Download Chapter 10 Guided Reading Notes

<NOTE: Content rearranged for 2015-2016, but not dates>
Day 1: 10.1 Comparing Two Proportions
Read p609 and articles to introduce big ideas of chapter (comparing two groups), discuss projects
Read 612–615
What is meant by “the sampling distribution of the difference between two proportions”?
It describes the possible values of pˆ1  pˆ 2 and how often they will occur, where p̂1 and p̂2 are
the proportions of successes from two independent random samples or two treatments in a
randomized experiment.
What are the shape, center, and spread of the sampling distribution of pˆ1  pˆ 2 ? Are there any conditions
that need to be met?
See box on page 614
Discuss what the SD measures—how far the estimated difference in proportions will be from the
true difference in proportions, on average.
Note that the Normal condition isn’t met for the basketball experiment.
Alternate Example: Nathan and Kyle both work for the Department of Motor Vehicles, but they live in
different states. In Nathan’s state, 80% of the registered cars are made by American manufacturers. In
Kyle’s state, only 60% of the registered cars are made by American manufacturers. Nathan selects a
random sample of 100 cars in his state and Kyle selects a random sample of 70 cars in his state. Let
pˆ N  pˆ K be the difference in the sample proportion of cars made by American manufacturers.
(a) What is the shape of the sampling distribution of pˆ N  pˆ K ? Why?
(b) Find the mean of the sampling distribution. Show your work.
(c) Find the standard deviation of the sampling distribution. Show your work.
(a) Because 100(0.80) = 80, 100(1 – 0.80) = 20, 70(0.60) = 42, and 70(1 – 0.60) = 28 are all ≥ 10, the sampling
distribution of pˆ N  pˆ K is approximately Normal.
(b) The mean is  pˆ
N
 pˆ K
 0.80  0.60  0.20.
(c) Because there are at least 10(100) = 1000 cars in Nathan’s state and at least 10(70) = 700 cars in Kyle’s state,
the standard deviation is  pˆ
 pˆ K
N

0.80 1  0.80 
100

0.60 1  0.60 
= 0.0709.
70
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Read 616–619
mention projects, start thinking about ideas
What are the conditions for calculating a two-sample z interval for p1  p2 ?
See box on p616
Discuss splitting 1 sample into 2 groups—OK (happens in HW)
Reminder about the grid in back of book
What is the standard error of pˆ1  pˆ 2 ? How is this different than the standard deviation of pˆ1  pˆ 2 ?
What does this measure?
What is the formula for a two-sample z interval for p1  p2 ? Is this on the formula sheet?
Alternate Example: Gun Control
Have opinions changed about gun control? Gallup regularly asks random samples of U.S. adults their
opinion on a variety of issues. In a poll of 1011 U.S. adults in January 2013, 38% responded that they
“were dissatisfied with the nation’s gun laws and policies, and want them to be stricter.” In a similar
poll of 1011 adults in January 2012, only 25% agreed with this statement.
(a) Explain why we should use a confidence interval to estimate the change in opinion rather than just
saying that the percentage increased by 13 percentage points.
Because of sampling variability, the difference of 0.13 is unlikely to be correct.
(b) Use the results of these polls to construct and interpret a 90% confidence interval for the change in
the proportion of U.S. adults who would agree with the statement about gun laws.
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(c) Based on the interval, is there convincing evidence that opinions about gun control have changed?
Can you use your calculator for the Do step? Are there any drawbacks?
HW #23: page 629 (1–11 odd)
Day 2: Significance Tests for a Difference in Proportions
Read 619–624
On 620, make sure students can give the 2 explanations!
What are the conditions for conducting a two-sample z test for a difference in proportions?
Same as the CI!
When reading the hungry children example, talk about not accepting the null hypothesis
What standard error do we use for a 2 sample z test for a difference in proportions? What is the pooled
(combined) sample proportion? Why do we pool the sample proportions?
Highlight two-way table for example in book and make for subsequent examples.
List all three standard deviations/standard errors:
If we know the true proportions:
If we only have phats:
If we only have phats and must assume p1 = p2:
What is the test statistic for a two-sample z test for a difference in proportions? Is this on the formula
sheet? What does the test statistic measure?
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Alternate Example: Hearing loss
Are teenagers going deaf? In a study of 3000 randomly selected teenagers in 1988–1994, 15% showed
some hearing loss. In a similar study of 1800 teenagers in 2005–2006, 19.5% showed some hearing loss.
