Super-Hard SAT Data Analysis Question 11 | Free SAT Prep

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Question 11·50 Super-Hard SAT Problem Solving and Data Analysis Questions·Problem Solving and Data Analysis

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The table represents points in a scatterplot.

$x$	$y$
$0$	$11$
$1$	$17$
$2$	$29$
$3$	$54$
$4$	$101$
$5$	$197$

A researcher believes that, except for one measurement error, the data follow an exponential model of the form $m(x)=a(b)^x+c$ , where $a$ , $b$ , and $c$ are positive constants. Which choice gives the $y$ -value that is most likely due to measurement error?

When an exponential model includes a constant added outside the exponent, do not expect the original $y$ -values to have a constant ratio. Instead, examine consecutive differences. For a model $a(b)^x+c$ , the differences follow an exponential pattern with common multiplier $b$ . Use consistent points to build the model, then test each point for an outlier.

Hints

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Look at changes in the outputs

Find the differences between consecutive $y$ -values. A vertical shift, represented by $c$ , does not prevent an exponential pattern from appearing in the differences.

Use the exponential structure

For $m(x)=a(b)^x+c$ , each first difference is multiplied by the same factor to get the next first difference.

Verify the suspicious point

Use points that appear consistent to determine the model, then compare the model's predicted value at each $x$ with the table.

Desmos Guide

Prefer the difference method first

Algebra using consecutive differences is faster here because the model includes a vertical shift. Use Desmos to verify the resulting model visually.

Enter the data

Enter the $x$ -values and $y$ -values into a Desmos table so the six data points appear on the graph.

Graph the model

Graph $y=6(2)^x+5$ . Five of the six points lie on this curve; identify the plotted point that does not lie on the curve.

Step-by-step Explanation

Compare consecutive differences

The first differences between the listed $y$ -values are $6$ , $12$ , $25$ , $47$ , and $96$ . For $m(x)=a(b)^x+c$ , consecutive first differences should have a constant multiplier of $b$ .

Use the first three points to find the model

The first two differences are $6$ and $12$ , so the multiplier is $2$ . Thus $b=2$ .

Using the first two points gives $a+c=11$ and $2a+c=17$ . Subtracting shows that $a=6$ , so $c=5$ .

Check the point at $x=3$

The model is $m(x)=6(2)^x+5$ . Therefore,

m(3)=6(2)^3+5=53.

The table instead gives $54$ when $x=3$ . The values at $x=4$ and $x=5$ match the model, so $54$ is the measurement error.

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation