JAWS 2: Incident at Arched Rock
By Dave Cone
Originally published in Bay Currents, Nov 1993
This time we can joke about it-nobody got hurt. For the second year in a row,
autumn in Northern California was marked by a Great White Shark attack on a
coastal kayaker.
Rosemary Johnson is a lifelong ocean sports enthusiast now living in
Sonoma County. On Sunday October 10, she and three companions set off
from Goat Rock Beach south of Jenner for the short paddle southward to
Arched Rock (another Arch Rock lies north of Jenner). Rosemary was
paddling a borrowed blue Frenzy, which is a plastic sit-on-top boat just
nine feet long. Her friend Rick Larson, on his second kayak trip ever, was
on a Scupper, and one of the others paddled a Kiwi, which is also a very
short kayak.
As the four paddlers approached Arched Rock, Rosemary veered off from the group and headed around the rock.
Rick and the others soon followed at some distance. Rosemary felt a powerful jolt and flew off the top of her boat.
Rick saw a shark perhaps 16 feet long knock the boat entirely out of the water, and he believes Rosemary
flew 12 to 14 feet into the air. Rosemary landed in the water, and briefly felt something solid
underfoot. She thinks she landed on the shark. Wearing a wetsuit but no life vest, she had to swim in one direction to
nab her paddle, then back the other way to get to her boat. She remained calm, believing that she had merely struck a
rock. Her approaching companions were less than calm, having seen the "monster" that hit her boat. Rick states that
the shark's mouth encompassed almost the whole width of the Frenzy.
Rosemary hopped back on her kayak, but immediately capsized. She re-entered a second time and attempted to head
toward shore, but had difficulty controlling the boat. When she capsized again, her friends saw the bite marks on her
hull and realized that her boat was tippy and unmanageable because it was taking on water. They quickly improvised
a rescue, with Rosemary riding belly down and head aft on the front of Rick's Scupper-"the only boat that could take
me"-and the other paddlers towing the punctured boat. The group returned to the beach without further difficulty.
Back on dry land, the party reported the incident to Sonoma State Beaches rangers and inspected the boat. The
damage measures 20 by 15 inches and lies entirely below the waterline approximately beneath the seat. A jagged
crack runs perpendicular to the line of tooth marks. The pattern differs from the evidence left on Ken Kelton's boat,
which shows tooth marks on both deck and hull (see "Ano Nuevo Revisited," Bay Currents, Dec. '92). Ken's shark
held his boat for five to ten seconds, while Rosemary's contact was a collision rather than a grab-and-shake. Although
it's common for sharks to intentionally release their prey after the initial attack, it is also possible that this shark
simply bit more than it could chew, and Rosemary's boat popped out of its jaw like a watermelon seed between your
fingers.
A researcher at Bodega Bay Marine Lab told the state park folks the size of
the bite marks is consistent with a Great White up to 14 feet and
1500 pounds. Photos of the bite mark have been sent to shark expert Dr.
Mark Marks at Humboldt State for more detailed analysis.
Park Rangers posted the beaches from Russian River to Bodega Head with
shark warnings for five days after the attack, on the reasoning that Great Whites
typically feed in an area for a few days and then move on. Of the numerous
press accounts of the attack, the silliest was in the Santa Rosa Press-Democrat,
which stated that "(Head Ranger Brian) Hickey said rangers don't know if the
shark attacked on the first or last day it was in the Goat Rock area." Evidently
the reporter was surprised that Hickey didn't have a copy of the shark's MISTIX
reservation.
Rosemary and Rick visited the October 27 BASK general meeting, where
Rosemary was peppered with questions, a few of them pertinent, and awarded a
shark tee shirt and refrigerator magnet. Ken Kelton, BASK's own poster child
for shark survival, greeted Rosemary, saying "You and I are members of a very
exclusive club." Most of us are happy to see that club remain small.
How are such PREDICTIONS made?
TASK #1
Unit 6 Math II - BEST MODELS
How do these researchers determine the approximate size of the
sharks based on the bite marks alone?
