Lab Instructions_ Scientific Reasoning Act II Mission Memo (Spring A 2024 Onward)
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Lab Instructions: Scientific Reasoning Act II Mission Memo
Greetings Fellow Explorer:
The temperature of Phygaris continues to rise with the passing of the season. The baby
astelars are now experiencing higher temperatures than they would in a typical breeding
season. Trapped in their eggs, they have no way to avoid the rising temperatures.
We can choose to do nothing and hope they survive. Alternatively, we can intervene by
moving the eggs to a cooler beach. However, a decision of this magnitude should not be
made lightly; we must use evidence to choose the better option.
With the help of your fellow explorers, you measured temperature, water, and oxygen at
a second beach—close to the beach where the astelars laid their eggs. At the same time,
I deployed a research drone to measure the same variables at the first beach. The data
will enable you to predict the probability that an egg would hatch on each beach.
In addition to the physical conditions of the beaches, we should consider any biological
threats to the astelars. At the second beach, we discovered not only red tippers but also
digworms. These large, burrowing creatures might crush the astelar eggs while moving
through the sand. By combining our knowledge of the physical conditions with our
knowledge of the biological conditions, we can decide whether to leave the eggs at their
current location (Beach 1) or move them to the other location (Beach 2).
Use the following questions to guide your work:
●
Would more eggs hatch under the physical conditions on Beach 1 or under the
physical conditions on Beach 2? (Appendices 1 and 2)
●
Considering the physical and biological conditions of each beach, should we
leave the eggs at Beach 1 or move them to Beach 2? (Appendix 3)
Universally in your debt,
The AI
Note:
You will be using the program Microsoft Excel for this assignment. We have
provided links to Excel tutorials and transcripts where applicable, to help you answer the
questions.
Appendix 1
Would more astelars hatch at the mean temperature on Beach 1
or at the mean temperature on Beach 2?
At least 381 astelar pups must survive each year to keep the population from shrinking.
This season, only 2217 eggs were laid, which means that 17.2% of the astelars in these
eggs must hatch and enter the sea. Whether or not we decide to move the eggs, the
wrong decision could cause the extinction of the astelars.
On Earth, many variables affect the probability that an egg will hatch, including the
temperature, oxygen concentration, and water concentration of the surrounding
environment. For this reason, parents lay their eggs in places where these conditions
enable their offspring to survive. Even small deviations from ideal conditions can kill an
embryo before hatching—a catastrophe we must avoid in the Intergalactic Wildlife
Sanctuary.
You measured temperatures at two beaches: the beach where astelars laid their eggs
(Beach 1), and a beach to which we might move those eggs (Beach 2). First, we will
explore how temperature affects the probability that an egg will hatch, also called
hatching success
(%). Then, we will determine which beach has temperatures that should
enable more eggs to hatch.
The following steps will enable us to answer the question "Would more astelars hatch at
the mean temperature on Beach 1 or at the mean temperature Beach 2?":
Step 1: Anticipate your analysis
: Determine what you should observe if the temperature
on Beach 1 or Beach 2 was better for astelars. This step will help us identify the evidence
we need to build an argument in Step 3.
Step 2: Predict whether more astelars will hatch at the mean temperature on Beach 1
or at the mean temperature on Beach 2
: Use the temperatures of each beach to predict
hatching success. First, we will use data for past breeding seasons to relate hatching
success to the mean temperature. Then, we will estimate the mean temperature of each
beach in the current breeding season. Finally, we will use the linear relationship between
temperature and hatching success to predict the hatching success in the current
breeding season. This step provides the evidence needed to build an argument on
whether or not to move the astelar eggs.
Step 3: Weigh the evidence and conclude whether the mean temperature is better for
astelar eggs on Beach 1 or Beach 2
: Construct an argument to answer the question
“Would more astelars hatch at the mean temperature on Beach 1 or at the mean
temperature Beach 2?” Your argument should draw on your calculations in Steps 1 and 2
Step 1: Anticipate your analysis.
Excel tutorial:
●
#12 Making a Scatter Plot
;
#12 Making a Scatter Plot transcript
To construct a sound argument, one must anticipate the evidence needed to support a
claim. For Appendix 1, you can choose between two claims:
1)
more astelars will hatch at
the mean temperature on Beach 1 than at the mean temperature on Beach 2; or
2)
more
astelars will hatch at the mean temperature on Beach 2 than at the mean temperature on
Beach 1.
To determine which claim is better supported, we must know the relationship between
temperature and hatching success. Relationships enable us to predict how the value of
an independent variable affects the value of a dependent variable, such as how
temperature affects hatching success. Step 2 of this appendix explains the mathematics
and interpretation of a linear relationship. But first, we’ll use our general understanding of
linear relationships to anticipate the evidence needed to support each of the potential
claims. Please remember that the independent variable is the cause. Its value is
independent of other variables in your study. The dependent variable is the effect. Its
value depends on changes in the independent variable.
In each of the figures below (A-C), the y-axis represents the hatching success (%), with
higher values indicating a higher probability of hatching. The x-axis represents the
temperature of the sand, with higher values indicating a higher temperature. The solid
black line represents a hypothetical linear model of the relationship between
temperature and hatching success. The mean temperature of each beach is marked by
an arrow pointing to the x-axis.
