Question 25·50 Super-Hard SAT Problem Solving and Data Analysis Questions·Problem Solving and Data Analysis

0 / 50 Questions completed

The scatterplot shows the daily energy produced by a solar panel system for different amounts of sunlight. The solid line is a line of best fit. Each choice gives the coordinates of a plotted data point.

Which choice identifies the data point for which the observed energy production is the greatest percent greater than the energy production predicted by the line of best fit?

A percent increase from a model prediction uses the prediction as its starting amount. On a scatterplot, a dot gives the observed value, while the line of best fit gives the prediction at the same input. Use two marked points on the line to build its rule, then compare the candidates in Desmos. The largest vertical gap needn't be the largest percent.

Hints

Hint 1
The slope of a line is its change in energy divided by its change in sunlight hours. Use the two open circles on the line, not two filled data dots, to find the rate for this model.
Hint 2
A prediction comes from the line at a dot's sunlight amount, not from the dot itself. Once you have the line's rule, what does it predict at each of the four listed sunlight amounts?
Hint 3
A percent increase divides the extra energy by the predicted energy, then multiplies by $100$ . A larger gap in kWh can be a smaller percent if it starts from a larger prediction.

Step-by-step

Compare each dot with the given line
Step 1
Find the line's rate
The open circles at $(2,21)$ and $(14,93)$ lie on the line of best fit; the filled dots are observations. The slope is the change in energy divided by the change in sunlight hours. Type $\frac{93-21}{14-2}$ in Desmos. It prints $6$ , so the line predicts $6$ more kWh per additional hour.
Step 2
Write a rule for the prediction
The line predicts $21$ kWh at $2$ hours and rises $6$ kWh per hour after that, so its prediction function is $p(x)=6(x-2)+21$ , where $x$ is sunlight hours. Type that rule in Desmos; its graph passes through both open circles. Don't fit a new line to the filled dots: the question already gives you one.
Step 3
Get the four predicted amounts
Put the choices' sunlight hours in a list: type $a=[4,6,8,10]$ , then $p(a)$ . Desmos shows $[33,45,57,69]$ kWh in that order. These are predictions from the line, not the energy amounts observed at the dots.
Step 4
Set up the percent comparison
A percent increase compares the extra amount with the amount you started from:
$\text{percent greater}=\frac{\text{observed}-\text{predicted}}{\text{predicted}}\times100.$
Here, the predicted amount is the percent's base. For $(8,70)$ , that means $\frac{70-57}{57}\times100$ . Dividing by the observed $70$ , or comparing kWh gaps without dividing, answers a different question.
Step 5
Compare the percentages
In the same order as $a$ , type $b=[45,50,70,80]$ for the observed kWh, then $\frac{100(b-p(a))}{p(a)}$ . Desmos shows about $[36.4,11.1,22.8,15.9]$ . The greatest percent is the first entry, at $4$ hours: the observed $45$ kWh is about $36.4\%$ greater than the predicted $33$ kWh. Choice D.

Question 25·50 Super-Hard SAT Problem Solving and Data Analysis Questions·Problem Solving and Data Analysis

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this