One measurement that is considered is the width of the mouth.
First, researchers have to collect data on several sharks and then
make a scatter plot. Use the Numbers below to make a scatter plot.
Mouth
Width (inches)
Length of
Shark (feet)
16"
18"
12"
17"
11"
15"
17"
16"
21"
12.3'
16.2'
10.4'
13.6'
8.2'
11.7'
14.7'
13.6'
16.9'
1. Which variable above (Mouth Width or
Shark Length) should be the
independent variable (the input, x)?
Why?
2. Which variable above (Mouth Width or
Shark Length) should be the dependent
variable (the output, y)? Why?
3. Make a SCATTER PLOT of the
data at the right.
4. A trend line is a single straight line that
best goes through the ‘center’ of all of the
data points.
(DO NOT JUST CONNECT THE POINTS)
Fit a TREND LINE through the
data points. (Using a piece of uncooked spaghetti
works well to estimate placement.)
5. Using this trend line, predict the size of a shark based on the width of the mouth.
Next, give the approximate size of a shark that has a mouth width of 23".
HINT: Use your Trend Line
??
6. Is your prediction in question #5 the exact same as everyone else in your class?
If it is NOT the same, whose prediction do you think is more accurate?
23”
Who created a better trend line? Discuss.
7. What should be the properties of the BEST possible trend line using the given data? Discuss.
8. When 2 sets of data have a relatively linear relationship the data is either described as having a
POSTIVE or NEGATIVE correlation. The data correlation is POSITIVE if both data sets move in
the same direction (i.e. as one variable increases so does the other). The data correlation is
NEGATIVE if the data sets move in opposite directions (i.e. as one variable increases the other
decreases). What type of association does this data show?
9. Most trend lines that are considered to be a “good fit” will be balanced such that the total RESIDUAL above
and below the trend line is equal. RESIDUAL can be defined as the difference between the actual value (y)
and expected value(𝑦̂). A more succinct definition, RESIDUAL can be described as the vertical distance
each data point is away from the trend line (with signed difference for above and below the trend line).
Find the RESIDUALs for each of the TREND LINES below (the SCATTER PLOT is the same in each graph).
TREND LINE 2
TREND LINE 1
2
-2
1
Data
Point
Residual
Data
Point
P1
1
P1
P2
2
P2
P3
–2
P3
P4
P4
P5
P5
P6
P6
Sum of
Residuals
Sum of
Residuals
Residual
TREND LINE 4
TREND LINE 3
Data
Point
Data
Point
Residual
P1
P1
P2
P2
P3
P3
P4
P4
P5
P5
P6
P6
Sum of
Residuals
Sum of
Residuals
10. What do all 4 trend lies above have in common?
Residual
(optional: what is the approximate residual of your trend line from earlier)
11. To better analyze which trend line is best, it is common to consider comparing the sum of the squares of the
residuals. Which trend line do you think is the best based on this new information? Is it the one you expected?
TREND LINE 4
TREND LINE 3
TREND LINE 2
TREND LINE 1
Residual
Squared
Data
Point
Residual
Residual
Squared
Data
Point
P1
1
1
P1
P1
P1
P2
2
4
P2
P2
P2
P3
–2
4
P3
P3
P3
P4
P4
P4
P4
P5
P5
P5
P5
P6
P6
P6
P6
Sum
Sum
Sum
Sum
Residual
Residual
Squared
Data
Point
Data
Point
Residual
Residual
Residual
Squared
12. The line that minimizes the squares is called the LEAST SQUARES REGRESSION LINE. Most scientific
calculators are capable of determining the equation of this trend line. Consider again the data about sharks.
The following are the directions for the TI-83/84:
1) First, it will be helpful to turn on additional diagnostic information in your calculator.
CATALOG
SCROLL DOWN TO DianosticOn
…….…
2) Under the Stat menu, press
.
(This just resets the list menus)
3) Next, press
4) If there is OLD data already in the lists that needs to be cleared press the
up arrow,
to clear out
, to highlight L1 and then press
the old data. Do the same for L2 if it has OLD data that needs to be
cleared.
To clear out OLD
data, first highlight
L1 and press
CLEAR, ENTER.
5) Next, enter the Shark’s Mouth Size in L1 and the Length of the Shark in L2.
6) Return to the home screen by pressing
and then to calculate the
linear regression press
.
7) This represents the an equation of a line that minimizes the total residuals squared.
Fill in the blanks to complete the LEAST SQUARES REGRESSION LINE equation.
y =
a
x +