(These data are reported in Arizona Daily Star, August 18, 2010)
(a) Do these data give convincing evidence that the proportion of all teens with hearing loss has
increased?
(a) State: We will test H 0 : p1  p2 = 0 versus H a : p1  p2 > 0 at the 0.05 significance level
where p1 = the proportion of all teenagers with hearing loss in 2005–2006 and p2 = the
proportion of all teenagers with hearing loss in 1988–1994.
Plan: We should use a two-sample z test for p1  p2 if the conditions are satisfied.
 Random: The data came from independent random samples.
 10%: There were more than 10(1800) = 18,000 teenagers in 2005–2006 and 10(3000) =
30,000 teenagers in 1988–1994.
 Large Counts: n1 pˆ1 = 351, n1 1  pˆ1  = 1449, n2 pˆ 2 = 450, n2 1  pˆ 2  = 2550 all ≥ 10.
Do: pˆ C 
z=
450  351
= 0.167 (show two-way table!!)
3000  1800
 0.195  0.15   0
0.167 1  0.167 
1800

0.167 1  0.167 
= 4.05, P-value  0
3000
Conclude: Since the P-value is less than 0.05, we reject H 0 . We have convincing evidence that
the proportion of all teens with hearing loss has increased from 1988–1994 to 2005–2006.
(b) Between the two studies, Apple introduced the iPod. If the results of the test are statistically
significant, can we blame iPods for the increased hearing loss in teenagers?
(b) No. Since we didn’t do an experiment where we randomly assigned some teens to listen to
iPods and other teens to avoid listening to iPods, we cannot conclude that iPods are the cause. It
is possible that teens who listen to iPods also like to listen to music in their cars, and perhaps the
car stereos are causing the hearing loss.
Is it OK to use your calculator for the Do step? Are there any drawbacks?
Recommend that students use their calculator for the Do step in Chapter 10 (tests and intervals)
Remind students that X’s must be integers and not to use 2-samp z test
HW #24: page 631 (13–19 odd)
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Day 3: Inference for Experiments
Read 625–626
What mistake do students often make when defining parameters in experiments? How can you avoid it?
They use language that refers to the sample, such as the proportion who took… or the proportion
of people in the __ group….
It is better to use present or future tense!
Alternate Example: Cash for quitters
In an effort to reduce health care costs, General Motors sponsored a study to help employees stop
smoking. In the study, half of the subjects were randomly assigned to receive up to $750 for quitting
smoking for a year while the other half were simply encouraged to use traditional methods to stop
smoking. None of the 878 volunteers knew that there was a financial incentive when they signed up. At
the end of one year, 15% of those in the financial rewards group had quit smoking while only 5% in the
traditional group had quit smoking. Do the results of this study give convincing evidence that a financial
incentive helps people quit smoking compared to traditional methods? (These data are reported in
Arizona Daily Star, February 11, 2009)
Ask for Two explanations!
State: We will test H 0 : p1  p2 = 0 versus H a : p1  p2 > 0 at the 0.05 significance level
where p1 = the true quitting rate for employees like these who get a financial incentive to quit
smoking and p2 = the true quitting rate for employees like these who don’t get a financial
incentive to quit smoking.
Plan: We should use a two-sample z test for p1  p2 if the conditions are satisfied.
 Random: The treatments were randomly assigned.
 10%: not needed for experiments!
 Normal: n1 pˆ1 = 66, n1 1  pˆ1  = 373, n2 pˆ 2 = 22, n2 1  pˆ 2  = 417 are all at least 10.
Do: pˆ C 
66  22
= 0.100, z =
439  439
 0.15  0.05  0
0.11  0.1 0.11  0.1

= 4.94, P-value  0
439
439
Conclude: Since the P-value is less than 0.05, we reject H 0 . We have convincing evidence that
financial incentives help employees like these quit smoking.
Activity page 634–635
Read page 627
HW #25: page 630 (21, 23, 25–30)
Day 4: Midterm
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Day 5: Significance Tests for the Difference of Two Means
Read 634–639
Discuss Polyester Activity from previous lesson—we want to be able to anticipate
the shape, center, and spread of the sampling distribution of x1  x2 without a simulation.
What is meant by “the sampling distribution of the difference between two means”?
It describes the possible values of x1  x2 and how often they will occur,
What are the shape, center, and spread of the sampling distribution of x1  x2 ? Are there any conditions
that need to be met?
