Question

correlation coefficient & line of best fit
use for #11 - 12: the table shows the
minutes after a hot cup of coffee was
poured and its temperature in °f.

1. based on the correlation

coefficient for the data, what type of
linear association exists between the
number of minutes and
temperature?

1. which function best models the

data in the table?
use capital letter when typing responses:
11)
12)

Explanation:

Step1: Analyze the trend of data

As the number of minutes \(x\) increases, the temperature \(f(x)\) decreases. So, it is a negative association.

Step2: Check the strength of the negative association

We can calculate the correlation coefficient (or observe the closeness of data points to a line). The data points seem to follow a relatively strong linear - like decreasing trend.

Step3: For the line of best - fit

We use the formula for the equation of a line \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\) - intercept.
The \(y\) - intercept \(b\) is the value of \(y\) when \(x = 0\). From the table, when \(x = 0\), \(f(x)=180\) (approximate, for option B when \(x = 0\), \(f(0)=181\) which is close).
The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let's take two points \((0,180)\) (approximate) and \((1,175)\)
\(m=\frac{175 - 180}{1-0}=- 5\) (approximate). Option B has a slope of \(-4.7\) which is a better fit compared to other options.

Answer:

1. D
2. B