statistical questions non-statistical questions
TRANSCRIPT
Microsoft Word - Quarter 2 BookletWhat is a Statistical
Question?
What types of questions do we investigate in statistics? The sheet you are given has two different types of
questions: statistical and non-statistical. Cut out the questions and sort them into two different groups according
to a characteristic they have in common.
1. Describe the criteria you used to sort the questions? What is the main difference between your two
groups of questions?
2. How can we tell the difference between a statistical question and a non-statistical question?
3. Attach 3 each type of question to the table below to use as examples of statistical and non-statistical
questions. In the last box of each column, write your own examples of a statistical and a non-statistical
question.
Researchers at Columbia University randomly selected 1000 teenagers in the United States for a survey.
According to an ABC News article about the research, “Teenagers who eat with their families at least five times a
week are more likely to get better grades in school.”
4. What is the statistical question that the researchers were trying to answer?
5. Identify the population and sample.
6. Is this an observational study or an experiment? Explain.
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Call your name two or three times in a row
Such a funny thing for me to try to explain
How I'm feeling and my pride is the one to blame
'Cuz I know I don't understand
Just how your love can do what no one else can
Got me looking so crazy right now, your love's
Got me looking so crazy right now (in love)
Got me looking so crazy right now, your touch
Got me looking so crazy right now (your touch)
Got me hoping you'll page me right now, your kiss
Got me hoping you'll save me right now
Looking so crazy in love's
Got me looking, got me looking so crazy in love
When I talk to my friends so quietly
Who he think he is? Look at what you did to me
Tennis shoes, don't even need to buy a new dress
If you ain't there ain't nobody else to impress
The way that you know what I thought I knew
It's the beat my heart skips when I'm with you
But I still don't understand
Just how the love your doing no one else can
I'm Looking so crazy in love's
Got me looking, got me looking so crazy in love
Got me looking, so crazy, my baby
I'm not myself, lately I'm foolish, I don't do this
I've been playing myself, baby I don't care
'Cuz your love's got the best of me
And baby you're making a fool of me
You got me sprung and I don't care who sees
'Cuz baby you got me, you got me, so crazy baby
HEY!
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1. Quickly circle a random sample of 5 words from the lyrics to Crazy in Love on the previous page. Write
them below. How many letters in each word?
2. What is the average word length of your sample? ________.
3. Add your average to the class’ dotplot. Copy the class dotplot below.
4. Find a new sample of 5 words using a random number generator. Add your average to the class’ dotplot.
Copy the class dotplot below.
5. How is the dotplot from #4 different than the dotplot for #3? Which do you think is a better estimator of
the true mean word length?
6. What do you think the true mean word length is for “Crazy in Love”?
7. It is known that Beyonce wrote the lyrics for all of the Destiny’s child songs. The average word length for
these songs is 3.64 letters. Based on your samples, do you have good evidence that Beyonce did not write
the lyrics for “Crazy in Love”. Explain.
5
In June 2008 Parade magazine posed the following question: “Should drivers be banned from using all cell
phones?” Readers were encouraged to vote online at www.parade.com. The July 13, 2008, issue of Parade
reported the results: 2407 (85%) said “Yes” and 410 (15%) said “No.”
8. What type of sample did the Parade survey obtain?
9. Explain why this sampling method is biased.
10. Is 85% likely to be greater than or less than the percentage of all adults who believe that cell-phone use
while driving should be banned? Why?
To help eliminate bias, a reporter from Parade decides she will go out and ask people in person if they think
drivers should be banned from using cell phones. She lives close to the local high school so she goes to the
parking lot at 3:00 pm and asks the first 100 people she sees.
11. What type of sample did the reporter obtain?
12. Explain why this sampling method is biased.
13. How could Parade magazine avoid the bias described above?
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Voluntary Response
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Simple Random Samples (SRS) 1. Copy the class dotplot from your notes from yesterday (the random sample, not the convenience sample).
2. Describe in words the process that we used yesterday to select a random sample. Be detailed.
3. Describe in words how we could use the “names from a hat” method to take a random sample. Be
detailed.
4. Take a random sample of 10 words from Crazy in Love. Find the mean of your sample and put it on the
class’ dotplot. Copy the dotplot below.
5. What happens when we increase the sample size?
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What proportion of students at your school use Twitter? To find out, you decide to survey a simple random
sample of students from your school. In your sample of 50 students, 76% say they use Twitter.
6. Describe how you could select an SRS of 50 students using a random number generator.
7. Will your sample result (76%) be exactly the same as the true population proportion for all students?
Explain.
8. Which would be more likely to yield a sample result closer to the true population value: an SRS of 50
students or an SRS of 100 students? Explain.
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Important IdeasImportant IdeasImportant IdeasImportant Ideas How to create a SRS with slips of paper
How to create a SRS with a Random Number Generator
Sampling Variability
Margin of Error
An SRS was conducted to find out how much television students watch on the weekend. Twenty-five students
were selected and asked “How many hours of TV did you watch this weekend?” Their responses are listed below.
5, 4, 3, 7, 5, 2, 5, 4, 8, 3, 6, 6, 5, 4, 6, 7, 5, 5, 3, 4, 6, 5, 7, 6, 4
1. What kind of data is this, categorical or quantitative?
2. Use your calculator. What is the mean? Do you expect that this mean is the same as the true population
mean? Explain.
Because of sampling variability, we know that if we continue to sample students in groups of 25 we will get
different means. Some will be above, and some will be below our sample mean of 5. To get a better idea of
what this looks like, we will use an applet to simulate taking 30 samples and finding each mean.
3. In the applet, we will scroll down to Perform Inference and choose “Simulate sample mean” from the
drop-down menu. Add 30 samples and Perform Simulation. Sketch the dotplot below.
4. Draw and label vertical lines on your dotplot to mark each of the following:
a. The mean of 5.
b. One standard deviation above and below the mean of 5. (You can find the SD listed under the dot plot.)
c. Two standard deviations above and below the mean of 5.
5. What percentage of the sample means are within two standard deviations above or below the mean?
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6. Write an interval that includes all the data two standard deviations above and below the mean.