See box on page 638
Discuss what the SD measures—how far the estimated difference in means will be from the true
difference in means, on average.
Read 639–640
What is the standard error of x1  x2 ? Is this on the formula sheet? How do you interpret this value?
What is the formula for the two-sample t statistic? Is this on the formula sheet? What does it measure?
What are the conditions for performing inference about 1  2 ?
What distribution does the two-sample t statistic have? Why do we use a t statistic rather than a z
statistic? How do you calculate the degrees of freedom?
Please use the calculator for the do step!
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Read 644–649
Alternate Example: Leaking Helium
After buying many helium balloons only to see them deflate within a couple of days, Erin and Jenna
decided to test if helium-filled balloons deflate faster than air-filled balloons. To find out, they bought
60 balloons of the same type and randomly divided them into two piles of 30, filling the balloons in the
first pile with helium and the balloons in the second pile with air. Then, they measured the
circumference of each balloon immediately after being filled and again three days later. The average
decrease in circumference of the helium-filled balloons was 26.5 cm with a standard deviation of 1.92
cm. The average decrease of the air-filled balloons was 2.1 cm with a standard deviation of 2.79 cm.
(a) Why was it important that they used the same type of balloons? What is this called in experiments?
(b) Do these data provide convincing evidence that helium-filled balloons deflate faster than air-filled
balloons?
(c) Interpret the P-value you got in part (a) in the context of this study.
(a) Control: Using the same type of balloon eliminates a possible source of variability, making
the standard deviations smaller and increasing the power.
(b) State: We want to perform a test of H 0 :  H   A = 0 versus H a :  H   A > 0 at the 5%
level of significance where  H = the mean decrease in circumference of helium-filled balloons
after three days and  A = the mean decrease in circumference of air-filled balloons after three
days.
Plan: If the conditions are met, we will conduct a two-sample t test for  H   A .
 Random: The data came from two groups in a randomized experiment.
o 10%: Not necessary because there was no sampling without replacement.
 Normal/Large Sample: nH = 30 ≥ 30 and n A = 30 ≥ 30.
Do: Test statistic: t 
 26.5  2.1  0 = 39.46
1.922 2.792

30
30
P-value: With df = 30 – 1 = 29, P-value  0. Using technology: With df = 51.4, P-value  0.
Conclude: Because the P-value of approximately 0 is less than  = 0.05, we reject H 0 . There is
convincing evidence that helium-filled balloons deflate faster than air-filled balloons.
(c) Assuming that the mean decrease in circumference is the same for helium-filled and air-filled
balloons, there is an approximately 0 probability of getting a difference of 24.4 cm or more by
chance alone.
Is it OK to use your calculator for the Do step? Are there any drawbacks?
Always say “NO” to pooling when doing inference for means.
HW #26 page 654 (31, 33, 45, 51)
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Day 6: Confidence Intervals for the Difference of Two Means / Projects
Read 641–643
What is the formula for the two-sample t interval for 1  2 ? What are the conditions for this interval
to be valid? Is this formula on the formula sheet?
Alternate Example Chocolate Chips
Ashtyn and Olivia wanted to know if generic chocolate chip cookies have as many chocolate chips as
name-brand chocolate chip cookies, on average. To investigate, they randomly selected 10 bags of
Chips Ahoy cookies and 10 bags of Great Value cookies and randomly selected 1 cookie from each bag.
Then, they carefully broke apart each cookie and counted the number of chocolate chips in each. Here
are their results:
Chips Ahoy: 17, 19, 21, 16, 17, 18, 20, 21, 17, 18
Great Value: 22, 20, 14, 17, 21, 22, 15, 19, 26, 18
(a) Construct and interpret a 99% confidence interval for the difference in the mean number of chocolate
chips in Chips Ahoy and Great Value cookies.
(b) Does your interval provide convincing evidence that there is a difference in the mean number of
chocolate chips?
(a) State: We want to estimate CA  GV at the 99% confidence level where CA = the true mean
number of chocolate chips in Chips Ahoy cookies and GV = the true mean number of chocolate
chips in Great Value cookies.
Plan: If the conditions are met, we will calculate a two-sample t interval for CA  GV .
 Random: The data come from independent random samples.
o 10%: There are more than 10(10) = 100 Chips Ahoy cookies and more than
10(10) = 100 Great Value cookies.
 Normal/Large Sample: Because there is no obvious
skewness or outliers in the graphs below, it is safe
to use t procedures.