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1.
Select the figure that best illustrates the pattern that one should observe
if more
astelars hatch at the mean temperature on Beach 1 than at the mean
temperature on Beach 2.
Excel tutorial:
●
#12 Making a Scatter Plot
;
#12 Making a Scatter Plot transcript
a.
Figure A
b.
Figure B
c.
Figure C
2.
Explain your answer to the previous question. Minimally your explanation should
include one factor that caused you to select one figure over the others.
a.
Figure C demonstrates that a higher percentage of hatching success
occurred in lower soil temperatures. The mean temperature of Beach 1 has
a lower temperature than Beach 2 and has a higher hatching success rate.
Step 2: Predict whether more astelars will hatch at the mean
temperature on Beach 1 or at the mean temperature on Beach 2.
Excel tutorial:
●
#13 Modeling a Linear Relationship
;
#13 Modeling a Linear Relationship transcript
Because Beach 2 is cooler than Beach 1, we might expect more astelars to hatch if we
move the eggs. But how many more astelars would hatch on Beach 2 than would hatch
on Beach 1? Would moving the eggs to Beach 2 outweigh the risks of keeping the eggs
on Beach 1?
To answer these questions, we must know the relationship between temperature and
hatching success. Relationships enable us to predict how the value of an independent
variable, such as temperature, affects the value of a dependent variable, such as the
probability of hatching.
This relationship can be described by the following linear model:
𝜇
=
aT
+
b
𝜇
(greek letter mu) is the mean (or expected) probability of hatching for a given
temperature;
a
is the slope of the linear relationship between temperature and the mean
hatching success;
T
is the temperature (°C); and
b
is the intercept of the linear
relationship between temperature and the mean hatching success.
[Note: This linear
model is the same as the one commonly used in algebra courses, where y = ax + b,
except that we are using the symbols T and
𝜇
instead of x and y, respectively.]
A linear model is an extension of a normal probability distribution in which the mean of
the dependent variable depends on the value of the independent variable. As with any
normal probability distribution, we can also calculate a standard deviation, which tells us
our uncertainty about the mean.
Microsoft Excel has functions for estimating the slope, intercept, and standard deviation
of a linear model.
To estimate the
slope
, use the
slope
function:
=slope(dependent observations, independent observations),
To estimate the
intercept
, use the
intercept
function:
=intercept(dependent observations, independent observations),
To estimate the
standard deviation
, use the
steyx
function:
=steyx(dependent observations, independent observations).
For example, consider the five pairs of observations in the spreadsheet below:
A
B
Observation #
Independent variable
Dependent variable
1
5
25
2
3
64
3
7
24
4
4
57
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5
6
35
For these data, you would calculate the slope, intercept, and standard deviation of the
linear relationship by entering the following functions in Excel:
Slope =slope(B2:B6, A2:A6)
Intercept =intercept(B2:B6, A2:A6)
Standard deviation =steyx(B2:B6, A2:A6)
When specifying cells in the function, be sure to specify the columns for the variables in
the correct order (dependent variable first, independent variable second). Also, be sure
that the starting row and ending row of each variable spans all of the data. In this
example, we would specify rows 2 through 6 because the data occur only in those rows.
If we round to the nearest whole number, we should obtain the following values for slope,
intercept, and standard deviation of the linear relationship in the example above:
Slope =slope(B2:B6, A2:A6) =
-10
Intercept =intercept(B2:B6, A2:A6) =
92
Standard deviation =steyx(B2:B6, A2:A6) =
10
Please remember that probabilities can be represented in decimal format (0.87) or
percent format (87%). You should be able to interchangeably represent probabilities in
both formats.
Directions
: For questions 3-6, download the Excel file titled, “Act II: A Big Decision
Workbook” from your Canvas assignment and refer to the sheets titled “Question 3” and
“Q4-6 Mean Temp Analysis.” This data set is a record of the mean temperature and the
probability of astelar hatching success in past breeding seasons (N = 37 breeding
seasons). Use Excel for calculations, modeling, and graphing.
3.
Use the sheet (tab) labeled “Question 3” to create a plot of a linear relationship
between the mean temperature and hatching success in past breeding seasons.
This plot should follow the formatting guidelines listed below.
Excel tutorials:
●
#9 Saving Plots as Images
;
#9 Saving Plots as Images transcript
●
#12 Making a Scatter Plot
;
#12 Making a Scatter Plot transcript
Formatting Instructions
●
Chart type: X Y (Scatter)
●
Quick layout: Layout 1 - Scatter
●
Chart title: Hatching Success and Mean Soil Temperature Relationship
●
Y-axis title: “Hatching Success (%)”; Font size = 18
●
Y-axis numbers: Font size = 14
●
X-axis title: “Mean Soil Temperature (°C)”; Font size 18
●
X-axis numbers: Font size = 14
●
Provide a trendline illustrating the relationship between the mean soil
temperature and hatching success
4.
Estimate the slope of the linear relationship between the mean temperature and
the probability of hatching in past breeding seasons. Round your answer to the
nearest tenth of a decimal place; for example, if you calculate the value as 0.385,
round your answer to 0.4.