Everything we have done so far has been to estimate a margin of error for a sample mean. We can also find a
margin of error for a sample proportion. Try the example below.
How many students text during class? A student created an anonymous survey asking students whether or not
they text in class. Of the 50 students who responded, 64% said they text in class.
7. Is this a categorical or quantitative variable?
8. How many students responded yes? How many said no?
We now use the applet to enter the data in 1 Categorical Variable. Enter the category names (“Yes” and “No”)
with the number of students who answered each. Click Begin analysis.
Scroll down to Perform Inference and choose “Simulate sample proportion” from the drop-down menu. Keep
“Yes” as the category to indicate as success and change the third drop-down menu to the observed proportion.
Note: Leave the Hypothesized proportion blank. Add 100 samples.
9. What is the standard deviation of the sample proportions?
10. Calculate the margin of error.
11. Interpret the margin of error.
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Many people can roll their tongues, but some can’t. Javier is interested in determining the proportion of students
at his school that can roll their tongue. In a random sample of 100 students, Javier determines that 70 can roll their
tongue. The dotplot below shows the proportion of people who can roll their tongue in each of 500 random
samples of size 100 from a population where 70% can roll their tongue.
Distribution of Simulated Proportion
z samples Mean SD
500 0.701 0.048
12. Use the results of the simulation to approximate the margin of error for Javier’s estimate of the proportion
of students at his school that can roll their tongue.
13. Interpret the margin of error.
14. Javier’s biology teacher claims that 75% of people can roll their tongue. According to Javier’s study, is this
claim plausible? Explain.
15. Explain how Javier could decrease the margin of error.
16. for the two streams similar or different? Justify your answer.
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How to decrease margin of error
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1. Which one of the following is a valid statistical
question?
b. How many points did LeBron James score in
the 2014 NBA playoffs?
customer service number, how long will you
be on hold?
on groceries last month?
Albuquerque residents support the local minor
league baseball team, the Isotopes. She stands
outside the stadium before a game and
interviews the first 20 people who enter the
stadium, asking them to rate their enthusiasm for
the team on a 1 (lowest) to 5 (highest) scale.
Which of the following best describes the results
of this survey?
be some sampling variability, but we can
expect it to produce quite accurate results.
b. This is a random sample, but the size of the
sample is too small to produce reliable
results.
likely to underestimate support for the team.
d. This is a convenience sample and is likely to
overestimate the level of support for the
team.
questions next to many of its news stories. Simply
click your response to join the sample. One recent
question was “Do you plan to diet this year?”
More than 30,000 people responded, with 68%
saying “Yes.” You can conclude that
a. about 68% of Americans planned to diet.
b. the poll used a convenience sample, so the
results tell us little about the population of all
adults.
results tell us little about the population of all
adults.
conclusion.
survey results, Sam and Joshan randomly
selected 60 students from their school’s student
list and divided them at random into two groups
of 30. By email, they asked the students the
following questions about a nearby convenience
store, the Quick Shop:
think the prices at the Quick Shop are
fair, given that the typical markup is
25%?
of the Quick Shop, do you think the
prices are fair, given that the typical
markup is 25%?
Mark-Up Convenience Total
Total 30 30 60
study? Justify your answer.
responded “fair” in each of the two treatment
groups and the difference in proportions
(Convenience – Markup).
university wants to evaluate the satisfaction level
of students living in one of the dormitories on
campus. She plans to interview a random sample
of 50 of the 816 residents of the building.
a. Explain how you could use a random
number generator to select a simple random
sample of 50 residents of the dormitory.
b. During each interview, the director asks, “If
you could make the choice again, would you
still choose this dorm?” In the sample of 50,
66% of the residents answered “Yes.” If the
director took a second random sample of 50
students, would this percent be the same?
Explain.
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Sampling and Surveys Directions:
1. Decide on a question you would like to ask 20 Park High School students. Determine if this question is a
mean or a proportion question.
2. Next, you should determine a type of bias you’d like to create when you ask students these questions.
3. Create a scenario (that is do-able) where you will ask 20 students these questions in a biased manner.
4. Next, collect your answers in this biased manner. What was the conclusion from your study?
5. You will now collect answers from 20 students using a SRS.
6. Compare your answers. How did your answer change?
7. Create a poster showing the question, describing method of data collection and comparing the answers
received from each type of data collection.
What is the question you will ask?
Is this a mean or proportion question? Explain.
Which bias will you be creating? How.
Collect your biased data here.
Collect your unbiased data here.
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with integrity.
time in places around the school
they were not supposed to be.
Student did not collect data and/or
was a nuisance to others while
collecting data.
clear.
missing components.
poorly.
Comparisons
The differences are clearly
minor success.
Answers Student completed worksheet
completely. Answers are well
Observational Studies and Experiments
Would you fall for the placebo effect? Watch this video, then complete the rest of the questions.
1. Describe what you saw in the video. Why do you think the people in the video got stronger?
2. Mr. Miles is trying out a new facial hair ointment to make his beard grow faster for No Shave November.
At the end of November, his beard is 3 inches long. He concludes since his beard grew 3 inches, the
ointment must work. What is wrong with his reasoning?
3. In a recent medical study testing the effectiveness of a new medication, researchers gave participants
either the new medication or a sugar pill. When participants came to get the pills, they were told whether
they had been assigned to the medication or to the sugar pills. At the conclusion of the study, researchers
found the medication was more effective than the sugar pill. Are the results valid?
4. Identify any possible variables that could be the cause of the correlations below:
a. Those people who tend to carry boxes of matches in their pockets also have a higher risk of cancer.
b. People who drink diet sodas tend to have a higher body mass index (weight to height ratio).
c. Students who eat dinner with their family often are more likely to have better grades.
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What happens when physicians study themselves?
Does regularly taking aspirin help protect people against heart attacks? The Physicians’ Health Study I was a
medical experiment that helped answer this question. The subjects in this experiment were 21,996 male
physicians. Half of these subjects took an aspirin tablet every other day and the remaining subjects took a dummy
pill that looked and tasted like the aspirin but had no active ingredient. After several years, 239 of the control
group but only 139 of the aspirin group had suffered heart attacks. This difference is large enough to provide
convincing evidence that taking aspirin does reduce heart attacks.
