Transcript AP Test Prep – Part 2

AP Statistics Test Prep
Table of Contents
Part 1: Test Overview
Part 2: Test Content
Part 3: 2013 Released Qs
Part 4: Specific Concepts
Part 5: Stats in a Nutshell
Part 6: Example Questions
Part 7: Formulas
Part 8: Calculator Shortcuts
Part 9: Study Guide by Chapter
© Richard Newton 2013
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AP Statistics Test Prep
Part 1: Test Overview
AP Test Prep – Part 1: Test Overview
AP Stats Test is May 10th at Noon!
AP Statistics Exam Study Materials!
Barron’s “AP Statistics”
Princeton Review’s “Cracking the AP”
Kaplan’s “5 Steps to a 5”
Part 1: Multiple Choice
50% of overall Grade
40 Questions
90 Minutes
Part 2: Free Response
50% of overall Grade
6 Questions (5 and 1)
90 Minutes
Breakdown:
Each Multiple Choice Question is 1.25% of the overall score
Each of the first 5 free response questions is 7.5% of the overall score
The final question is 12.5% of the overall score
There is no penalty for guessing.
AP Statistics Test Prep
Part 2: Test Content
AP Test Prep – Part 2: Test Content
The course can be divided into 4 major topics:
1. Exploratory Analysis
2. Planning a Study
3. Probability
4. Inference
You will see 1 Free Response on each of these and 1 “Investigative
Task” which is basically where you hit all of them in a single
question.
Being able to look at a multiple choice or free response question and
put it into 1 of these categories is one of the best skills you can learn.
AP Test Prep – Part 2: Test Content
Exploratory Analysis
Ch.1) Categorical vs. Quantitative Data
Types of Graphs
Describe the Distribution: SOCS
Ch.2) Density Curves (Area = 1)
Normal Curves
z-scores (Percentiles)
Testing Normal:
5 Number Summary Box Plot
Use histogram to check normal
68-95-99.7 Test
Normal Probability Plot
Ch.3) LSRL (y = a + bx, r, r2, residual)
Ch.4) Marginal Distributions
Conditional Distributions
Simpsons Paradox
Exponential Regression
Power Regression
- Look back at the various types of graphs and
be familiar with which can be used with
categorical vs. quantitative. You can expect to
see at least 2 questions on stem plots – possibly
with a back to back stem plot. You will see 2+
histograms as well. It is possible that you will
see an O-give but unlikely you would have to
convert it backward into a histogram.
- Make sure you can find the area under a
density curve if it is blocky. They’ll likely ask
you to check if a data set is normal.
- Make sure you’re able to create and interpret a
LSRL line – in particular, refresh your memory
on r, r2, and residuals!
- Marginal and Conditional Distributions just
means you’re able to interpret 2 way tables and
answer questions about the data.
AP Test Prep – Part 2: Test Content
Planning a Study
Ch.5) Designing Observational Studies
Stratified Random Sample
Multistage Sampling Design
Bias
Voluntary Response
Convenience Sampling
Leading Question
Non Response Bias
Response Bias
Designing Experiments
Matched Pairs
Block Design
Control and Placebo
Simulations
- If you split your observational study up by
strata, you are stratifying it.
- Be sure that you understand the specific
types of bias!
“The design is biased if it systematically favors
certain outcomes.”
- If you have each participant experience both
treatments (1 after the other), you’ve used a
Matched Pairs design. You could also attempt
to match each participant with a very similar
participant so that for each treatment, you
have similar experimental units. This would
be an attempt at a Matched Pairs design but I
don’t think they’ll ask about this.
- Make sure you can diagram experiments,
especially if they are blocked.
- Using a placebo is an attempt to create a
control group to which you can compare your
results.
- Use randomInt to conduct simulations.
AP Test Prep – Part 2: Test Content
Probability
Ch.6) Probability
And (*)vs. Or (+)
Replacement
without Replacement
Disjoint (not overlapping)
Non Disjoint (overlapping)
Ch.7) Discrete (integer columns)
Continuous
z-scores
Combining Means
Combining Variance
Ch.8) Binomial Distributions
Geometric Distributions
Permutations
Combinations
Counting Principal
- Remember the Bridge and Moat!