Excel tutorial:
●
#13 Modeling a Linear Relationship
;
#13 Modeling a Linear Relationship
transcript
Slope = -2.8
5.
Estimate the intercept of the linear relationship between the mean temperature
and the probability of hatching in past breeding seasons. Round your answer to
the nearest hundredth of a decimal place; for example, if you calculate the value
as 0.385, round your answer to 0.39.
Excel tutorial:
●
#13 Modeling a Linear Relationship
;
#13 Modeling a Linear Relationship
transcript
Intercept = 126
6.
Estimate the standard deviation of the linear relationship between the mean
temperature and hatching success in past breeding seasons. Round your answer
to the nearest hundredth of a decimal place; for example, if you calculate the
value as 0.385, round your answer to 0.39. Do not round intermediate
calculations.
Excel tutorial:
●
#13 Modeling a Linear Relationship
;
#13 Modeling a Linear Relationship
transcript
Standard deviation = 2.74
Now that you know the slope of and intercept of the linear relationship, you can predict
hatching success at any temperature. Specifically, we want to know the probability of
hatching at the mean temperature of each beach. Therefore, we must first estimate the
mean temperature of each beach. Then, we can enter the mean temperature as the
independent variable in our linear model and predict the probability of hatching.
For instructions about how to estimate the mean of a normal probability distribution, refer
back to Step 2 in Appendix 1 of the Act 1 Mission Memo (Excel tutorial
#10 Estimating
Parameters of a Normal Probability Distribution
;
#10 Estimating Parameters of a Normal
Probability Distribution transcript
).
Directions
: For questions 7-8, use the Act II: A Big Decision Workbook and refer to the
sheet labeled, “Q7-8 Temp at Nest Depth Analysis,” containing the temperatures on
Beach 1 and Beach 2, measured at the mean depth of an astelar nest. Use these data to
calculate the mean temperature of each beach.
Excel tutorial:
●
#10 Estimating Parameters of a Normal Probability Distribution
;
#10 Estimating
Parameters of a Normal Probability Distribution transcript
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7.
Use a normal probability distribution to estimate the mean temperature on Beach
1. Round your answer to the nearest tenth of a decimal place; for example, if you
calculate the value as 0.35, round your answer to 0.4. Do not round intermediate
calculations.
Mean temperature on Beach 1 = 22.6
8.
Use a normal probability distribution to estimate the mean temperature on Beach
2. Round your answer to the nearest tenth of a decimal place; for example, if you
calculate the value as 0.35, round your answer to 0.4. Do not round intermediate
calculations.
Mean temperature on Beach 2 = 15.9
Use the linear model to calculate the expected probability of hatching at the mean
temperature of each beach (Excel tutorial
#14 Predicting the Value of a Variable from a
Linear Model
;
#14 Predicting the Value of a Variable from a Linear Model transcript
). For
example, assume that the relationship between temperature (
T
) and the expected
hatching success (
𝜇
) is
𝜇
=
aT
+ b
where
a
= 0.01 per °C
b
= 0.22
If the mean temperature of a beach were 10°C, you would calculate the expected
hatching success as follows:
𝜇
= (0.01 °C
-1
)(10°C) + 0.22
= 0.10 + 0.22
= 0.32
Directions
: For questions 9-10, use the Act II: A Big Decision Workbook and refer to the
sheets titled “Q4-6 Mean Temp Analysis” and “Q9-10 Hatching Prob (Temp).” Use the
linear relationship you calculated in questions 4 and 5 to determine the expected
hatching success (
𝝁
) at the mean temperature of each beach (
T
).
Excel tutorial:
●
#14 Predicting the Value of a Variable from a Linear Model
;
#14 Predicting the
Value of a Variable from a Linear Model transcript
9.
Calculate the expected hatching success at Beach 1 when the mean temperature
on Beach 1 equals the value that you estimated in question 7. Express your answer
as a percentage, rounding the value to the nearest tenth of a decimal place. For
example, if you calculate a probability of 72.44, report a value of 72.4. Do not
round intermediate calculations.
Expected hatching success at Beach 1 = 62.1%
10. Calculate the expected hatching success at Beach 2 when the mean temperature
on Beach 2 equals the value that you estimated in question 8. Express your
answer as a percentage rounding the value to the nearest tenth of a decimal
place. For example, if you calculate a probability of 72.44, report a value of 72.4.
Do not round intermediate calculations.
Expected hatching success at Beach 2 = 81%
Step 3: Weigh the evidence and conclude whether the mean
temperature is better for astelar eggs on Beach 1 or Beach 2.
You are ready to answer the question “Would more astelars hatch at the mean
temperature on Beach 1 or at the mean temperature on Beach 2?” Be sure to provide
your reasoning, highlighting the relevant evidence supporting your claim.
11. Select the claim that is better supported by the evidence.
a.
More astelars will hatch at the mean temperature on Beach 1 than at the
mean temperature on Beach 2.
b.
More astelars will hatch at the mean temperature on Beach 2 than at the
mean temperature on Beach 1.
c.
The same number of astelars will hatch at the mean temperature on Beach
1 and the mean temperature on Beach 2.