5. Why was it necessary to perform an experiment rather than simply asking the doctors whether or not
they take aspirin regularly?
6. Explain why it was necessary for the experiment to include a control group that didn’t receive aspirin.
7. Was blinding used in this experiment? Explain why this is an important consideration.
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Placebo Effect
Control Group
.
Many golfers started using longer putters (anchored putting), until it was recently banned. Are anchored putters
actually better than traditional putters?
Suppose that we have 30 students willing to participate in our experiment to test short putters vs. long putters.
We decide to have 15 students use the short putter and 15 students use the long putter.
1. Suppose we let students pick which putter they wanted to use. Why might this be a problem?
Instead of letting students choose which putter they will use, we can use random assignment to select 15 students
for the long putter and 15 students for the short putter.
2. Explain how you could use slips of paper to do random assignment.
3. Explain how you could use a random number generator to do random assignment.
4. What is the purpose of using random assignment?
5. What other variables will we attempt to keep the same during this experiment (list several)? Why?
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Researchers in Canada performed an experiment with university students to examine the effects of multitasking
on student learning. The 40 participants in the study were asked to attend a lecture and take notes with their
laptops. Half of the participants were randomly assigned to complete other online tasks not related to the lecture
during that time. These tasks were meant to imitate typical student Web browsing during classes. The remaining
students simply took notes with their laptops. At the end of the lecture, all participants took a comprehension test
to measure how much they learned from it. The results: students who were assigned to multitask did significantly
worse (11%) than students who were not assigned to multitask.
6. Describe how the researchers could have carried out the random assignment.
7. Why was it important that the researchers randomly assigned treatments to the students?
8. Identify one variable that the researchers kept the same for all subjects. Provide two reasons why it was
important for the researchers to keep this variable the same.
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What is Random Assignment
Why Use Random Assignment
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.
How do we know if the experimental results are different due to the treatments or if the difference occurred
purely by chance?
1. Below is a diagram that details how one group conducted an experiment to test if the long or short putter
was better. Use the diagram to describe how the experiment was conducted.
Below are the results of the experiment showing how far each ball was from the hole (in cm) after the
putt.
Mean
Long
Putter 2 3 0 2 5 4 2 1 6 3
Short
Putter 2 8 10 9 6 5 6 5 0 6
2. Find the mean distance and standard deviation from the hole for each putter. Calculate the difference in
means for the putters (short – long)? Explain in context.
20 randomly selected senios
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How do we know whether or not the difference in means is large enough to say one putter is better? Could the
difference have occurred purely by chance? In order to decide if this result is statistically significant, we will use
the applet to simulate many experiments to see how often this result occurs by chance.
3. Enter the data: (Directions for the TI84).
Press S, then select 1:Edit
Press :, to highlight the list header (L1)
Press C, m>>>, to move to the Prob menu and select 6:randNorm
Enter the mean and standard deviation for the long putter, and 100 trials.
Repeat with the short putter in L2.
4. Simulate difference in two means:
Highlight the third list header (L3)
Press `2 (æ) - `1 () to find the difference of the simulation trials
5. Create a Histogram of the difference of the trial means: Sketch the plot below.
6. Calculate a percentage: What percentage of the histogram is greater than or equal to the difference in
means from our experiment?
7. Make a decision: Do you think the difference in means we found from our experiment is due to the
treatments or has it occurred purely by chance?
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To see if fish oil can help reduce blood pressure, males with high blood pressure were recruited and randomly
assigned to different treatments. Seven of the men were randomly assigned to a 4-week diet that included fish oil.
Seven other men were assigned to a 4-week diet that included a mixture of oils that approximated the types of fat
in a typical diet. At the end of the 4 weeks, each volunteer’s blood pressure was measured again and the
reduction in diastolic blood pressure was recorded. These differences are shown in the table below. Note that a
negative value means that the subject’s blood pressure increased.
Fish oil: 8 12 10 14 2 0 0
Mixture: −6 0 1 2 −3 −4 2
8. Outline a completely randomized design for this experiment.
9. Calculate the mean reduction for each group and the difference in mean reduction (fish oil – mixture).
10. One hundred trials of a simulation were performed to see what differences in means are likely to occur
due only to chance variation in the random assignment, assuming that the type of oil doesn’t matter. Use
the results of the simulation below to determine if the difference in means from part (b) is statistically
significant. Explain your reasoning.
Completely Randomized Designs
.
Some students at your school claim that listening to music while studying will help improve their GPA. Design a
study to help discover if this claim is true.
Here are four proposed studies for investigating the question of the day. Suppose we found that the mean GPA
of students who listen to music is significantly lower than the mean GPA of students who didn’t listen to music.
What conclusions could we make?
1. Get all the students in your statistics class to participate in a study. Ask them whether or not they study
with music on and divide them into two groups based on their answer to this question.
Random sample?__________ Random assignment?__________
Conclusion:
2. Select a random sample of students from your school to participate in a study. Ask them whether or not
they study with music on and divide them into two groups based on their answer to this question.
Random sample?__________ Random assignment?__________
Conclusion:
3. Get all the students in your statistics class to participate in a study. Randomly assign half of the students
to listen to music while studying for the entire semester and have the remaining half abstain from
listening to music while studying.
Random sample?__________ Random assignment?__________
Conclusion:
4. 4. Select a random sample of students from your school to participate in a study. Randomly assign half of
the students to listen to music while studying for the entire semester and have the remaining half abstain
from listening to music while studying.
Random sample?__________ Random assignment?__________
Conclusion:
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Do abandoned children placed in foster homes do better than similar children placed in an institution? The
Bucharest Early Intervention Project found that the answer is a clear “Yes.” The subjects were 136 young children
abandoned at birth and living in orphanages in Bucharest, Romania. Half of the children, chosen at random, were
placed in foster homes. The other half remained in the orphanages. (Foster care was not easily available in
Romania at the time and so was paid for by the study.)
5. Did this study use random sampling?
6. Did this study use random assignment?
7. What conclusion(s) can we draw from this study? Explain.
8. The children in this study were too young to provide informed consent. Does this make this study
unethical? Explain.
