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```Problem Solving and
Quantitative Skills in AP Biology
Use this document as
a resource to help
guide you.
Up until now, there is strong possibility that
you have you used the word hypothesis in
only one way.
Now we have a null hypothesis that we
either reject or fail to reject
And an alternate hypothesis that we
reject or fail to reject.
Example: ( on a very simplified scale)
My null hypothesis at this point:
There is no difference in the statistics skill
level between members of our class that
reviewed statistics during the summer and
those that didn’t review.
Now, if we come back to school and there is
a significant difference we would state:
Our null hypothesis has been rejected.
GET IT??
Please don’t let “math” be the enemy,
Yes, you will struggle a little, but that is how
you learn. I have found that all the tech tricks
with graphs, programs, functions, etc, actually
make it kinda cool.
 We will work on this during the entire school year during our
investigation time.
 This material, along with other supporting statistics resources
will remain available to help you
each time.
We will build our
expertise over time,
this will not
happen overnight.
Statistics really helps
makes things clear.
You have probably asked yourself after many labs during
your lifetime,
“What did our results mean?”,
“How do we know if it really has any meaning at all?”,
“How could I test this to see if my experiment had any real
meaning?”
I have noticed sometimes students get bogged down in the
method, equipment, etc. And after the lab have no idea what
they just did. I will try my best to introduce lab procedures
vs investigations separately so that doesn’t happen.
For some it is a new Vocabulary
 http://www.fiserscience.com/statistics.html
 – I have a whole page of words I think you might need as a
reference place.
 Statistics is the science of conducting studies to collect, organize,
summarize, analyze, and draw conclusions from data.
Descriptive statistics consists of the collection,
organization, summarization, and presentation of data.
Inferential statistics consists of generalizing from samples to
populations, performing estimations and hypothesis tests,
determining relationships among variables, and making
predictions.
Probability: the chance of the event occurring. Use in
inferential statistics.
Descriptive Statistics
More vocabulary:
 A population consists of all subjects (human or
otherwise) that are being studied.
A sample is a group of subjects selected from a
population.
Discrete variables assume values that can be counted.
Continuous variables can assume an infinite number of
values between any two specific values. They are
obtained by measuring. They often include fractions and
decimals
Qualitative vs. Quantitative
Marital status of nurses in hospital.
Time it takes to run a marathon.
Weights of lobsters in a tank in a restaurant.
Colors of automobiles in a shopping center
parking lot.
Ounces of ice cream in ice cream float.
Capacity of the NFL football stadiums.
Ages of people living in a personal care home.
Discrete vs. Continuous
 Number of pizzas sold by Pizza Express each day.
 Relative humidity levels in operating rooms at
local hospitals.
 Number of bananas in a bunch at several local
supermarkets.
 Lifetime (in hours) of 15 Ipod batteries.
 Weights of the backpacks of first graders on a
school bus.
 Number of students each day who make
appointments with a math tutor at a local
college.
 Blood pressures of runners in a marathon.
1. Graphing
Effective graphs convey
summary or descriptive
statistics as part of the
display.
It is not practical to think you
can count everything.
Most of the time, that would simply
be impossible.
 So we take samples.
 Statistics takes samples and make inferences
that represent the true population.
Sampling follows predictable patterns.
It’s theoretical.
Mean, Median, Mode
(Central tendencies)
Mean means
average, so the
first time you
think I’m
mean….just know
I’m average. =)
To get the mean:
Standard Deviation

