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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 retrieve 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 it's maw 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 data below to make a scatter plot.
Mouth
16" 18" 12" 17" 11" 15" 17" 16" 21"
Width (inches)
Length of
12.3' 16.2' 10.4' 13.6' 8.2' 11.7' 14.7' 13.6' 16.9'
Shark (feet)
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)
Draw a TREND LINE through the
data points. (Using a piece of uncooked spaghetti
works well to estimate placement.)
HINT: Use your Trend Line
5. Using this trend line, we can now predict the size of a shark based on the
width of a shark's mouth. Give the approximate size of a shark that has a ??
mouth width of 23".
6. Is your prediction in question #5 the exact same as everyone else in your class?
23”
If it is NOT the same, whose prediction do you think is more accurate?
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 expected value
( ) and actual value (y). 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 1 TREND LINE 2
Data Data
Residual Residual
Point Point
P1 1 P1
P2 2 P2
P3 –2 P3
2 -2
P4 P4
P5 P5
1
P6 P6
Sum of Sum of
Residuals Residuals
TREND LINE 3 TREND LINE 4
Data Data
Residual Residual
Point Point
P1 P1
P2 P2
P3 P3
P4 P4
P5 P5
P6 P6
Sum of Sum of
Residuals Residuals
10. What do all 4 trend lies above have in common? (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 1 TREND LINE 2 TREND LINE 3 TREND LINE 4
Data Residual Data Residual Data Residual Data Residual
Residual Residual Residual Residual
Point Squared Point Squared Point Squared Point Squared
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
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) Press . (This just resets the list menus)
3) Press
4) If there is OLD data already in the lists that needs to be cleared press the
To clear out OLD
up arrow, , to highlight L1 and then press to clear out data, first highlight
the old data. Do the same for L2 if it has OLD data that needs to be L1 and press
cleared. 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 =
x +
a b
Use this equation to reattempt your prediction of a shark with a 23” mouth.
y =
(23) + =
a b
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 is 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)?
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 Weak None Weak Strong
r: 0
Perfect No Perfect
Negative Linear Positive
Linear Relationship Linear
Relationship Relationship
1 x x y y
The formula for correlation coefficient is given by the equation: r is is , where
n 1 x y
x1 , y1 , 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. The TI-
83/84 automatically calculates this result as long as the Diagnostic option is On.
How strong of a linear relationship is shown by the data about sharks?
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.
Height of Distance
Ramp (cm) 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 =
x +
a b
3. How strong of a linear relationship is shown by the matchbox car data?
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 b
Actually try releasing the car with the ramp at 9 cm high.
( )
How close was your prediction? Calculate the percent error .
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).
=
( )+
a b
Actually try releasing a car from the height you determined above.
How many centimeters did your car get to 900 cm?
COMPUTER COST
The following show the value of a Pentium based computer system over a 7 year period.
Year 93' 94' 95' 96' 97' 98' 99'
value of $3800 $3200 $2500 $2100 $1350 $900 $450
computer
1. First determine an appropriate range and scale for the plot.
2. Make a Scatter Plot.
3. Draw a trend line
4. What type of association does the data show?
5. What is the correlation coefficient?
6. Estimate the cost of a Pentium computer in 1997.5 (interpolation or extrapolation)
7. Estimate the cost of a Pentium computer in 2009. (interpolation or extrapolation)
(careful this problem may suffer from the y2k bug)
8. Do you think the computer will ever be free? Explain your reasoning
9. Do you think the price will ever be negative? Is that possible? Explain.
10. 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 $2.50 $4.00 $5.50 $7.00 $8.50 $10.00
Number 33 59 91 117 101 48
Sold
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 7. At what price do you think they would
should charge to sell the most lipsticks? 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 Enter the data
from the chart into
ii. If there is OLD data already in the lists that needs to be cleared press the up arrow, L1 and L2
, to highlight L1 and then press to clear out the old data. Do the
same for L2 if it has OLD data that needs to be cleared.
iii. Next, enter all of the Cost in L1 and the Number Sold in L2. Select each of the
iv. After entering the data, press and select all of the options shown following options
by moving your
in the screen at the right. To do this move the cursor to the appropriate option ( , cursor to each and
, )and press . To change the Xlist to L1 if needed move the cursor to Pressing ENTER .
Xlist and press and to the Ylist and press .
v. Finally, press . To make further adjustments to the graph window press .
vi. Additionally, you can type the equation you calculated earlier in the to see the scatter plot and regression equation.
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 Pizza Rolls Pop Tarts Gold Fish Ck Kudo CC Dorritos Oreo Ice Cream Barron Pizza Toaster Strudel
Calories 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?
y = .x +
(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
trend line by selecting 3:Med-Med under the STAT menu. y = .x +
(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 Med-Med Line
Prediction using Least Squares Line
g. Is your prediction in problem #1c. an example of an Interpolation OR Extrapolation
Circle One
Corvette Value
The table displays data for value of 1953 Corvette (EX122) in actual dollars over the years
Year 1953 (new) 1955 (used) 1960(used) 1963(used) 1968(used) 1970(used) 1975(used) 1978(used)
Value $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?
y = .x3 + .x2 + .x + .
(a) (b) (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)
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