Define random sample:
Why do we take random samples?
What conclusion can be made for a study that uses random sampling?
Define random assignment:
Why do we do random assignment?
What conclusion can be made for a study that uses random assignment?
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unemployment by randomly dialing phone
numbers until it had gathered responses from
1000 adults in the state. In the survey, 19% of
those who responded said they were not
currently employed. In reality, only 6% of the
adults in the state were not currently employed
at the time of the survey. Which of the following
best explains the difference in the two
percentages?
We shouldn’t expect the results of a random
sample to match the truth about the
population every time.
that they are unemployed.
survey included only adults and did not
include teenagers who are eligible to work.
d. The difference is due to nonresponse. Adults
who are employed are less likely to be
available for the sample than adults who are
unemployed.
2. A study of treatments for angina (pain due to low
blood supply to the heart) compared bypass
surgery, angioplasty, and use of drugs. The study
looked at the medical records of thousands of
angina patients whose doctors had chosen one of
these treatments. It found that the average
survival time of patients given drugs was the
highest. What do you conclude?
a. This study provides convincing evidence
that drugs prolong life and should be the
treatment of choice.
because the patients were volunteers.
c. We can’t conclude that drugs prolong life
because this was an observational study.
d. We can’t conclude that drugs prolong life
because no placebo was used.
3. Some studies of the relationship between car
color and frequency of accidents have found that
red cars are more likely to be in accidents than
black cars, despite how visible they are. Some
experts warn that we should not conclude red
cars are less safe than black cars because of
possible confounding. Which of the following
best describes what this means?
a. There are too many variables involved in
traffic accidents to isolate the effect of car
color.
b. The accidents used in the study might not be
a random sample of all accidents.
c. It is not possible to separate the effect of car
color from the type of people who choose to
drive red cars.
d. It can be hard to determine if red cars or black
cars are more visible on modern highways.
4. A new headache remedy was given to a group of
25 subjects who had headaches. Four hours after
taking the new remedy, 20 of the subjects
reported that their headaches had disappeared.
From this information, you conclude
a. that the remedy is effective for the treatment
of headaches.
comparison.
5. One hundred volunteers who suffer from
attention deficit hyperactivity disorder (ADHD)
are available for a study. Fifty are randomly
assigned to receive a new drug that is thought to
be particularly effective in treating ADHD. The
other 50 are given a commonly used drug. A
psychiatrist evaluates the symptoms of all
volunteers after four weeks to determine if there
has been substantial improvement in symptoms.
The study would be double-blind if
a. neither drug had any identifying marks on it.
b. none of the volunteers were allowed to see
the psychiatrist, and the psychiatrist is also
not allowed to see the volunteers during the
evaluation session.
knew which treatment any person had
received.
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effectiveness of different insecticides in
controlling pests and their impact on the
productivity of tomato plants. What is the best
reason for randomly assigning treatments
(spraying or not spraying) to the farms?
a. Random assignment allows researchers to
generalize conclusions about the
b. Random assignment will tend to average out
all other variables, such as soil fertility, so
that they are not confounded with the
treatment effects.
other variables, like soil fertility.
d. Random assignment eliminates chance
variation in the responses.
7. Which of the following is not a benefit of keeping
other variables the same in an experiment?
a. Keeping other variables the same helps
prevent confounding.
variability in the response variable.
c. Keeping other variables the same makes it
easier to get statistically significant results if
one treatment is more effective than the
other.
the need for random assignment.
8. Every hour, the quality control manager at a
factory making tortilla chips selects a random
sample of 20 bags of chips and weighs the
contents. If the manager is convinced that the
mean weight of all bags produced that hour
differs from the target of 16 ounces, the
production line will be stopped so the problem
can be identified. In the current sample of 20
bags, the mean weight is 15.94 ounces, and this
estimate has a margin of error of 0.12 ounces.
Which of the following conclusions is best?
a. The manager should stop the production line
because the mean of 15.94 is less than 16.
b. The manager should stop the production line
because the margin of error is greater than 0.
c. The manager should not stop the production
line because the difference between 15.94 and
16 is less than the margin of error.
d. The manager should not stop the production
line because the margin of error is very small.
Section II: Free Response
survey results, Sam and Joshan randomly
selected 60 students from their school’s student
list and divided them at random into two groups
of 30. By email, they asked the students the
following questions about a nearby convenience
store, the Quick Shop:
think the prices at the Quick Shop are
fair, given that the typical markup is
25%?
of the Quick Shop, do you think the
prices are fair, given that the typical
markup is 25%?
Question Wording
Total 30 30 60
study? Justify your answer.
responded “fair” in each of the two treatment
groups and the difference in proportions
(Convenience – Markup).
performed to see what differences in
proportions would happen due only to
chance variation in the random assignment,
assuming that the wording of the question
didn’t matter. Use the results of the
simulation below to determine if the
difference in proportions from part (b) is
statistically significant. Explain your
conclusions can we draw? Explain.
32
university wants to evaluate the satisfaction level
of students living in one of the dormitories on
campus. She plans to interview a random sample
of 50 of the 816 residents of the building.
a. Explain how you could use a random
number generator to select a simple random
sample of 50 residents of the dormitory.
b. During each interview, the director asks, “If
you could make the choice again, would you
still choose this dorm?” In the sample of 50,
66% of the residents answered “Yes.” If the
director took a second random sample of 50
students, would this percent be the same?
Explain.
might make the estimate from part (b)
unreliable. Would increasing the sample size
solve this problem? Explain.
turns out that elephants dislike bees. They
recognize beehives and avoid them. Can this
information be used to keep elephants away from
trees? Will elephant damage be less in trees with
hives? Will empty hives even keep elephants
away? Researchers want to design an experiment
to answer these questions using 72 acacia
trees.106
and the response variable.
a completely randomized design for this
experiment. Include a description of how the
treatments should be assigned.
experiment in which subjects were told they were
“teachers” and were instructed to give electric
shocks to other subjects (“students”) when the
latter made mistakes on a word-matching test.
What the “teachers” didn’t know was that the
“students” were Milgram’s assistants, who faked
their discomfort at receiving nonexistent shocks.