b
Use this equation to reattempt your prediction of a shark with a 23” mouth.
y =
a
(23) +
=

b
Now, was this close to your original prediction?
13. When a prediction is made between two given data points the prediction is called an INTERPOLATION.
When a prediction is made outside the range of given data points the prediction is an EXTRAPOLATION.
Which type of prediction was used when you predicted the length of a shark with a 23” wide mouth?
14. In the movie JAWS the shark was approximately 35 feet in
length, based on the equation you just calculated, how wide
would his mouth be? (careful the length might be represented by x)
Would it be large enough to bite the back end of a boat
(consider even a small boat is 48 inches wide)?
35 =
a

x +
b
15. A calculation called the correlation coefficient (r) is used to measure the extent to which the data for the
two variables show a linear relationship. The closer the value is to 1 or –1 the stronger the linear
relationship.
Strong
r:
Perfect
Negative
Linear
Relationship
Weak
Weak
None
Strong
0
No
Linear
Relationship
Perfect
Positive
Linear
Relationship
The TI-83/84 automatically calculates the correlation coefficient as long as the Diagnostic option is On. The
formula for correlation coefficient is given by the equation: r  1
 x i  x  y i  y 
 , where  x1 , y1  ,

 s 
s
 x  y 

n 1 
 x1 , y1  , …,  x1 , y1 
are the data points, x is the mean of the x-values, y is the mean of the y-values, sx is
the standard deviation of the x-values, and s y is the standard deviation of the y-values.
How strong of a linear relationship is shown by the data about sharks?
(Your calculator displays the correlation coefficient r =…..when it calculated the least squares trend line)
Matchbox Measures Task
1. Using a matchbox car and tracks, create a table of values and
scatter plot that shows the distance the car travels
based on how high the starting ramp (as shown
at in the diagrams at the right). Start with a
single book to create the initial ramp and
continue increasing so that you have at least 4 data points.
Determine which measure should be the independent and dependent
variable.
Create an appropriate scale for each of the axes.
1000
960
920
880
840
800
760
720
680
640
600
560
520
480
440
400
360
320
280
240
200
160
120
80
40
Height of
Ramp (cm)
1
2 3
4 5
6
7
8
9 10 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
Distance
Traveled (cm)
2. Using your calculator, fill in the blanks to complete the LEAST SQUARES
REGRESSION LINE equation for the data collected with the match box car.
y =
3.
a
x +

b
How strong of a linear relationship is shown by the matchbox car data (use the
correlation coefficient to support your answer)?
4. Using your LEAST SQUARES REGRESSIONS LINE, make a prediction how far
the car will travel if the ramp is 9 cm. Is this an INTERPOLATION or an
EXTAPOLATION?
9cm
=
a

(9
cm
)+
b
Actually try releasing the car with the ramp at 9 cm high.
How close was your prediction?
Calculate the percent error
(𝐴𝑐𝑡𝑢𝑎𝑙 𝑉𝑎𝑙𝑢𝑒−𝑃𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑 𝑉𝑎𝑙𝑢𝑒)
𝐴𝑐𝑡𝑢𝑎𝑙 𝑉𝑎𝑙𝑢𝑒
.
5. How high would the ramp need to be for the car to travel a distance of 900 cm before coming to a rest?
(Careful, you may need to solve for your solution).
900cm =
a