Do: For these data, xCA = 18.4, sCA = 1.78, xGV = 19.4,
sGV = 3.60. Using df = 10 – 1, the critical value for 99% confidence is t* = 3.250. Thus, the
1.782 3.602

= –1  4.13= (−5.13, 3.13). Using
10
10
technology: Using df = 13.145, (–4.81, 2.81).
Conclude: We are 99% confident that the interval from −4.81 to 2.81 captures the true
difference in the mean number of chocolate chips in Chips Ahoy and Great Value cookies.
(b) Because the interval includes 0, there is not convincing evidence that there is a difference in
the mean number of chocolate chips in Chips Ahoy and Great Value chocolate chip cookies.
confidence interval is 18.4  19.4   3.250
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When doing two-sample t procedures, should we pool the data to estimate a common standard
deviation? Is there any benefit? Are there any risks?
Benefit: slightly more power due to slightly larger df. Risk, if equal variance condition isn’t met,
then the p-value can be quite a bit off.
What about a two-sample test for a difference in proportions? Why do we pool for this test??
HW #27: page 654 (35, 37, 43, 47, 49) Note: 35 is about conditions only
Day 7: 10.2 Using t Procedures Wisely (half-day)
Read 650–651
Should you use two-sample t procedures with paired data? Why not? How can you know which
procedure to use?
Can you create a label for each row? I.e., would it mess up the data if I scrambled one of the
lists? If so, then the data are paired and should use the differences.
Illustrate using the caffeine and blood pressure from top of p645 (unpaired) and caffeine
dependence from p587 (paired)
Alternate Example: Testing with distractions
Suppose you are designing an experiment to determine if students perform better on tests when there are
no distractions, such as a teacher talking on the phone. You have access to two classrooms and 30
volunteers who are willing to participate in your experiment.
(a) Design an experiment so that a two-sample t test would be the appropriate inference method.
(a) On 15 index cards write “A” and on 15 index cards write “B.” Shuffle the cards and hand
them out at random to the 30 volunteers. All 30 subjects will take the same reading
comprehension test. Subjects who receive A cards will go to a classroom with no distractions,
and subjects who receive B cards will go to a classroom that will have the proctor talking on the
phone during the test. At the end of the experiment, compare the mean score for subjects in room
A with the mean score for subjects in room B.
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(b) Design an experiment so that a paired t test would be the appropriate inference method.
(b) Using the same procedure in part (a), divide the subjects into two rooms and give them the
same reading comprehension test. One room will be distraction free, and the other room will
have a proctor talking on the phone. Then, after a short break, give all 30 subjects a similar
reading comprehension test but have the distraction in the opposite room. At the end of the
experiment, calculate the difference in the two reading comprehension scores for each subject
and compare the mean difference with 0.
(c) Which experimental design is better? Explain.
(c) The experimental design in part (b) is better since it eliminates an important source of
variability—the reading comprehension skills of the individual subjects. MORE POWER!
(d) What is the purpose of random assignment in this experiment?
(d) To make sure one environment (treatment) isn’t favored by always going first (or second)
HW #28: page 659 (53–62)
Day 8: Review Chapter 10
FRAPPY!
HW #29: page 664 Chapter 10 Review Exercises
Day 9: Review Chapter 10
Discuss projects again
HW #30: page 666 Chapter 10 AP Practice Test
Day 10: Chapter 10 Test
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Final Project: Description
Purpose: The purpose of this project is for you to actually do statistics. You are to formulate a
statistical question, design a study to answer the question, conduct the study, collect the data, analyze the
data, and use statistical inference to answer the question. You are going to do it all!!
Topics: You may do your study on any topic, but you must be able to include all 6 steps listed above.
Make it interesting and note that degree of difficulty is part of the grade. No projects involving surveys
of humans will be allowed. Special bonus for projects that use regression!
Group Size: You may work alone or with a partner for this project.
Proposal (25 points): To get your project approved, you must be able to demonstrate how your study
will meet the requirements of the project. In other words, you need to clearly and completely
communicate your statistical question, the explanatory and response variables, your null and alternative
hypothesis, the test you will use to analyze the results, and how you will collect the data so the
conditions for inference will be satisfied. You must include at least two graphs of imaginary data: one
graph for a study when there is a clear significant answer to the research question of interest and another
graph for a study whose data are ambiguous. Also, make sure that your study will be safe and ethical if
you are using human subjects (anonymous, able to quit at any time, informed consent). If your proposal
isn’t approved, you must resubmit the proposal for partial credit until it is approved.