Milgram showed that people were willing to
administer what they thought were serious
shocks rather than disobey someone in authority.
In what way did this experiment violate the
principles of ethical research?
.
What is the probability of getting an odd product from two dice?
We’re going to play a game to answer this question. You and your partner must decide who will be “Odds” and
who will be “Evens”. Then you will roll two dice and multiply the numbers. If the product is odd, the odd
person wins and vice versa for evens. Play 20 times, keeping track of how many wins each person has.
1. How many times did the odds win? Write this as a fraction out of 20 and turn it to a percentage.
Maybe the odds just had a run of bad luck. Let’s see how the rest of the class did with odds. Write the number of
odds wins for your group in the table on the board.
2. Find the total percent of rolls that were odd products for the whole class. How does this compare to your
group’s results?
3. To determine the true probability of rolling an odd product, we should list out all possible products that
we could get. Complete the table below to show all possible products.
4. Use your table to find the probability of rolling an odd product.
5. Which was closer to the percentage you found in #4, your group data
or the classroom data? Why do you think that is?
6. Explain what the probability of rolling odds means in this setting.
1 2 3 4 5 6
1
2
3
4
5
6
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New Jersey Transit claims that its 8:00 a.m. train from Princeton to New York has probability 0.9 of arriving on
time. Assume for now that this claim is true.
7. Explain what probability 0.9 means in this setting.
8. The 8:00 a.m. train has arrived on time 5 days in a row. What’s the probability that it will arrive on time
tomorrow? Explain.
9. A businessman takes the 8:00 a.m. train to work on 20 days in a month. He is surprised when the train arrives
late in New York on 3 of the 20 days. Should he be surprised? Describe how you would carry out a simulation
to estimate the probability that the train would arrive late on 3 or more of 20 days if New Jersey Transit’s
claim is true. Do not perform the simulation.
10. The dotplot below shows the number of days on which the train arrived late in 100 repetitions of the
simulation. What is the resulting estimate of the probability described in Question 3? Should the businessman
be surprised?
Probability
.
What is the probability of being a male student with blue eyes?
To answer today’s question, we will randomly select 10 students to come up front. Use those students’
information to answer the following questions.
1. In any given class, there are males and females who have blue, brown or green eyes. Create a table or
tree diagram that shows all possible combinations of gender with eye color.
2. Using the 10 students chosen, find each of the following probabilities:
P(Male) = P(Blue Eyes) =
P(Female) = P(Brown Eyes) =
P(Green Eyes) =
P(Other Eyes) =
3. Find each of the following probabilities and explain why your answer makes sense.
P(Male or Female) = P(Blue or Brown Eyes) =
4. Find each of the following probabilities and explain why your answer makes sense.
P(Male or Blue Eyes) = P(Female or Brown Eyes) =
5. Find each of the following probabilities and explain why your answer makes sense.
P(Not Green Eyes) = P(Not Male) =
37
Choose an American adult at random. Define two events:
A = the person has a cholesterol level of 240 milligrams per deciliter of blood
(mg/dl) or above (high cholesterol)
B = the person has a cholesterol level of 200 to <240 mg/dl (borderline high
cholesterol)
According to the American Heart Association, P(A) = 0.16 and P(B) = 0.29.
6. Explain why events A and B are mutually exclusive.
7. Say in plain language what the event “A or B” is.
8. Find P(A or B).
9. Let C be the event that the person chosen has normal cholesterol (below 200 mg/dl). Find P(C).
38
Sample Space
VS .
What is the probability that a randomly selected person from the class uses Facebook? Uses Twitter? Uses both?
Neither?
1. Collect class data to fill in the following two-way table.
Twitter No Twitter Total
Total
2. Suppose that we randomly choose a student from class. Find the following probabilities.
P(Facebook) = P(Twitter) =
P(no Facebook) = P(no Twitter) =
3. Now we will find some probabilities for two events occurring.
P(Facebook AND Twitter) = P(Twitter AND no Facebook) =
P(Facebook AND no Twitter) = P(no Facebook AND no Twitter) =
4. Is P(Facebook OR Twitter) = P(Facebook) + P(Twitter)? Why or why not?
5. Take the values from the two-way table and put them in the following Venn Diagram.
a. Where is “Twitter AND Facebook”
b. Where is “Twitter OR Facebook”
TwitterFacebook
40
What is the relationship between educational achievement and home ownership? A random sample of 500 U.S.
adults was selected. Each member of the sample was identified as a high school graduate (or not) and as a
homeowner (or not). The two-way table displays the data. Suppose we choose a member of the sample at
random. Define event G: is a high school graduate and event H: is a homeowner.
High School Graduate Not a High School
Graduate
Not a homeowner 89 71
6. Explain why P(G or H) ≠ P(G) + P(H). Then find P(G or H).
7. Make a Venn diagram to display the sample space of this chance process.
8. Write the event “is not a high school graduate but is a homeowner” in symbolic form.
41
Two-Way Tables
Venn Diagram
.
Definition: Two events are independent if knowing whether or not one event has occurred does not change the
probability that the other event will occur.
Are the events “Female” and “prefers English” independent?
1. Collect class data to fill in the following two-way table and Venn Diagram.
English Math Total
Female
Male
Total
2. Suppose that we randomly choose a student from class. Find the following probabilities.
P(Female) = P(English) =
P(Female AND no English) = P(no Female AND no English) =
3. Find P(Female OR English).
4. What is the probability that a student prefers English, given that they are a female? Write as a percent.
5. What is the probability that a student prefers English, given that they are a male? Write as a percent.
6. Are the events “Female” and “prefers English” independent? Explain.
Female English
43
Students at the University of New Hampshire received 10,000 course grades in a recent semester. The two-way
table below breaks down these grades by which school of the university taught the course. The schools are Liberal
Arts, Engineering and Physical Sciences (EPS), and Health and Human Services.
Grade Level
Engineering and Physical Sciences 368 432 800
Health and Human Services 882 630 588
Choose a University of New Hampshire course grade at random. Define a low grade as one that is below a B.