12. Summarize the evidence that supports your claim, including how you determined
whether more astelars would hatch at Beach 1 or Beach 2. Use quantitative
evidence when possible.
a.
After calculating the percentage of hatching success for both Beach 1 and
Beach 2, it can be concluded that more astelars would hatch at Beach 2.
Using the mean temperature and hatching success of each beach, and
using prior knowledge that cooler temperatures results in higher hatching
success, the results show that Beach 2 would see more astelars hatching.
Appendix 2
Would more astelars hatch at the mean concentration of water
and oxygen on Beach 1 or at the mean concentration of oxygen
or water Beach 2?
In addition to temperature, you measured the concentrations of water and oxygen at the
two beaches. Now that we have quantified the thermal effects of moving the eggs to
Beach 2 vs keeping them on Beach 1, we must determine whether the concentrations of
oxygen or water on Beach 2 will be worse for astelars than the concentrations on Beach
1.
We will follow three steps to answer the question “Would more astelars hatch at the
mean concentration of water (or oxygen) on Beach 1 or Beach 2?”:
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Step 1: Anticipate your analysis
: Determine what you should observe if the mean
concentration of water (or oxygen) is better for astelar eggs on Beach 1 or Beach 2. This
step will help us identify the evidence needed to build an argument in Step 3.
Step 2: Predict whether more astelars will hatch at the mean concentration of oxygen
(or water) on Beach 1 or at the mean concentration of oxygen (or water) on Beach 2
:
Use the concentrations of oxygen and water at each beach to predict hatching success.
First, we will use data for past breeding seasons to relate the probability of hatching to
the mean concentration of oxygen or water. Then, we will estimate the mean
concentration of oxygen and water at each beach in the current breeding season. Finally,
we will use these relationships to predict the probability of hatching in the current
breeding season. This step provides the evidence needed to build an argument in Step
3, when we will conclude whether the mean concentrations of oxygen and water on
Beach 1 or Beach 2 would enable more astelars to hatch.
Step 3: Weigh the evidence and conclude whether the mean concentrations of oxygen
and water are better for astelar eggs on Beach 1 or Beach 2
: Construct an argument to
answer the question “Would more astelars hatch at the mean concentrations of oxygen
and water on Beach 1 or at the mean concentrations of oxygen and water on Beach 2?”
Your argument should draw on your calculations in Steps 1 and 2.
Step 1: Anticipate your analysis.
Excel tutorial:
●
#12 Making a Scatter Plot
;
#12 Making a Scatter Plot transcript
To construct a sound argument, one must anticipate the evidence needed to support a
claim. In this step, we will use our knowledge of linear relationships to anticipate the
evidence needed to support each of the potential claims. Because similar types of
evidence are needed to support a claim about oxygen concentration or water
concentration, we will focus on oxygen concentration here.
You can choose between two claims about oxygen concentration:
1)
more astelars will hatch at the mean oxygen concentration on Beach 1 than at the
mean oxygen concentration on Beach 2; or
2)
more astelars will hatch at the mean oxygen concentration on Beach 2 than at the
mean oxygen concentration on Beach 1.
To determine which claim is better supported, we must know the relationship between
the oxygen concentration in soil and the probability of hatching success.
In each of the figures below (A-C), the y-axis represents the probability of hatching, which
ranges from 0 to 1; a greater value indicates a greater probability of hatching. The x-axis
represents the oxygen concentration of the soil, with a greater value indicating a greater
oxygen concentration. The solid black line represents a hypothetical linear model of the
relationship between oxygen concentration and the probability of hatching. The mean
oxygen concentration of each beach is marked by an arrow pointing to the x-axis.
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13. Select the figure that best illustrates the pattern that one should observe
if more
astelars hatch at the mean oxygen concentration on Beach 2 than at the mean
oxygen concentration on Beach 1.
a.
Figure A
b.
Figure B
c.
Figure C
14. Explain your answer to the previous question. Minimally your explanation should
include one factor that caused you to select one figure over the others.
a.
Figure B shows that a higher soil concentration means a higher percentage
for hatching success. The mean soil concentration for Beach 2 is higher
than Beach 1, and is therefore more likely to hatch successfully.
Step 2: Predict whether more astelars will hatch at the mean
concentration of oxygen or water on Beach 1 or at the mean
concentration of oxygen or water on Beach 2.
Excel tutorials:
●
#12 Making a Scatter Plot
;
#12 Making a Scatter Plot transcript
●
#13 Modeling a Linear Relationship
;
#13 Modeling a Linear Relationship transcript
●
#14 Predicting the Value of a Variable from a Linear Model
;
#14 Predicting the
Value of a Variable from a Linear Model transcript
On Earth, the concentrations of water and oxygen affect the hatching success of eggs.
While you were measuring these variables on the two beaches, I used data from past
breeding seasons to model the relationship between the mean water concentration of a
nest and the hatching success of eggs. Similarly, I modeled the relationship between the
mean oxygen concentration and the probability of hatching.
Using these relationships, we can determine which beach has the physical conditions to
enable more astelars to hatch.
Part 1: Oxygen concentration
The plot below shows the relationship between oxygen concentration and hatching
success.