The average value away from the average.
Math is fun. Good website.
http://www.mathsisfun.com/data/standard-deviation.html
There are some simple standard deviation problems at
The bottom of the page.
Categories
The data you collect to will generally fall into three Categories:
 Parametric: Normal data. Fits normal curve or distribution.
Generally in a decimal form. The bell curve!
 Examples: plant height, body temperature, response rate.
(The prefix para means side by side. So close in comparison, not like
an outlier. If para means normal, does paranormal activity mean
normal normal activity? )
 Nonparametric: not normal. Does not fit a normal
distribution. May include outliers. Can be ordered. Can be
big, medium, small (qualitative)
 Frequency or count data: How many fits in this category? Flies you
counted with normal wings, etc.
Questions to divide into 2 groups.
 Is A different than B?
 How are A and B correlated?
 To design a good investigation, the
type of data needed to answer your
question has to be decided during the
planning stages. What kind of
question is being asked and what
kind of data will be required to
answer it?
5 types of graphs
1. Bar Graphs
2. Scatter plots
3. Box and whiskers plot
4. Histograms
5. Line Graphs
College Board AP Investigations
that need a graph:
 1. Artificial Selection – AP exam 2014
 4. Diffusion and Osmosis
 5. Photosynthesis
 6. Cellular Respiration
 9. Biotechnology: Restriction Enzyme Analysis of DNA
 12. Fruit Fly Behavior – AP exam 2013
 13. Enzyme Activity
Bar Graphs
 Use to compare two samples of categorical or count
data. You will need to compare the calculated means
with error bars. Example: qualitative: eye color.
Always use Sample Standard error bar. (Sample error of the
mean) (SEM)
Bar graph with only the
mean plotted.
Scatterplots
 Used to explore associations between two variables
visually. One variable is measured against another. It is
looking for trends or associations. Plotting of individual
data points on an x-y plot.
Scatterplots
Sine wave-like
Bell-curved
Box-and-whisker plot
 Allows comparison of two samples of nonparametric data
(data that does not fit a normal distribution)
 Medians and quartiles
1. Lowest value, 2: Q1, 3: the
Median; 4. Q3, 5. highest value
Of set.
Histograms – use this as a
preliminary step to see shape.
 Frequency diagram. Used to display the distribution of data,
providing a representation of the central tendencies (mean,
Median, Mode) and the spread of the data. Can then see if
parametric or non.
 Involves measurement data and data distribution.
 Requires setting up “bins” – which are uniform range
intervals that cover the entire range of the data. Then the
number of measurements that fit in each bin (range of unit)
are counted and graphed. If enough measurements are
made, the data can show the normal distribution, or bellshaped distribution, on a histogram.
 Quantitative, uses averages, spreads from low to high;
continuous data such as time, or measurements; tenths
show minor changes.
Sample size should be at least 30.
Record High Temperatures
Class
Boundaries
Frequency
99.5104.5
2
104.5109.5
8
109.5114.5
18
114.5119.5
13
119.5124.5
7
124.5129.5
1
129.5134.5
1
Ipad app: Numbers
+ Create new spreadsheet: blank
Enter just as you see on table.
Grab all. Create chart. Add details.
Mail symbol. Open in other app then
Google Drive for home or print.
Sunny & Shady Leaves
Histogram - nonparametric
Line Graphs
Can use multiple variables.
Usually change over time;
line suggests trend
Good
Graph
reminders:
Title
Easy to
identify
lines and
bars
Clearly
labeled
with units.
X-axis independent variable
Y-axis dependent variable
Uniform intervals
More than one condition: different lines:style,color
Indicated on a key
Don’t start at origin if your data doesn’t.
Standard Error of the Means bars (SEM)
Taken from the Handbook of
Biological Statistics
Good Resource

http://udel.edu/~mcdonald/statintro.html

I find that a systematic, step-by-step approach is the best way to analyze biological data.
The statistical analysis of a biological experiment may be broken down into the following
steps:

Specify the biological question to be answered.

Put the question in the form of a biological null hypothesis and alternate hypothesis.

Put the question in the form of a statistical null hypothesis and alternate hypothesis.

Determine which variables are relevant to the question.

Determine what kind of variable each one is.

Design an experiment that controls or randomizes the confounding variables.

Based on the number of variables, the kind of variables, the expected fit to the parametric
assumptions, and the hypothesis to be tested, choose the best statistical test to use.

If possible, do a power analysis to determine a good sample size for the experiment.

Do the experiment.

Examine the data to see if it meets the assumptions of the statistical test you chose
(normality, homoscedasticity, etc.). If it doesn't, choose a more appropriate test.

Apply the chosen statistical test, and interpret the result.