7. Find P(low grade | Engineering). Interpret this probability in context.
8. Find P(low grade | Liberal Arts). Interpret this probability in context.
9. Are events “low grade” and “engineering” independent? Justify your answer.
44
Probability of NO
What types of questions do we investigate in statistics? The sheet you are given has two different types of
questions: statistical and non-statistical. Cut out the questions and sort them into two different groups according
to a characteristic they have in common.
1. Describe the criteria you used to sort the questions? What is the main difference between your two
groups of questions?
2. How can we tell the difference between a statistical question and a non-statistical question?
3. Attach 3 each type of question to the table below to use as examples of statistical and non-statistical
questions. In the last box of each column, write your own examples of a statistical and a non-statistical
question.
Researchers at Columbia University randomly selected 1000 teenagers in the United States for a survey.
According to an ABC News article about the research, “Teenagers who eat with their families at least five times a
week are more likely to get better grades in school.”
4. What is the statistical question that the researchers were trying to answer?
5. Identify the population and sample.
6. Is this an observational study or an experiment? Explain.
3
Call your name two or three times in a row
Such a funny thing for me to try to explain
How I'm feeling and my pride is the one to blame
'Cuz I know I don't understand
Just how your love can do what no one else can
Got me looking so crazy right now, your love's
Got me looking so crazy right now (in love)
Got me looking so crazy right now, your touch
Got me looking so crazy right now (your touch)
Got me hoping you'll page me right now, your kiss
Got me hoping you'll save me right now
Looking so crazy in love's
Got me looking, got me looking so crazy in love
When I talk to my friends so quietly
Who he think he is? Look at what you did to me
Tennis shoes, don't even need to buy a new dress
If you ain't there ain't nobody else to impress
The way that you know what I thought I knew
It's the beat my heart skips when I'm with you
But I still don't understand
Just how the love your doing no one else can
I'm Looking so crazy in love's
Got me looking, got me looking so crazy in love
Got me looking, so crazy, my baby
I'm not myself, lately I'm foolish, I don't do this
I've been playing myself, baby I don't care
'Cuz your love's got the best of me
And baby you're making a fool of me
You got me sprung and I don't care who sees
'Cuz baby you got me, you got me, so crazy baby
HEY!
4
1. Quickly circle a random sample of 5 words from the lyrics to Crazy in Love on the previous page. Write
them below. How many letters in each word?
2. What is the average word length of your sample? ________.
3. Add your average to the class’ dotplot. Copy the class dotplot below.
4. Find a new sample of 5 words using a random number generator. Add your average to the class’ dotplot.
Copy the class dotplot below.
5. How is the dotplot from #4 different than the dotplot for #3? Which do you think is a better estimator of
the true mean word length?
6. What do you think the true mean word length is for “Crazy in Love”?
7. It is known that Beyonce wrote the lyrics for all of the Destiny’s child songs. The average word length for
these songs is 3.64 letters. Based on your samples, do you have good evidence that Beyonce did not write
the lyrics for “Crazy in Love”. Explain.
5
In June 2008 Parade magazine posed the following question: “Should drivers be banned from using all cell
phones?” Readers were encouraged to vote online at www.parade.com. The July 13, 2008, issue of Parade
reported the results: 2407 (85%) said “Yes” and 410 (15%) said “No.”
8. What type of sample did the Parade survey obtain?
9. Explain why this sampling method is biased.
10. Is 85% likely to be greater than or less than the percentage of all adults who believe that cell-phone use
while driving should be banned? Why?
To help eliminate bias, a reporter from Parade decides she will go out and ask people in person if they think
drivers should be banned from using cell phones. She lives close to the local high school so she goes to the
parking lot at 3:00 pm and asks the first 100 people she sees.
11. What type of sample did the reporter obtain?
12. Explain why this sampling method is biased.
13. How could Parade magazine avoid the bias described above?
6
Voluntary Response
7
Simple Random Samples (SRS) 1. Copy the class dotplot from your notes from yesterday (the random sample, not the convenience sample).
2. Describe in words the process that we used yesterday to select a random sample. Be detailed.
3. Describe in words how we could use the “names from a hat” method to take a random sample. Be
detailed.
4. Take a random sample of 10 words from Crazy in Love. Find the mean of your sample and put it on the
class’ dotplot. Copy the dotplot below.
5. What happens when we increase the sample size?
8
What proportion of students at your school use Twitter? To find out, you decide to survey a simple random
sample of students from your school. In your sample of 50 students, 76% say they use Twitter.
6. Describe how you could select an SRS of 50 students using a random number generator.
7. Will your sample result (76%) be exactly the same as the true population proportion for all students?
Explain.
8. Which would be more likely to yield a sample result closer to the true population value: an SRS of 50
students or an SRS of 100 students? Explain.
9
Important IdeasImportant IdeasImportant IdeasImportant Ideas How to create a SRS with slips of paper
How to create a SRS with a Random Number Generator
Sampling Variability
Margin of Error
An SRS was conducted to find out how much television students watch on the weekend. Twenty-five students
were selected and asked “How many hours of TV did you watch this weekend?” Their responses are listed below.
5, 4, 3, 7, 5, 2, 5, 4, 8, 3, 6, 6, 5, 4, 6, 7, 5, 5, 3, 4, 6, 5, 7, 6, 4
1. What kind of data is this, categorical or quantitative?
2. Use your calculator. What is the mean? Do you expect that this mean is the same as the true population
mean? Explain.
Because of sampling variability, we know that if we continue to sample students in groups of 25 we will get
different means. Some will be above, and some will be below our sample mean of 5. To get a better idea of
what this looks like, we will use an applet to simulate taking 30 samples and finding each mean.
3. In the applet, we will scroll down to Perform Inference and choose “Simulate sample mean” from the
drop-down menu. Add 30 samples and Perform Simulation. Sketch the dotplot below.
4. Draw and label vertical lines on your dotplot to mark each of the following:
a. The mean of 5.
b. One standard deviation above and below the mean of 5. (You can find the SD listed under the dot plot.)
c. Two standard deviations above and below the mean of 5.
5. What percentage of the sample means are within two standard deviations above or below the mean?
11
6. Write an interval that includes all the data two standard deviations above and below the mean.
Everything we have done so far has been to estimate a margin of error for a sample mean. We can also find a
margin of error for a sample proportion. Try the example below.