- Most sequential probability questions are
done with replacement (unless it says
otherwise).
- Diagram – tree or chart – probability
questions wherever possible. Most can be
reasoned out with a diagram and fractions
alone.
- Remember that discrete involves integer
columns which makes them easier to work
with but that means > and ≥ are different.
- Continuous is composed of infinitely small
columns and unless it’s rectangular or
normal, you don’t have the tools to calculate
the areas.
- BinomPDF is a single column while
BinomCDF is a single column and all the
columns to the left.
AP Test Prep – Part 2: Test Content
Inference
Ch.9) Sampling Distributions
Central Limit Theorem
Ch.10) Inferences when σ is known (z)
Type 1 Error Type 2 Error
Ch.11) Inferences when σ is unknown (t)
Ch.12) Proportions
1propZtest
2propZtest
Ch.13) Chi-Squared GOF
Chi-Squared Homogeneity
Ch.14) LSRL Significance Test
Confidence Intervals:
They ONLY want to see:
1. The interval (either notation is fine)
2. What the interval means
“We are 95% confident the true mean (or
proportion) is between ____ and ____.”
- The sampling distribution is normal because
of the Central Limit theorem if the rules of
thumb are met. This is the principal which
allows us to calculate probabilities and
estimate standard deviation.
ALWAYS: Talk about the conditions!!
1) Is the SRS from the population of interest?
2) How big is the sample size? Is that big
enough to do the test you’re doing?
You will get marked off big time if you don’t!
- You will only see t tests (see chapter 11
middle board), prop tests (ch.12), chi-squared
(ch.13), and LSRL tests (ch.14).
- You can do this entire part with a calculator
and get full credit! Just check the conditions,
explain the hypothesis, show the t score and
probability, and then clearly state if the Null
or Alternative is supported!
AP Test Prep – Part 2: Test Content
Other topics that are not tested:
Covariance (combining variance when they are not independent)
Baye’s Theorem
Pooled t-tests
Chapter 14.2 – LSRL Confidence Intervals
Read the
Question
Re-Read the
Question
Exploratory
Analysis
Categorical vs. Quantitative
SOCS
Center vs. Spread
LSRL
y  a  bx
y  ab
x
y  10 x
a
Designing
Studies
Significance Tests
and Confidence
Intervals
Z-Tests vs. T-Tests
Samples vs. Proportions
Confidence Intervals
Chi-Squared
LSRL
b
Observational Studies vs. Experiments
Simulations
Probability
Mutual Exclusivity
Independence vs. Dependence
And vs. Or
Discrete vs. Continuous
Binomial vs. Geometric
AP Statistics Test Prep
Part 3: 2013 Released Qs
AP Test Prep – Part 3: 2013 Released Qs
We are 95% confident that the true mean for lead levels in crows to be between 4.4157 and 5.3843 ppm
At first I thought I had to do a 1propZinterval,
but I realized since they gave me the mean
mean and standard deviation of a single
sample, I had to do a t-interval.
AP Test Prep – Part 3: 2013 Released Qs
Pop > 10n
Football game attendees might be more school spirited which might cause bias in the results and overestimate
the proportion which are satisfied with the appearance of the university buildings and grounds.
Using the list of 70,000 names, assign each student a number from 1 to 70,000, then have the number generator generate
500 unique random numbers from 1-70,000. The students who match those numbers are selected for the survey.
At first I thought this was saying the campuses were gender specific, but both campuses are mixed.
If the campuses are different, such that the results of the survey when stratified by location is expected to give
different results, then it would be important to survey each campus individually rather than by gender.
AP Test Prep – Part 3: 2013 Released Qs
Simple z-score question.
i)
z
x


850  840
7.9
 1.266
P  % 100  89.8%  10.2%
Combined mean = 840g
840g – 20g for the carton = 820g for the dozen eggs,
divided by 12 = 68.33 grams per egg
mean of X = 68.33g
ii) Combined standard deviation = 7.9g, so Combined variance = 62.41g
Container standard deviation = 1.7g, so variance = 2.89g
62.41 – 2.89 = 59.52g variance combined for 12 eggs
59.52 / 12 = 4.96g variance for a single egg
2.23g standard deviation for a single egg
standard deviation of X = 2.23g
AP Test Prep – Part 3: 2013 Released Qs
H0: No association
between age and
fruit/vegi
consumption.
