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Sample submitted solutions:
From: Katie, age 20
School: North Carolina State University,
Raleigh, North Carolina
1. Which triplet's backyard pond most
closely matches the male:female ratio in
the lake? Justify your answer with
data you collected. In your explanation,
tell us which display(s) of data you used
to help you and how it helped. Remember,
Angel's pond has a 1:1 ratio, Molly's pond
has a 3:1 ratio [three times as many males
as females], and Gar's pond has a 1:2 ratio
[twice as many females as males].
Angel's pond with a ratio of 1:1 most closely
matches that of the lake. In the data that
I collected there were only a few more males
than females, resulting in having percentages
like 55:45, 54:46, and 56:44, all three
of which came up twice each. This data came
from an overall total of 700 scoops(200,200,100,100,50,50).
With this data, there is definitely not
a 3:1 ratio of males to females. And since
there are more males coming up than females,
I do not think that it could possibly be
a 1:2 ratio of males to females. Thus, the
only logical answer, as it seems to me,
would be for a ratio closest to 1:1.
2. What is your best estimate of the
percent of fish in Lake Mark that are male?
Describe the strategy you used to determine
this estimate.
My best estimate of percentage of male fish
would be around 55%. I determined this by
taking the average of the 55,55,54,54,56,56
different percentages that came from the
data.
3. How confident are you that your estimate
is close to the actual percent of fish that
are male? What makes you feel confident
about your estimate, and what would make
you feel more confident?
I am fairly certain, around 90%, that my
estimate is close to the actual percent
of fish in the lake. I think that having
the data show so close results every time,
that the data has to be fairly close to
accurate. I would feel more confident with
more scooping to further my hypothesis.
Bonus: Suppose we knew that Lake Mark
had just been filled with 350 male and 650
female fish. About how many male and female
fish might be caught if you scooped a fish
10 times? What if you scooped a fish 700
times? How confident are you that your prediction
would be close to an actual sample? Explain
your reasoning.
I think that with only 10 scoops the prediction
would not be justified enough because of
such little sampling, so I think that the
scooping would show an almost even trend,
with a few more female fish than males.
With 700 scoops, I think that the true ratio
would be shown, or close to it, showing
nearly twice as many females and males.
I think that my prediction is fairly accurate,
around maybe 80%. I definitely think that
there is a possibility of the outcome being
different from that of my prediction.
From: Matt,
age 21
School: North Carolina State University,
Raleigh, North Carolina
1. Which triplet's backyard pond most
closely matches the male:female ratio in
the lake? Justify your answer with
data you collected. In your explanation,
tell us which display(s) of data you used
to help you and how it helped. Remember,
Angel's pond has a 1:1 ratio, Molly's pond
has a 3:1 ratio [three times as many males
as females], and Gar's pond has a 1:2 ratio
[twice as many females as males].
The ratio in the lake most closely resembles
Angel's ratio of 1:1. The data I collected
showed the amount to be 55% male and 45%
female in the lake. This is closest to Angel's
ratio which would be 50% male and 50% female.
The closest other pond would be Gar's which
was approximately 33% male and 66% female.
The data I collected was first with 10 scoops.
It was 5 male and 5 female. Then I tried
1000 scoops and the lake had 54.3% male
and 45.7% female. I then tried 5000 scoops
to make sure my percentages were correct.
When I tried 5000 I got 55.3% male and 44.7%
female. Now I was sure I had the right amount.
To compare the ratios I multiplied all the
ponds so the total of the ratio was 100.
Angel's ratio became 50:50, Gar's became
33:66 and Mollie's became 75:25. Then I
compared my ratio of 55:45 and saw that
Angel's ratio was the closest.
2. What is your best estimate of the
percent of fish in Lake Mark that are male?
Describe the strategy you used to determine
this estimate.
As I discussed in #1, I used the ratios
totaling 100 to find the percent. The lake
was approx. 55% male and 45% female. I used
the ratios I found and the percent given
in the table to come up with this estimate.
3. How confident are you that your estimate
is close to the actual percent of fish that
are male? What makes you feel confident
about your estimate, and what would make
you feel more confident?
I am very confident because I scooped so
many fish. When I scooped 1000 and then
5000, this gave me a very broad look at
the lake. I am confident because with this
many trials, the chance of having data that
is falsly represented decreases. In other
words, the more scoops I make the closer
to the actual percent I will get. The only
way to be more certain is to take even more
scoops, perhaps 50,000.
Bonus: Suppose we knew that Lake Mark
had just been filled with 350 male and 650
female fish. About how many male and female
fish might be caught if you scooped a fish
10 times? What if you scooped a fish 700
times? How confident are you that your prediction
would be close to an actual sample? Explain
your reasoning.
If we scooped ten fish, we could expect
3 male and 7 female or 4 male and 6 female.
However, with such a small number, the amounts
could be unrepresentative of the amount.
In other words, we could get 8 male and
2 female. On the other hand if we scooped
700 fish, we would be more likely to get
closer to the ratio that is represented
in the pond. I would expect approximately
245 males and 455 females. This is because
this ratio is equivalent to the 350:650
that was placed in the lake. Therefore,
we can use this ratio to estimate the number
of each sex that will be taken out with
any number of scoops. Therefore, given any
number of scoops, you could be confident
that your prediction was fairly close to
the correct amount because you had equivalent
ratios. The reason these ratios are equivalent
is that the male amount is the same part
of the total fish. In other words, the percent
of male and female fish is the same in each
ratio and the ratio of males to females
is also the same.
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Reflections:
For question 1, some students were unwilling
even to evaluate the 1-to-1 ratio as a possibility
if the scoops did not produce an exact 1-to-1
ratio. They were able to compare the other
values but did not recognize that a 57%
male population was closer to the 1:1 ratio
than to 1:2. They seemed to have a better
grasp of what 1-to-1 means both visually
and mathematically, thus making it easy
to discount based upon preliminary, although
slightly inaccurate, data. Students generally
picked a sample size and then evaluated
it from 3 to 5 times. Not changing the sample
size, and missing the leveling trend on
male percentages, affected the following
questions.
For question 3, many students did not base
their confidence levels on the leveling
effect of numerous scoops with large samples.
Answers were generally vague, and stated
that it would have been better if we knew
how many fish were in the pond, or if they
could all be counted. Students either used
large samples from the start, or never even
noticed that the percentages changed when
larger samples were scooped. This prevented
a high confidence rating. Not many students
seemed to see the data trend change over
sample size.
The bonus caused some confusion for a group
of students from the Caroline Davis school.
They did not know whether there were fish
in the pond before more were added. The
number of students that tried the bonus
was significantly lower because of this.
Others were able to complete the mathematical
percentages expected or they figured that
the larger number of scoops would be more
accurate. Only one student received credit
for the bonus for understanding the entire
concept.
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