Given the chart of linear relationship above, you can use the data given in the “DATA-
Soil Conditions Hatching” tab of the Act II: A Big Decision Workbook to determine slope
and intercept to predict the probability of hatching at any oxygen concentration.
Specifically, we want to know the expected probability of hatching at the mean oxygen
concentration of each beach. Therefore, we must enter the mean oxygen as the
independent variable in our linear model and calculate the probability of hatching
success as the dependent variable.
Using the data you collected in the sanctuary, I estimated the mean and standard
deviation of oxygen concentration for each beach:
●
Beach 1
has a mean oxygen concentration of 12.9% and a standard deviation of
0.5%.
●
Beach 2
has a mean oxygen concentration of 12.0% and a standard deviation of
0.2%
Directions
: For questions 15-16, use the Act II: A Big Decision Workbook and refer to the
sheet titled “Q15-16 Hatching Prob (Oxygen).” Use the linear relationship to calculate the
expected probability of hatching at the mean oxygen concentration of each beach. For
instructions about how to use a linear relationship to calculate the expected probability of
hatching, see Step 2 in Appendix 1 of this Mission Memo.
Excel tutorial:
●
#14 Predicting the Value of a Variable from a Linear Model
;
#14 Predicting the
Value of a Variable from a Linear Model transcript
15. Calculate the expected probability of hatching on Beach 1 when the mean oxygen
concentration on Beach 1 equals 12.9%. Round your answer to the nearest tenth of
a decimal place; for example, if you calculate the value as 38.425%, round your
answer to 38.4%. Do not round intermediate calculations.
Expected hatching success on Beach 1 = 70.9%
16. Calculate the expected probability of hatching on Beach 2 when the mean oxygen
concentration on Beach 2 equals 12.0%. Round your answer to the nearest tenth
of a decimal place; for example, if you calculate the value as 38.425%, round your
answer to 38.4%. Do not round intermediate calculations.
Expected hatching success on Beach 2 = 64.4%
Part 2: Water concentration
The plot below shows the relationship between water concentration and hatching
success.
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Having modeled the linear relationship between water concentration and the probability
of hatching and having estimated the slope, intercept, and standard deviation of this
model, we are ready to figure out which beach would be better for the eggs.
To do this, we must estimate the expected probability of hatching at each beach given its
mean water concentration. Using the data that you collected in the sanctuary, I estimated
the mean and standard deviation of water concentration for each beach:
●
Beach 1
has a mean water concentration of 9.2% and a standard deviation of
0.9%.
●
Beach 2
has a mean water concentration of 13.2% and a standard deviation of
2.1%.
Directions
: For questions 17 and 18, use the Act II: A Big Decision Workbook and refer to
the sheet titled “Q17-18 Hatching Prob (Water)” to calculate the linear relationship
between water concentration and hatching success. Then, determine the expected
probability of hatching at the mean water concentration of each beach. For instructions
about how to use a linear relationship to calculate the expected probability of hatching,
see Step 2 in Appendix 1 of this Mission Memo
Excel tutorial:
●
#14 Predicting the Value of a Variable from a Linear Model
;
#14 Predicting the
Value of a Variable from a Linear Model transcript
17. Calculate the expected probability of hatching on Beach 1 when the mean water
concentration equals 9.2%. Express your answer as a decimal, rounded to the
nearest tenth of a decimal place; for example, if you calculate the value as
38.425%, round your answer to 38.4%.
Expected hatching success on Beach 1 = 74.4%
18. Calculate the expected hatching success on Beach 2 when the mean water
concentration equals 13.2% Express your answer as a decimal, rounded to the
nearest tenth of a decimal place; for example, if you calculate the value as
38.425%, round your answer to 38.4%. Do not round intermediate calculations.
Expected hatching success on Beach 2 = 74.3%
Step 3: Weigh the evidence and conclude whether the mean
oxygen (or water) concentration is better for astelar eggs on
Beach 1 or Beach 2.
We’re now ready to answer the question “Would more astelars hatch at the mean oxygen
(or water) concentration on Beach 1 or Beach 2?” Be sure to provide your reasoning,
highlighting the relevant evidence supporting your claim.
Part 1: Oxygen concentration
19. Select the claim that is better supported by the evidence.
a.
More astelars will hatch at the mean oxygen concentration on Beach 1 than
at the mean oxygen concentration on Beach 2.
b.
More astelars will hatch at the mean oxygen concentration on Beach 2 than
at the mean oxygen concentration on Beach 1.
c.
The same number of astelars will hatch at the mean water concentration on
Beach 1 and the mean oxygen concentration on Beach 2.
20.Summarize the evidence that supports your claim, including how you determined
where more astelars would hatch at Beach 1 or 2. Use quantitative evidence when
possible.
Part 2: Water concentration
21. Select the claim that is better supported by the evidence.
a.
More astelars will hatch at the mean water concentration on Beach 1 than at
the mean water concentration on Beach 2.
b.
More astelars will hatch at the mean water concentration on Beach 2 than
at the mean water concentration on Beach 1.
c.
The same number of astelars will hatch at the mean water concentration on
Beach 1 and the mean water concentration on Beach 2.