x +
Actually try releasing a car from the height you determined above.
How many centimeters did your car get to 900 cm?
b
COMPUTER COST
The following show the value of a Pentium based computer system over a 7 year period.
Year
value of
computer
93'
94'
95'
96'
97'
98'
99'
$3800
$3200
$2500
$2100
$1350
$900
$450
1. First determine an appropriate range and scale for the plot and make a scatter plot.
2. Draw a trend line
3. What type of association does the data show?
4. What is the correlation coefficient?
5. Estimate the cost of a Pentium computer
in 1997.5 (interpolation or extrapolation)
6. Estimate the cost of a Pentium computer
in 2009. (interpolation or extrapolation)
(careful this problem may suffer from the y2k bug)
7. Do you think the computer will ever be free? Explain your reasoning
8. Do you think the price will ever be negative? Is that possible? Explain.
9. Do you think there should be a domain restriction of the use of the trend lines (i.e.
just using the trend line for a limited number of years)?
LIP STICK CHARGE
A company in California is test marketing a new line of lipsticks. The lipstick only costs the company
$0.90 to make due to the volume production. The company located several different cities with
approximately the same demographics and sold the exact same lipstick at different prices. They wanted
to know which price would yield the largest profit. The following table shows the prices at which they
were sold and the number sold at that price over a period of 3 months.
Cost
Number
Sold
$2.50
$4.00
$5.50
$7.00
$8.50
$10.00
33
59
91
117
101
48
1. First determine an
appropriate range and scale
for the plot.
2. Make a Scatter Plot.
3. Draw a trend line or curve if
more appropriate.
4. What type of association
does the data show? (Is it
linear?)
The TI-83/84 is capable of
calculating quadratic, cubic,
and quartic regression
equations.
5. Explain why you think the
data looks the way it does.
6. At what price do you think the company
should charge to sell the most lipsticks?
7. At what price do you think they would
make the largest profit?
8. Explain either why the prices you selected are the same in #6 and #7 or why they are different.
9. (OPTIONAL) Make a graph of the data on your calculator and on the grid.
i.
Press
ii.
If there is OLD data already in the lists that needs to be cleared press the up arrow,
to clear out the old data. Do the
, to highlight L1 and then press
same for L2 if it has OLD data that needs to be cleared.
Next, enter all of the Cost in L1 and the Number Sold in L2.
Select each of the
following options
After entering the data, press
and select all of the options shown
by moving your
in the screen at the right. To do this move the cursor to the appropriate option (
, cursor to each and
Pressing ENTER .
, )and press
. To change the Xlist to L1 if needed move the cursor to
Xlist and press
and to the Ylist and press
.
Finally, press
. To make further adjustments to the graph window press
.
Additionally, you can type the equation you calculated earlier in the
to see the scatter plot and regression equation.
iii.
iv.
v.
vi.
Enter the data
from the chart into
L1 and L2
JUNK FOOD Calories and Fat Grams
1. The table displays data for Nutrition Guides of a single serving of particular foods. Find the equation of the best trend
line. What is your prediction as to how many Fat Grams a serving size of 200 Calories would have?
Food
Calories
Pizza Rolls
Pop Tarts
Gold Fish Ck
Kudo CC
Dorritos
Oreo Ice Cream Barron Pizza
Toaster Strudel
230
220
150
130
140
160
340
190
Fat Grams
11
4
6
5
7
8
18
8
a. What type of association does the data suggest?
b.
Create a scatter plot on the graph at the right and an
approximate trend line.
c. Plot the data on the graph and using the TI-83/84. Then, find the
best possible line that fits the data using the TI-83/84's linear
regression (4:linReg) command. What is the equation of the line?
.x +
y =
(a)
(b)
d. When a data point or two falls out of line from the rest of the data
that point is usually referred to as an outlier. Outliers can adversely affect the least squares regression line
so that it doesn’t accurately represent the majority of the data (similar to how an outlier effects the mean).
Would you describe any of the points in this set as an outlier?
e. When scatter plots contain outliers usually the median-median trend line is used instead of the least squares
regression line (just as the medians are more resistant to the effects of outliers the median-median line is
more resistant to outliers of a scatter plot). Calculate the median-median trend line.
Calculating the equation of the median-median trend line requires that the data points are ordered from
smallest to largest first coordinate and then separates the data into three equal, or nearly equal, groups with at
least 1/3 of the data points in each of the first and last groups. The median x-values and y-values of each
group are calculated. These medians, from smallest x-values to largest, are named x1 , y1 , x2 , y2 , x3 , y3  .
Then a line through the first and third medians is found. Finally, a line parallel to this line, 1/3 of the distance
between the line and the remaining median is formed. The resulting line is of the following form
y y
y  y2  y3  ax1  x2  x3 
y  ax  b, a  3 1 , b  1
. The TI83/84 is also capable of calculating this
x3  x1
3
.x +
trend line by selecting 3:Med-Med under the STAT menu.
y =
(a)
(b)
f. Using each of your equations, predict the number of Fat Grams in an average serving of food that has
280 calories
Prediction using Least Squares Line
g. Is your prediction in problem #1c. an example of an
Prediction using Med-Med Line
Interpolation OR Extrapolation
Circle One
Corvette Value
The table displays data for value of 1953 Corvette (EX122) in actual dollars over the years
Year
Value
1953 (new)
1955 (used)
1960(used)
1963(used)
1968(used)
1970(used)
1975(used)
1978(used)
$3600
$2400
$1100
$1000
$3200
$4500
$7600
$18,500
a. Plot the data on the graph and using the TI-83. Then, find
the best possible line that fits the data using the TI-83's
cubic regression (6:CubicReg) command. What is the
equation of the line?
.x3 +
y =
(a)
.x2 +
(b)
.x +
(c)
.
(d)
b. Using your equation, what might be a good estimate for the
value of the car in 2005?
and 2008?
c. Are your predictions in problem #2b. examples of an
Interpolation OR Extrapolation
Circle One
d. In 2005, a 1953(EX122) of which only 300 were made sold on auction for $130,000 and in 2008,
the model sold for $440,000. Were your cubic regression models still reasonable?
e. How well does the data fit a cubic equation? (Support your answer with the coefficient of
determination)