Communicate your results effectively, usually with a graph or table.
Chapter 2: Data Analysis
 Labs using DATA ANALYSIS:
 1. ARTIFICIAL SELECTION – FRQ 2014
 2. MATHEMATICAL MODELING: HARDY-WEINBERY
 3. COMPARING DNA SEQUENCES TO UNDERSTAND
EVOLUTIONARY RELATIONSHIPS WITH BLAST
 4. CELL DIVISION: MITOSIS AND MEIOSIS
Data analysis
 Descriptive statistics: summarize data. Shows variation
in the data, standard errors, best-fit functions, and confidence
that sufficient data have been collected.
Inferential statistics: involves making conclusions, using
experimental data to infer parameters in the natural
population. Relies on probability.
Most measurements are continuous, infinite number over a
period of time. Examples: size, time, height, weight,
absorbance.
Others are:
Count data: qualitative, categorical, example: number of hairs,
etc. Requires different statistical tools.
Run through of the steps.
 Good idea to do this first, before you design your own
experiment.
 Most investigations will allow to have an initial runthrough with all of class doing the same thing.
Components of Data Analysis
Bias
How can we be certain if the statistic we use is giving an accurate
estimate of?
Think of what we want to estimate (the parameter) as the target and the
way we estimate it (the statistic) as the arrow.
Our statistic is called unbiased if it is
close to being on target. Our statistic is
called biased if it is not close to being on
target.
Bias
We determine bias of a statistic by comparing the center of its sampling
distribution to the parameter being estimated.
Sample means and proportions are both unbiased estimators.
Sample Standard Error
 Allows for inference of how sample mean matches up to the true
population.
 A distribution of the sample means helps define
boundaries of confidence in our sample.
∓ 1 SE describes 67% confidence range.
∓ 2 SE defines 95% certainty
Confidence limits contains the true population mean.
95% confident: larger range to contain the population mean.
(95% of the data found here)
67% confident: narrower range to contain the population mean.
SE – inference in which to draw conclusions
SD – just looking at data
95% is trade off for never being 100% sure.
Inference/Confidence Intervals/
Empirical Rule/ 67-95-99.7 Rule
Confidence Intervals: Use to determine if
two populations are different from one another.
Parametric or Nonparametric:
Time for a group activity.
Make a histogram.
What’s in your tool bag?
So, we made a histogram
Looked for our shape.
Parametric or nonparametric?.
 What type of graph will we use?
 If parametric, we will use our sample size, mean
standard deviation, and standard error to analyze.
If nonparametric, we will use medians, modes, quartiles,
and range . Often use a box and whiskers plot.
These are the tools we will use for each type to analyze and
describe it.
Error Bars
Another example:
Body Temperature
What kind of data has been
collected?
Explore the sample:
Is it parametric?
What type of graph?
What is in your tool bag?
Chapter 3: Hypothesis Testing
 Necessary in the following Investigations:
 1. Artificial Selection
 2. Mathematical Modeling: Hardy-Weinberg
 3. Comparing DNA: BLAST
 4. Diffusion and Osmosis
 5. Photosynthesis
 6. Cellular Respiration
 7. Cell Division: Mitosis and Meiosis
 8. Biotechnology: Bacterial Transformation
 9. Biotechnology: RE of DNA
 10. Energy Dynamics
 11. Transpiration
 12. Fruit-Fly Behavior
 13. Enzyme Activity
Hypothesis
 A hypothesis is a statement explaining that a causal
relationship exists between and underlying factor
(variable) and an observable phenomenon.
 You might propose a tentative explanation and this
would be called your working hypothesis.
 ABSOLUTE PROOF IS NOT POSSIBLE, SO WE FOCUS ON
TRYING TO REJECT A NULL HYPOTHESIS.
 A null hypothesis is a statement that the variable has no
causal relationship.
 An alternative to the null hypothesis would be that there
is a size difference, etc.
Crucial to understand.
We say we reject the null, not that the
investigation has proven the alternative
hypothesis.
A hypothesis is a testable statement
explaining some relationship between
cause and effect.
A prediction is a statement of what you
think will happen.
 It is good to have more than
one working hypotheses. The
investigator can remain more
open scientifically and not tied
to one single explanation.
Hypothesis Testing
Asks the question:
Is there something to these
measurements?
Or
Is the effect real?
Hypothesis testing
 The data analysis tools used, such as SE, mean,
variance, median, etc. are now used to calculate
comparative test statistics.
 These tests determine a probability and distribution of
the samples – the null distribution.
 A critical value of the probability of whether the results
or even more extreme results occur by chance alone, if
the null hypothesis is true (probability value, or p-value
of less than 5%)
4 possible outcomes of
Hypothesis Testing
5% critical values in Biology
Decision point in biology.
1% or 0.1% more critical.