22.Summarize the evidence that supports your claim, including how you determined
where more astelars would hatch at Beach 1 or 2. Use quantitative evidence when
possible.
a.
After calculating the percentages of successful hatching at each beach, it
can be determined that Beach 1 would hatch more astelars. The
percentage of successful hatchings from the water concentration for Beach
1 was 74.4%, whereas the percentage for Beach 2 was 74.3%, making only
a 0.1% difference. The calculations for the oxygen concentration, however,
showed that the percentage for successful hatching on Beach 1 was 70.9%,
and the percentage for Beach 2 was 64.4%, showing a much larger
difference.
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Appendix 3
Considering the physical and biological conditions of each
beach, should we leave the eggs at Beach 1 or move them to
Beach 2?
For the population of astelars to persist, at least 381 pups must survive each season.
Given that 2217 eggs were laid this season, at least 17.2% of the astelars in these eggs
must hatch and enter the sea. Because predators will eliminate some pups before they
reach the sea, much more than 17.2% of the eggs must hatch for 17.2% of the pups to
enter the sea. To be safe, we should aim for the greatest possible hatching success.
Thanks to your careful analysis, we know the expected hatching success at each beach,
given the expected temperature, water concentration, and oxygen concentration. Now,
we can use this information to decide whether hatching success would be greater under
the physical conditions on Beach 1 or Beach 2.
Still, physical conditions are not the only variables that affect the survival of astelar eggs.
Certain biological variables, such as the number of predators, are also important. A major
predator of astelars, the red tippers, occur at both beaches. However, we discovered an
additional threat at Beach 2—digworms.
Digworms are large burrowing creatures that occur on Beach 2, but not on Beach 1.
Although digworms do not eat astelar eggs, a burrowing digworm can easily crush an
astelar egg, killing the embryo inside. Consequently, our decision to move the astelars to
Beach 2 should be informed by the potential impact of this biological variable on
hatching success.
Follow these three steps to determine whether more astelars would hatch on Beach 1 or
Beach 2:
Step 1: Determine which physical variable has the greatest impact on hatching
success:
You measured three physical variables—temperature, water, and oxygen—on
each beach. Unfortunately, the linear model that you know how to build can only
incorporate one independent variable. Therefore, you must choose the variable that has
the greatest impact on hatching success. This step would enable us to decide how many
eggs would survive the physical conditions of the beach, before we account for the
impact of digworms in Step 2.
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Step 2: Determine how much the digworms will decrease hatching success on Beach
2:
Here, you will reduce the expected hatching success on Beach 2 to account for the
impact of digworms. The hatching success for Beach 1 will remain the same as calculated
in Step 1, because no digworms occur on this beach. This step provides the evidence
needed to build an argument in Step 3, when we will answer the question “Should we
leave the eggs on Beach 1 or move them to Beach 2?”
Step 3: Weigh the evidence and conclude whether we should leave the astelars on
Beach 1 or move them to Beach 2:
Construct an argument to answer the question
“Should we leave the eggs at Beach 1 or move them to Beach 2?” Your argument will
draw on your answers and calculations in Steps 1 and 2.
Step 1: Determine which physical variable has the greatest
impact on hatching success.
Excel tutorial:
●
#14 Predicting the Value of a Variable from a Linear Model
;
#14 Predicting the
Value of a Variable from a Linear Model transcript
In Appendices 1 and 2, you used a linear model to calculate the expected hatching
success, given the value of a dependent variable (the mean temperature, water
concentration, or oxygen concentration). Now, you must decide which of the three
physical variables has the greatest impact on hatching success.
Given how curious humans can be you're probably wondering why we have to choose
one of the physical variables. Why can't we use all of them?
The answer illustrates the difference between reality and a model. In reality, the hatching
success of astelar eggs depends on the temperature, oxygen concentration,
and
water
concentration of the environment. In our model, however, we can only include one
independent variable. For example, the independent variable on the x-axis of a linear
model could be either temperature or oxygen concentration, but not both. Biologists
have more complex models that deal with multiple independent variables; however, we
haven't learned how to use such models yet. Therefore, we have to use a model that
focuses on the most important variable.
To identify the most important physical variable, let’s review the expected hatching
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successes that you calculated in Appendices 1 and 2.
23.In the table below, provide the expected hatching success for each beach given
the mean temperature, water concentration, and oxygen concentrations on each
beach, based on the linear model for each physical variable. Recall that you
calculated these values in Appendices 1 and 2, so refer back to these sections to
complete the table. Be sure to report all values a decimal between 0 and 1,
rounded to the nearest tenth of a decimal place. For example, if the calculated
value were 0.382, you should have rounded this value to 0.4.
Expected probability of
hatching on Beach 1
(nesting beach)
Expected probability of
hatching on Beach 2
(alternative beach)
Mean temperature
(°C)
62.1%
81%
Mean oxygen
concentration (%)
70.9%
64.4%
Mean water
concentration (%)
74.4%
74.3%
As the table in question 23 shows, we know the expected hatching success for each
physical variable measured at the two beaches. Now, we must choose one of these
variables to consider when deciding how the physical environment will affect hatching
success.