How many students text during class? A student created an anonymous survey asking students whether or not
they text in class. Of the 50 students who responded, 64% said they text in class.
7. Is this a categorical or quantitative variable?
8. How many students responded yes? How many said no?
We now use the applet to enter the data in 1 Categorical Variable. Enter the category names (“Yes” and “No”)
with the number of students who answered each. Click Begin analysis.
Scroll down to Perform Inference and choose “Simulate sample proportion” from the drop-down menu. Keep
“Yes” as the category to indicate as success and change the third drop-down menu to the observed proportion.
Note: Leave the Hypothesized proportion blank. Add 100 samples.
9. What is the standard deviation of the sample proportions?
10. Calculate the margin of error.
11. Interpret the margin of error.
12
Many people can roll their tongues, but some can’t. Javier is interested in determining the proportion of students
at his school that can roll their tongue. In a random sample of 100 students, Javier determines that 70 can roll their
tongue. The dotplot below shows the proportion of people who can roll their tongue in each of 500 random
samples of size 100 from a population where 70% can roll their tongue.
Distribution of Simulated Proportion
z samples Mean SD
500 0.701 0.048
12. Use the results of the simulation to approximate the margin of error for Javier’s estimate of the proportion
of students at his school that can roll their tongue.
13. Interpret the margin of error.
14. Javier’s biology teacher claims that 75% of people can roll their tongue. According to Javier’s study, is this
claim plausible? Explain.
15. Explain how Javier could decrease the margin of error.
16. for the two streams similar or different? Justify your answer.
13
How to decrease margin of error
14
1. Which one of the following is a valid statistical
question?
b. How many points did LeBron James score in
the 2014 NBA playoffs?
customer service number, how long will you
be on hold?
on groceries last month?
Albuquerque residents support the local minor
league baseball team, the Isotopes. She stands
outside the stadium before a game and
interviews the first 20 people who enter the
stadium, asking them to rate their enthusiasm for
the team on a 1 (lowest) to 5 (highest) scale.
Which of the following best describes the results
of this survey?
be some sampling variability, but we can
expect it to produce quite accurate results.
b. This is a random sample, but the size of the
sample is too small to produce reliable
results.
likely to underestimate support for the team.
d. This is a convenience sample and is likely to
overestimate the level of support for the
team.
questions next to many of its news stories. Simply
click your response to join the sample. One recent
question was “Do you plan to diet this year?”
More than 30,000 people responded, with 68%
saying “Yes.” You can conclude that
a. about 68% of Americans planned to diet.
b. the poll used a convenience sample, so the
results tell us little about the population of all
adults.
results tell us little about the population of all
adults.
conclusion.
survey results, Sam and Joshan randomly
selected 60 students from their school’s student
list and divided them at random into two groups
of 30. By email, they asked the students the
following questions about a nearby convenience
store, the Quick Shop:
think the prices at the Quick Shop are
fair, given that the typical markup is
25%?
of the Quick Shop, do you think the
prices are fair, given that the typical
markup is 25%?
Mark-Up Convenience Total
Total 30 30 60
study? Justify your answer.
responded “fair” in each of the two treatment
groups and the difference in proportions
(Convenience – Markup).
university wants to evaluate the satisfaction level
of students living in one of the dormitories on
campus. She plans to interview a random sample
of 50 of the 816 residents of the building.
a. Explain how you could use a random
number generator to select a simple random
sample of 50 residents of the dormitory.
b. During each interview, the director asks, “If
you could make the choice again, would you
still choose this dorm?” In the sample of 50,
66% of the residents answered “Yes.” If the
director took a second random sample of 50
students, would this percent be the same?
Explain.
15
Sampling and Surveys Directions:
1. Decide on a question you would like to ask 20 Park High School students. Determine if this question is a
mean or a proportion question.
2. Next, you should determine a type of bias you’d like to create when you ask students these questions.
3. Create a scenario (that is do-able) where you will ask 20 students these questions in a biased manner.
4. Next, collect your answers in this biased manner. What was the conclusion from your study?
5. You will now collect answers from 20 students using a SRS.
6. Compare your answers. How did your answer change?
7. Create a poster showing the question, describing method of data collection and comparing the answers
received from each type of data collection.
What is the question you will ask?
Is this a mean or proportion question? Explain.
Which bias will you be creating? How.
Collect your biased data here.
Collect your unbiased data here.
16
with integrity.
time in places around the school
they were not supposed to be.
Student did not collect data and/or
was a nuisance to others while
collecting data.
clear.
missing components.
poorly.
Comparisons
The differences are clearly
minor success.
Answers Student completed worksheet
completely. Answers are well
Observational Studies and Experiments
Would you fall for the placebo effect? Watch this video, then complete the rest of the questions.
1. Describe what you saw in the video. Why do you think the people in the video got stronger?
2. Mr. Miles is trying out a new facial hair ointment to make his beard grow faster for No Shave November.
At the end of November, his beard is 3 inches long. He concludes since his beard grew 3 inches, the
ointment must work. What is wrong with his reasoning?
3. In a recent medical study testing the effectiveness of a new medication, researchers gave participants
either the new medication or a sugar pill. When participants came to get the pills, they were told whether
they had been assigned to the medication or to the sugar pills. At the conclusion of the study, researchers
found the medication was more effective than the sugar pill. Are the results valid?
4. Identify any possible variables that could be the cause of the correlations below:
a. Those people who tend to carry boxes of matches in their pockets also have a higher risk of cancer.
b. People who drink diet sodas tend to have a higher body mass index (weight to height ratio).
c. Students who eat dinner with their family often are more likely to have better grades.
18
What happens when physicians study themselves?
Does regularly taking aspirin help protect people against heart attacks? The Physicians’ Health Study I was a
medical experiment that helped answer this question. The subjects in this experiment were 21,996 male
physicians. Half of these subjects took an aspirin tablet every other day and the remaining subjects took a dummy
pill that looked and tasted like the aspirin but had no active ingredient. After several years, 239 of the control
group but only 139 of the aspirin group had suffered heart attacks. This difference is large enough to provide
convincing evidence that taking aspirin does reduce heart attacks.
