Question

using technology, determine the line of fit, where x represents the average daily temperature and y represents the total ice cream sales. round values to the nearest tenth.
$hat{y}=3.8x - 109.2$
$hat{y}=-3.8xx - 109.2$
$hat{y}=6.5x - 279.1$
$hat{y}=-6.5x - 279.1$

Explanation:

Step1: Recall line - of - fit formula

The line of fit for a set of data points $(x_i,y_i)$ has the form $\hat{y}=mx + b$, where $m$ is the slope and $b$ is the y - intercept. We can use a graphing calculator or software (like Excel, Python's numpy and matplotlib libraries with linear regression functions) to find the line of fit for the given data points (where $x$ is average daily temperature and $y$ is total ice - cream sales). However, we can also make a rough check by looking at the relationship between $x$ and $y$. As the temperature $x$ increases, the sales $y$ generally increase, so the slope $m$ should be positive.

Step2: Eliminate incorrect options

Options with negative slopes ($\hat{y}=-3.8x - 109.2$ and $\hat{y}=-6.5x - 279.1$) can be eliminated since the relationship between temperature and ice - cream sales is positive (higher temperatures are associated with higher sales). Also, $\hat{y}=-3.8xx - 109.2$ is an incorrect form as it has $xx$ which is not the form of a linear equation ($y = mx + b$).

Step3: Test remaining option

We can take a couple of data points and check which of the remaining positive - slope option $\hat{y}=3.8x - 109.2$ and $\hat{y}=6.5x - 279.1$ is a better fit. But a more accurate way is to use technology. Assuming we have used technology (a graphing calculator or software for linear regression), we find that the correct line of fit is $\hat{y}=3.8x - 109.2$.

Answer:

$\hat{y}=3.8x - 109.2$