Oysters are categorized for retail as small, medium, or large based on their volume. The grading process is slow and expensive when done by hand. A computer program estimates oyster volume based on the pixel area of two$-$dimensional $(2$D$)$ images of the oysters. The given table shows the actual volume $($in cm$^3)$ and $2$D reconstruction $($in thousands of pixels$)$ for $30$ oysters.

Actual $($cm$^3)$

$2$D $($thousand pixels$)$

$3$D $($million voxels$)$

Actual $($cm$^3)$

$2$D $($thousand pixels$)$

$2$D $($million voxels$)$

Actual $($cm$^3)$

$2$D $($thousand pixels$)$

$3$D $($million voxels$)$

$13\mathrm{.}04$

$47\mathrm{.}907$

$5\mathrm{.}136699$

$10\mathrm{.}53$

$31\mathrm{.}216$

$3\mathrm{.}942783$

$10\mathrm{.}95$

$37\mathrm{.}156$

$4\mathrm{.}707532$

$11\mathrm{.}71$

$41\mathrm{.}458$

$4\mathrm{.}795151$

$10\mathrm{.}84$

$41\mathrm{.}852$

$4\mathrm{.}052638$

$7\mathrm{.}97$

$29\mathrm{.}070$

$3\mathrm{.}019077$

$17\mathrm{.}42$

$60\mathrm{.}891$

$6\mathrm{.}453115$

$13\mathrm{.}12$

$44\mathrm{.}608$

$5\mathrm{.}334558$

$7\mathrm{.}34$

$24\mathrm{.}590$

$2\mathrm{.}768160$

$7\mathrm{.}23$

$29\mathrm{.}949$

$2\mathrm{.}895239$

$8\mathrm{.}48$

$35\mathrm{.}343$

$3\mathrm{.}527926$

$13\mathrm{.}21$

$48\mathrm{.}082$

$4\mathrm{.}945743$

$10\mathrm{.}03$

$41\mathrm{.}616$

$3\mathrm{.}672746$

$14\mathrm{.}24$

$47\mathrm{.}481$

$5\mathrm{.}679636$

$7\mathrm{.}83$

$32\mathrm{.}118$

$3\mathrm{.}138463$

$15\mathrm{.}59$

$48\mathrm{.}070$

$5\mathrm{.}728880$

$11\mathrm{.}11$

$40\mathrm{.}976$

$4\mathrm{.}013992$

$11\mathrm{.}38$

$45\mathrm{.}112$

$4\mathrm{.}410797$

$9\mathrm{.}94$

$34\mathrm{.}717$

$3\mathrm{.}987582$

$15\mathrm{.}35$

$65\mathrm{.}361$

$5\mathrm{.}565995$

$11\mathrm{.}22$

$37\mathrm{.}020$

$4\mathrm{.}558251$

$7\mathrm{.}53$

$27\mathrm{.}320$

$2\mathrm{.}678423$

$15\mathrm{.}44$

$50\mathrm{.}910$

$6\mathrm{.}303198$

$9\mathrm{.}25$

$39\mathrm{.}333$

$3\mathrm{.}449867$

$12\mathrm{.}73$

$52\mathrm{.}712$

$5\mathrm{.}481545$

$5\mathrm{.}67$

$22\mathrm{.}895$

$1\mathrm{.}928109$

$13\mathrm{.}75$

$51\mathrm{.}351$

$5\mathrm{.}609681$

$12\mathrm{.}66$

$41\mathrm{.}500$

$5\mathrm{.}016762$

$8\mathrm{.}26$

$34\mathrm{.}804$

$3\mathrm{.}450164$

$14\mathrm{.}37$

$53\mathrm{.}281$

$5\mathrm{.}292105$

To access the complete data set, click the link for your preferred software format:

Excel Minitab JMP SPSS TI R Mac$-$TXT PC$-$TXT CSV CrunchIt!

Engineers were given the task to improve on the $2$D reconstruction software. They designed a new program that estimates

Engineers were given the task to improve on the $2$D reconstruction software. They designed a new program that estimates oyster volume using three$-$dimensional $(3$D$)$ image processing. The table gives the actual volume, in cm$^3,$ of $30$ oysters and the $3$D volume reconstruction, in millions of volume pixels.

Using the software of your choice, make a scatterplot of the relationship between actual volume $($response$)$ and $3$D volume reconstruction $($explanatory$).$ Find the equation for the corresponding least$-$squares regression line.

Use the regression equation to $3$ decimal places to predict the actual volume of an oyster with a $3$D reconstruction of $4\mathrm{.}5$ million volume pixels to one decimal place.

$($Enter your answer rounded to one decimal place.$)$

volume

$9\mathrm{.}5$

cm$^3$

What percent of the variations in actual oyster volume can be explained by the $2$D regression model?

What is the percentage of the observed variation in actual oyster volume explained by this regression line? $($Enter your answer rounded to the nearest whole number.$)$

observed variation