Ha: There is an
association between
age and fruit/vegi
consumption.
Because we’re given a 2-way table, I know that I need to do a Chi-Squared
Test of Homogeneity. I need to first generate my Expected values. You can
see those below. Then I need to do a Chi-Squared Homogeneity T-test.
Yes, at a probability of 1.12% there is significant evidence at the 5% level to
support the alternative hypothesis. There appears to be an association
between age group and fruit/vegi consumption.
AP Test Prep – Part 3: 2013 Released Qs
This is an observational study, meaning that treatments were not applied.
Thus, lurking variables could be at work so we would not want to make any
conclusions about causation.
If the situation were replicated and made into an experiment, control groups
could be established to attempt to mitigate the impact of lurking variables.
This would allow the researchers to make stronger conclusions about
causation between meditation and blood pressure.
AP Test Prep – Part 3: 2013 Released Qs
b) There are a few issues –
#1, the sample size of 11 is too
small, it should be atleast 15
#2, To do a matched pairs
design like this and set the u of
the differences to zero for the
Null, you want to have the
same sample size
Values over here indicate the
proportion with high bp was
higher for non meditators
Values over here indicate the
proportion with high bp was
higher for meditators
The simulated distribution favors the left side, where meditators have a lower proportion of
individuals with high bp than non meditators. For this reason, it might be reasonable to conclude that
there is a relationship between meditation and blood pressure.
AP Test Prep – Part 3: 2013 Released Qs
AP Test Prep – Part 3: 2013 Released Qs
AP Test Prep – Part 3: 2013 Released Qs
AP Statistics Test Prep
Part 4: Specific Concepts
AP Test Prep – Part 4: Specific Concepts
Things that are important to remember:
Type 1 Error: Rejecting the Null when it’s actually true - P(type 1 error) = alpha
Type 2 Error: Accepting the Null when it’s actually false –
P(type 2 error) is difficult to calculate (and not on the test, thankfully)
Power = 1 – P (type 2 error)
High power is good – it means you’re unlikely to commit a type 2 error
Ways to increase power:
increase sample size or move alternative further away
Simulation: If you’re asked to explain how you would simulate something, make sure you
clearly identify what different number values will mean, example:
RandomInt(1,10)
1-3: Miss
4-10: make
10 trials, results are:
5, 3, 2, 8, 5, 10, 1, 1, 6, 7
4 misses, 6 makes
Experimental Result = 40% miss rate
Repeated this process 5 times and got: 40%, 60%, 10%, 30%, 40%
Mean Miss Rate after 5 trials of experiment: 36% miss rate
AP Test Prep – Part 4: Specific Concepts
Any t-test will fall into one of these categories:
1 Sample t test – Use Calculator, set μ = whatever was given
Matched Pairs – Use calculator (use 1 sample t test), set μ = zero
2 Sample T test – Use calculator, set μ1 = μ2
1 prop Z test – Use calculator, set p = whatever was given
2 prop Z test – Use calculator, set p1 = p2
Considerations:
SRS comes from population
of interest
N ≥ 15
SRS comes from population
of interest
Pop ≥ 10n
Np ≥ 10 or n(1-p) ≥10
X2 GOF – No shortcut in a TI83 - sum((L1-L2)2/L2)
X2 Homogeneity – use calculator, Matrix A and B – df = (r-1)(c-1)
SRS comes from population
of interest
LSRL – Use Calculator, df = n - 2
Observations are
independent
True relationship is linear
S the same everywhere
Residuals are normally
distributed
AP Test Prep – Part 4: Specific Concepts
Confidence Intervals
The formula is always the same:
 s 
C : x  t*

 n 
You can just use your calculator to calculate most of them. Just remember that df = n – 1
unless it’s LSRL and then its df = n – 2 (but you shouldn’t see a LSRL confidence interval).