Like life or death issues in
medical studies.
The suggested path for including
hypothesis testing:
 1. Choose a statistical test based on the question and
types of data. Next page table.
 2. State the null and alternative hypotheses
 3. Design and carry out the investigation.
 4. Conduct a data analysis, and present graphical and
tabular summaries of the data.
 5. Carry out the statistical tests. Include the sample
size, test statistic chosen, and p-values in the data
reports.
 6. Make a conclusion, always stating the amount of
evidence in terms of the alternative hypothesis.
Examples of Hypothesis Testing
 t– test
 Chi-Square
 Trends – linear regression and r
1. Establish null and alternate
H0 null hypothesis – There is no difference between the two true populations.
H1 alternate hypothesis – There is a significant difference between the two
population means.
2. Establish critical value – p value of 5%
3. Decision table for test
4. P-value
5. Analyze p value and critical value. Reject null. Or state results support
alternate hypothesis. Do not accept the alternative hypothesis.
T-test
Example two groups independent from each other:
Unpaired T test : determines how different two sample
populations are from one another by comparing means and
errors. The College Board manual doesn’t go into how to run
the test outside of an Excel program, so I don’t foresee this as
a problem to calculate, maybe just an understanding required.
One tail and two tail refer to both sides of the distribution or
just one.
The t-test can be used to determine how different two sample
populations are from one another by comparing means and
standard errors.
One sample T-test formula:
(sample mean-population mean)
The sample standard error
Then the t-test value is taken to a t-test table and you need
degrees of freedom and t value to estimate the p value.
d= n-1
Comparing two samples and
testing for differences
 1. Construct two histograms – to get a visual of
distribution, overlap, shape.
 2. Calculate descriptive statistics.
 3. Display in bar graphs
 4. Test for differences.
 5. Two sample t-test
Chi-Square Results
 Makes comparison between the data collected and what
we expect.
 We will use this test most often. Genetics, animal
behavior.
A p-value of 0.05 means that there is a 5% chance that the difference
between the observed and the expected data is a random difference and a
95% chance that the difference is real and repeatable. A significant
difference. If the p-value is greater than 0.05 we would fail to reject to the
null hypothesis.
This is the table given on your formula sheet. We need to
wrap your head around the difference of the tables and how the
application to the null is still the same.
Trends and Associations
Best-fit line, linear regression, and r
 Is there a general trend?
 A regression analysis graphs a best fit line that summarizes all the
points into a single linear equation.
Positive correlation: heart rate goes up when temp goes up. Trend
line is created from how far each point is from the mean x value and
from the mean y value.
The r-value provides an estimate of the degree of correlation.
r-value of 1 is positive 1:1 correlation
R-value of -1 is negative correlation
r-value of .25 equal R2 of 0.064 which means 6.4% of
variation (difference) is due to temp and not a strong correlation.
R-value does not imply causal relationship.
Two tailed t for this since 2 variables.
P value on last graph p 0.04. which means 96% confidence that heart
rate and body temperature are correlated, although not strongly so.
Chapter 4: Mathematical Modeling
 Labs using Mathematical Modeling
 1. Artifical Selection
 2. Mathematical Modeling: Hardy-Weinberg
 3. Comparing DNA: BLAST
 9. Biotechnology: RE Analysis of DNA
 10. Energy Dynamics
 11. Transpiration
Mathematical Modeling
 The process of creating mathematical or computer-based
representations of the structure and interactions of
complex biological systems.
 What can we measure?
 What should we measure?
 What are the relevant variables?
 What are the simplest informative models that we can
build?
 Deterministic models are calculated with fixed
probabilities.
 Stochastic models use a random number generator to
create a model that has a variable outcome and
contingencies, much as normal biological systems do.
 Example is Investigation 2: Mathematical modeling:
Hardy-Weinberg.
Example: Simple Model of
Population Growth
 Maybe we might try this sometime: Building a model.
FRQ writing
If you use statistics to answer the
question, use statistics to justify
your answer.
More resources:
http://www.radford.edu/jkell/statsgr
aphs.pdf
http://www.biologyforlife.com/graph
ing.html
Make some time for this!!
 www.randi.org
 www.badscience.net
 MythBusters
 Ted Talks “Battling Bad Science”
 http://www.ted.com/talks/ben_goldacre_battling_bad_s
cience
Data sets
for demo.
avg. →
set A
set B
set C
set D
14
17
17
3
15
18
18
6
13
15
15
7
17
20
20
12
15
18
18
9
12
13
13
13
11
9
9
12
15
18
18
8
6
16
16
15
10
7
7
10
12.8
15.1
15.1
9.5
```