from a formal calculation or from the graph). If we assume the line equation is $y=mx + b$, we substitute $x = 12$ into it.

Step5: Estimate the y - intercept

The y - intercept is the value of y when x = 0. We look at where the line of best - fit intersects the y - axis.

Answer:

1. Positive
2. (Sketch the line as described above)
3. Weak or close to 0
4. (Value obtained from the line of best - fit for x = 12)
5. (Value where the line of best - fit intersects the y - axis)