Poster (75 points):
The key to a good statistical poster is communication and organization. Make sure all components of
the poster are focused on answering the question of interest and that statistical vocabulary is used
correctly. The poster should include the following. It should not include the 4-steps.
 Title (in the form of a question).
 Introduction. In the introduction you should discuss what question you are trying to answer, why
you chose this topic, what your hypotheses are, and how you will analyze your data.
 Data Collection. In this section you will describe how you obtained your data. Be specific.
 Graphs, Summary Statistics and the Raw Data (if numerical). Make sure the graphs are well
labeled and easy to compare. Use the graphs and summary statistics to describe the evidence (if
any) for the alternative hypothesis. Then, clearly give the “two explanations” for why the data
seem to support the alternative hypothesis.
 Analysis. In this section, identify the inference procedure you used and discuss the conditions for
inference. Give the test statistic and p-value (with interpretation), along with the corresponding
confidence interval (with interpretation).
 Conclusion. In this section, you will state your conclusion. You should also discuss any possible
errors (e.g., Type I or Type II) or limitations to your conclusion, what you could do to improve
the study next time, and any other critical reflections.
 Live action pictures of your data collection in progress.
Presentation: You will be required to give a 5 minute oral presentation to the class.
EXTRA CREDIT: Write up your project for submission to the ASA Project Competition. There are
great prizes as well! See http://www.amstat.org/education/posterprojects/index.cfm for more details.
DUE DATES: Proposal: _________
Poster/Oral: ____________
Note: Late work will lost 20% per class period.
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Final Project: Rubric
Final Project
Introduction
Data Collection
Graphs and
Summary
Statistics
Analysis
Conclusions
Overall
Presentation/
Communication
4 = Complete
 Describes the context of the
research
 Has a clearly stated question of
interest
 Clearly defines the parameter of
interest and states correct
hypotheses
 Question of interest is of
appropriate difficulty
 Method of data collection is
clearly described
 Includes appropriate
randomization
 Describes efforts to reduce bias,
variability, confounding
 Quantity of data collected is
appropriate
3 = Substantial
 Introduces the context
of the research and has
a specific question of
interest
 Has correct parameter/
hypotheses OR has
appropriate difficulty
 Appropriate graphs are included
 Graphs are neat, clearly labeled,
and easy to compare
 Appropriate summary statistics
are included
 Preliminary answer with two
explanations are provided
 Appropriate graphs
and summary statistics
are included
 Graphs are neat,
clearly labeled, and
easy to compare or
preliminary answer
provided and
discussed
 Correct inference
procedure is chosen
 Lacks justification,
lacks interpretation, or
makes a calculation
error
 Correct inference procedure is
chosen
 Use of inference procedure is
justified
 Test statistic, P-value and
confidence interval are
calculated correctly
 P-value and confidence interval
is interpreted correctly
 Uses P-value to correctly answer
question of interest
 Discusses what inferences are
appropriate based on study
design
 Shows good evidence of critical
reflection (discusses possible
errors, limitations, etc.)
 Clear, holistic understanding of
the project
 Poster is well organized, easy to
read, and visually appealing
 Statistical vocabulary is used
correctly
 Oral presentation is organized
 Method of data
collection is clearly
described
 Some effort is made to
incorporate principles
of good data collection
 Quantity of data is
appropriate
 Makes a correct
conclusion
 Discusses what
inferences are
appropriate
 Shows some evidence
of critical reflection
 Clear, holistic
understanding of the
project
 Statistical vocabulary
is used correctly
 Poster or oral is
unorganized or has
other problems
2 = Developing
 Introduces the
context of the
research and has
a specific
question of
interest OR has
question of
interest and
hypotheses
 Method of data
collection is
described
 Some effort is
made to
incorporate
principles of
good data
collection
 Graphs and
summary
statistics are
included
1 = Minimal
 Briefly describes
the context of
the research
 Correct inference
procedure is
chosen
 Some
calculations or
interpretations
are correct
 Inference
procedure is
attempted
 Makes a partially
correct
conclusion (such
as accepting
null).
 Shows some
evidence of
critical reflection
 Poster is not well
done or
communication is
poor
 Makes a
conclusion
 Some evidence
of data
collection
 Graphs or
summary
statistics are
included
 Communication
and organization
are very poor
Note: A score of 0 is possible in each category.
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