How should we decide which physical variable has the greatest impact on hatching
success? The answer is simple...subtraction!
For each variable, subtract the expected probability of hatching on Beach 1 from the
expected probability of hatching on Beach 2. That calculation will tell you how different
the hatching success would be between the two beaches. The most important
environmental variable is the one that would cause the greatest difference in the
probability of hatching between beaches.
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For example, imagine two beaches called Beach A and Beach B. The mean temperature
on Beach A is 16°C, and the mean temperature on Beach B is 13°C.
Furthermore, assume that the linear relationship between temperature (
T
) and the
expected probability of hatching (
𝜇
) is
𝜇
=
aT
+
b
where the slope (
a
) equals -0.01 per °C, and the intercept (
b
) equals 1.00.
Using this linear model, the expected probability of hatching on Beach A (
𝜇
A
) would be
𝜇
A
= (-0.01 per °C) (16°C) + 1.00
= 0.84 (or 84.0%)
and the expected probability of hatching on Beach B (
𝜇
B
) would be
𝜇
B
= (-0.01 per °C) (13°C) + 1.00
= 0.87 (or 87.0%)
To summarize, we should expect 0.84 (or 84%) of the eggs to hatch on Beach A, but 0.87
(or 87%) of the eggs to hatch on Beach B.
Therefore, the difference in hatching success between beaches would be 0.84 - 0.87 =
-0.03. Or (84.0% - 87.0% = 3%)
Because we are interested in the magnitude of the difference in hatching success (rather
than its sign, positive or negative), we need to take the absolute value of the difference.
To obtain the absolute value of a number, you convert a negative value to a positive one.
For example, because the difference between the probabilities is were -0.034, we should
report an absolute value to 0.03.
The absolute value of a positive number is the same number. Therefore, if the value were
0.03, the absolute value would also be 0.03. Likewise, the absolute value of 0.00 would
be 0.00.
Directions
: Use the probabilities of hatching that you stated in question 23 (above) to
answer questions 24-26.
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24.Calculate the absolute value of the difference between the probability of hatching
on Beach 1 and the probability of hatching on Beach 2, caused by the mean
temperature of each beach. Express your answer as a percentage (%), rounded to
the nearest tenth of a decimal place; for example, if you calculate the value as
3.8218, round to 3.8. Do not round intermediate calculations.
Absolute value of the difference between the pr6.obabilities of hatching, caused
by the mean temperature of each beach = 18.9%
25.Calculate the absolute value of the difference between the probability of hatching
on Beach 1 and the probability of hatching on Beach 2, caused by the mean
oxygen concentration at each beach. Express your answer as a percentage (%),
rounded to the nearest tenth of a decimal place; for example, if you calculate the
value as 3.8218, round to 3.8. Do not round intermediate calculations.h.
Absolute value of the difference between the probabilities of hatching, caused by
the mean oxygen concentration at each beach = 6.5%
26.Calculate the absolute value of the difference between the probability of hatching
on Beach 1 and the probability of hatching on Beach 2, caused by the mean water
concentration at each beach. Express your answer as a percentage (%), rounded
to the nearest tenth of a decimal place; for example, if you calculate the value as
3.8218, round to 3.8. Do not round intermediate calculations.
Absolute value of the difference between the probabilities of hatching, caused by
the mean water concentration at each beach = 0.1%
Remember that we want to select the physical variable that has the greatest impact on
hatching success. Select the variable that would cause the greatest absolute difference
in expected probability of hatching.
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27. Which physical variable has the greatest impact on the expected probability of
hatching?
a.
Temperature
b.
Oxygen concentration
c.
Water concentration
Step 2: Determine how much the digworms will decrease
hatching success on Beach 2.
Now that we’ve determined the impact of physical variables on hatching success, we're
ready to consider the impact of a major biological variable—the presence of digworms.
An adult digworm can easily crush an astelar egg while burrowing through the sand.
Based on my simulations, an astelar egg on Beach 2 would have a 27% chance of being
crushed by a digworm before hatching; that's a probability of 0.27. An astelar egg on
Beach 1 has no chance of being crushed by a digworm, because no digworms occur on
Beach 1.
Clearly, the presence of digworms will decrease hatching success on Beach 2. But would
the benefit of a lower mean temperature on Beach 2 more than compensate for the
presence of digworms? In other words, should we expect a greater hatching success on
Beach 2 despite the presence of digworms?
To answer this question, we need to calculate probability of hatching at each beach in a
way that accounts for two independent processes:
1)
surviving the physical conditions at the beach (temperature, oxygen, or water), and
2)
surviving the biological conditions at the beach (the burrowing of digworms).
Let's return to our example on Beach A and Beach B from Step 1 (see above). In this
example, the expected probability of hatching at the temperature on Beach A was 0.84
and the expected probability of hatching at the temperature on Beach B was 0.87.
Now, let's assume that no digworms occur at Beach A, but some digworms occur at
Beach B. Given the absence of digworms at Beach A, the probability of a digworm
crushing an astelar egg at Beach A equals 0.00 (0%). However, the presence of
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digworms at Beach B means that the probability of a digworm crushing an astelar egg at
Beach B equals 0.15 (15%), which means the probability of surviving the borrowing of
digworms equals 1.00 - 0.15, or 0.85 (85%).