Remember the definition of a confidence interval – it is very specific:
“If you conducted the sampling process multiple times and created many additional
confidence intervals, 95% of the created intervals should contain the true mean.”
Or:
“We are 95% confident that the true mean is between these two values.”
AP Test Prep – Part 4: Specific Concepts
Combining Means and Variances
Let’s say that you and your friend are ordering 2 items at a restaurant – a burrito and a
chicken salad. You know that the following about the time (in minutes) it takes to make
each item:
Burrito
Chicken Salad
mean
2
3
Combined mean is:
Combined standard deviation is:
standard deviation
1
2
AP Statistics Test Prep
Part 5: Stats in a Nutshell
AP Test Prep – Part 5: Stats in a Nutshell
AP Test Prep – Part 5: Stats in a Nutshell
AP Test Prep – Part 5: Stats in a Nutshell
AP Test Prep – Part 5: Stats in a Nutshell
AP Statistics Test Prep
Part 6: Example Questions
AP Test Prep – Part 6: Example Questions
AP Test Prep – Part 6: Example Questions
AP Test Prep – Part 6: Example Questions
AP Test Prep – Part 6: Example Questions
AP Test Prep – Part 6: Example Questions
AP Test Prep – Part 6: Example Questions
AP Test Prep – Part 6: Example Questions
AP Test Prep – Part 6: Example Questions
AP Test Prep – Part 6: Example Questions
AP Test Prep – Part 6: Example Questions
AP Test Prep – Part 6: Example Questions
AP Test Prep – Part 6: Example Questions
AP Test Prep – Part 6: Example Questions
AP Test Prep – Part 6: Example Questions
AP Test Prep – Part 6: Example Questions
AP Statistics Test Prep
Part 7: Formulas
AP Test Prep – Part 7: Formulas
AP Test Prep – Part 7: Formulas
AP Test Prep – Part 7: Formulas
AP Test Prep – Part 7: Formulas
AP Statistics Test Prep
Part 8: Calculator Shortcuts
AP Test Prep – Part 8: Calculator Shortcuts
AP Test Prep – Part 8: Calculator Shortcuts
AP Test Prep – Part 8: Calculator Shortcuts
AP Test Prep – Part 8: Calculator Shortcuts
AP Test Prep – Part 8: Calculator Shortcuts
AP Test Prep – Part 8: Calculator Shortcuts
AP Statistics Test Prep
Part 9: Study Guide by Chapter
AP Test Prep – Part 9: Study Guide by Chapter
Chapter 1 – Exploring Data
AP Test Prep – Part 9: Study Guide by Chapter
Chapter 2 – Normal Distributions – z scores
AP Test Prep – Part 9: Study Guide by Chapter
Chapter 3 – Examining Relationships - LSRL
AP Test Prep – Part 9: Study Guide by Chapter
Chapter 4 – 2 Variable Data
AP Test Prep – Part 9: Study Guide by Chapter
Chapter 5 – Producing Data
AP Test Prep – Part 9: Study Guide by Chapter
Chapter 5 continued – Producing Data
AP Test Prep – Part 9: Study Guide by Chapter
Chapter 6 - Probability
AP Test Prep – Part 9: Study Guide by Chapter
Chapter 7 – Random Variables
AP Test Prep – Part 9: Study Guide by Chapter
Chapter 8 – Binomial and Geometric Distributions
AP Test Prep – Part 9: Study Guide by Chapter
Chapter 9 – Sampling Distributions
AP Test Prep – Part 9: Study Guide by Chapter
Chapter 10 – Z Tests
AP Test Prep – Part 9: Study Guide by Chapter
Chapter 11 – T tests
AP Test Prep – Part 9: Study Guide by Chapter
Chapter 11 continued – T tests
AP Test Prep – Part 9: Study Guide by Chapter
Chapter 12 – Prop tests
AP Test Prep – Part 9: Study Guide by Chapter
Chapter 13 – Chi Squared
AP Test Prep – Part 9: Study Guide by Chapter
Chapter 14 – LSRL test