How would we calculate the probability that an astelar survives both the temperature of
the beach and the burrowing of digworms? The key is that these two events occur
independently of each other; in other words, the probability of surviving a certain
temperature does not depend on the probability of surviving the burrowing of digworms.
If we want to know the probability that two independent events occur together, we use
the
product rule of probability
. According to the product rule, the probability of two
independent events occurring together, abbreviated as
, equals the product
𝑃(? 𝑎𝑛𝑑 ?)
of the probabilities of these events:
𝑃(? 𝑎𝑛𝑑 ?)
= 𝑃(?)∙𝑃(?)
Please remember that probabilities can be represented in decimal format (0.87) or
percent format (87%). You should be able to interchangeably represent probabilities in
both formats. For the product rule, if you use a hand calculator to do the product, you
should use the decimal format to perform the multiplication. If you use Excel, you need
to ensure that your values are formatted as percent values in the number formatting
toolbar.
Applying the product rule to our example, we can calculate the probability of an astelar
hatching on Beach A as follows:
= (84.4%) · (100%) = 0.84 (84%)
𝑃(? 𝑎𝑛𝑑 ?)
in which we multiply the probability of surviving the temperature at Beach A (0.84) by the
probability of surviving digworms (1.00, because no digworms occur on Beach A). You
should be able to interchangebly convert from the decimal probability (0.84) to the
percent probability (84%). When using Excel to multiply probabilities, please ensure that
your numbers are correctly formatted as percentages in the number formatting toolbar.
Similarly, we can calculate the probability of an astelar hatching on Beach B as follows:
= (86.8%) · (85%) = 0.74 (74%)
𝑃(? 𝑎𝑛𝑑 ?)
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in which we multiply the probability of surviving the temperature at Beach B (0.87) by the
probability of surviving digworms (0.85, because digworms occur on Beach B and have a
15% chance of crushing an astelar egg).
Although we chose temperature as the physical variable in this example, you should
choose whichever physical variable would cause the greatest difference in hatching
success between Beach 1 and Beach 2.
The probability of two or more events occurring together is referred to as a
joint
probability
. Using the product rule, calculate the joint probability that an astelar hatches
as the product of two independent probabilities:
1)
the probability that an astelar survives the mean temperature, oxygen
concentration,
or
water concentration of the beach (predicted with the linear
model in Appendix 1 or 2 relating the most important physical variable to hatching
success); and
2)
the probability that an astelar survives the burrowing of digworms (assumed to be
1.00 for Beach 1 and 0.73 for Beach 2).
Directions
: Use the product rule of probability to answer questions 28 through 29.
28.Calculate the joint probability that an astelar egg will hatch on Beach 1, accounting
for the impact of the most important physical variable (temperature, oxygen, or
water) and the impact of digworms. Express your answer as a percentage (%),
rounded to the nearest whole number; for example, if you calculate the value as
48.52%, round your answer to 49%. Do not round intermediate calculations.
Joint probability that an astelar egg will hatch on Beach 1, accounting for the
impact of the most important physical variable (temperature, oxygen, or water) and
the impact of digworms = 62%
29.Calculate the joint probability that an astelar egg will hatch on Beach 2,
accounting for the impact of the most important physical variable (temperature,
oxygen, or water) and the impact of digworms. Express your answer as a
percentage (%), rounded to the nearest whole number; for example, if you
calculate the value as 48.52%, round your answer to 49%. Do not round
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intermediate calculations.
Joint probability that an astelar egg will hatch on Beach 2, accounting for the
impact of the most important physical variable (temperature, oxygen, or water) and
the impact of digworms = 59%
Step 3: Weigh the evidence and conclude whether we should
leave the astelars on Beach 1 or move them to Beach 2.
Thanks to your efforts in Step 2 of Appendix 3, we are now ready to answer the question
“Should leave the astelars on Beach 1 or move them to Beach 2?” Be sure to provide
your reasoning, highlighting the relevant evidence supporting your claim.
Remember that our decision should be based on the joint probability you calculated in
Step 2 of Appendix 3 that an astelar hatches on Beach 1 and Beach 2 given: 1) the
probability that an astelar survives the mean temperature, oxygen concentration,
or
water
concentration of the beach (predicted with the linear model in Appendix 1 or 2 relating
the most important physical variable to hatching success); and 2) the probability that an
astelar survives the burrowing of digworms (assumed to be 1.00 for Beach 1 and 0.73 for
Beach 2).
30.Select the claim that is better supported by the evidence.
a.
We should leave the astelars on Beach 1
b.
We should move the astelars to Beach 2
31. Summarize the evidence that supports your claim, including how you determined
whether more astelars would hatch on Beach 1 or 2. Use quantitative evidence
when possible.
a.
Given all of the evidence, the conclusion is that the astelars should remain
on Beach 1. This is supported by the fact that Beach 1 has a higher
expected hatching success than Beach 2, given all of the most important
physical variables - in this case, temperature - and the presence of
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digworms. The joint probabilities of Beach 1 and Beach 2 are 62% and 59%
respectively, leaving a 18.9% gap that makes a large difference.
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