5. Why was it necessary to perform an experiment rather than simply asking the doctors whether or not
they take aspirin regularly?
6. Explain why it was necessary for the experiment to include a control group that didn’t receive aspirin.
7. Was blinding used in this experiment? Explain why this is an important consideration.
19
Placebo Effect
Control Group
.
Many golfers started using longer putters (anchored putting), until it was recently banned. Are anchored putters
actually better than traditional putters?
Suppose that we have 30 students willing to participate in our experiment to test short putters vs. long putters.
We decide to have 15 students use the short putter and 15 students use the long putter.
1. Suppose we let students pick which putter they wanted to use. Why might this be a problem?
Instead of letting students choose which putter they will use, we can use random assignment to select 15 students
for the long putter and 15 students for the short putter.
2. Explain how you could use slips of paper to do random assignment.
3. Explain how you could use a random number generator to do random assignment.
4. What is the purpose of using random assignment?
5. What other variables will we attempt to keep the same during this experiment (list several)? Why?
21
Researchers in Canada performed an experiment with university students to examine the effects of multitasking
on student learning. The 40 participants in the study were asked to attend a lecture and take notes with their
laptops. Half of the participants were randomly assigned to complete other online tasks not related to the lecture
during that time. These tasks were meant to imitate typical student Web browsing during classes. The remaining
students simply took notes with their laptops. At the end of the lecture, all participants took a comprehension test
to measure how much they learned from it. The results: students who were assigned to multitask did significantly
worse (11%) than students who were not assigned to multitask.
6. Describe how the researchers could have carried out the random assignment.
7. Why was it important that the researchers randomly assigned treatments to the students?
8. Identify one variable that the researchers kept the same for all subjects. Provide two reasons why it was
important for the researchers to keep this variable the same.
22
What is Random Assignment
Why Use Random Assignment
23
.
How do we know if the experimental results are different due to the treatments or if the difference occurred
purely by chance?
1. Below is a diagram that details how one group conducted an experiment to test if the long or short putter
was better. Use the diagram to describe how the experiment was conducted.
Below are the results of the experiment showing how far each ball was from the hole (in cm) after the
putt.
Mean
Long
Putter 2 3 0 2 5 4 2 1 6 3
Short
Putter 2 8 10 9 6 5 6 5 0 6
2. Find the mean distance and standard deviation from the hole for each putter. Calculate the difference in
means for the putters (short – long)? Explain in context.
20 randomly selected senios
24
How do we know whether or not the difference in means is large enough to say one putter is better? Could the
difference have occurred purely by chance? In order to decide if this result is statistically significant, we will use
the applet to simulate many experiments to see how often this result occurs by chance.
3. Enter the data: (Directions for the TI84).
Press S, then select 1:Edit
Press :, to highlight the list header (L1)
Press C, m>>>, to move to the Prob menu and select 6:randNorm
Enter the mean and standard deviation for the long putter, and 100 trials.
Repeat with the short putter in L2.
4. Simulate difference in two means:
Highlight the third list header (L3)
Press `2 (æ) - `1 () to find the difference of the simulation trials
5. Create a Histogram of the difference of the trial means: Sketch the plot below.
6. Calculate a percentage: What percentage of the histogram is greater than or equal to the difference in
means from our experiment?
7. Make a decision: Do you think the difference in means we found from our experiment is due to the
treatments or has it occurred purely by chance?
25
To see if fish oil can help reduce blood pressure, males with high blood pressure were recruited and randomly
assigned to different treatments. Seven of the men were randomly assigned to a 4-week diet that included fish oil.
Seven other men were assigned to a 4-week diet that included a mixture of oils that approximated the types of fat
in a typical diet. At the end of the 4 weeks, each volunteer’s blood pressure was measured again and the
reduction in diastolic blood pressure was recorded. These differences are shown in the table below. Note that a
negative value means that the subject’s blood pressure increased.
Fish oil: 8 12 10 14 2 0 0
Mixture: −6 0 1 2 −3 −4 2
8. Outline a completely randomized design for this experiment.
9. Calculate the mean reduction for each group and the difference in mean reduction (fish oil – mixture).
10. One hundred trials of a simulation were performed to see what differences in means are likely to occur
due only to chance variation in the random assignment, assuming that the type of oil doesn’t matter. Use
the results of the simulation below to determine if the difference in means from part (b) is statistically
significant. Explain your reasoning.
Completely Randomized Designs
.
Some students at your school claim that listening to music while studying will help improve their GPA. Design a
study to help discover if this claim is true.
Here are four proposed studies for investigating the question of the day. Suppose we found that the mean GPA
of students who listen to music is significantly lower than the mean GPA of students who didn’t listen to music.
What conclusions could we make?
1. Get all the students in your statistics class to participate in a study. Ask them whether or not they study
with music on and divide them into two groups based on their answer to this question.
Random sample?__________ Random assignment?__________
Conclusion:
2. Select a random sample of students from your school to participate in a study. Ask them whether or not
they study with music on and divide them into two groups based on their answer to this question.
Random sample?__________ Random assignment?__________
Conclusion:
3. Get all the students in your statistics class to participate in a study. Randomly assign half of the students
to listen to music while studying for the entire semester and have the remaining half abstain from
listening to music while studying.
Random sample?__________ Random assignment?__________
Conclusion:
4. 4. Select a random sample of students from your school to participate in a study. Randomly assign half of
the students to listen to music while studying for the entire semester and have the remaining half abstain
from listening to music while studying.
Random sample?__________ Random assignment?__________
Conclusion:
28
Do abandoned children placed in foster homes do better than similar children placed in an institution? The
Bucharest Early Intervention Project found that the answer is a clear “Yes.” The subjects were 136 young children
abandoned at birth and living in orphanages in Bucharest, Romania. Half of the children, chosen at random, were
placed in foster homes. The other half remained in the orphanages. (Foster care was not easily available in
Romania at the time and so was paid for by the study.)
5. Did this study use random sampling?
6. Did this study use random assignment?
7. What conclusion(s) can we draw from this study? Explain.
8. The children in this study were too young to provide informed consent. Does this make this study
unethical? Explain.
Define random sample:
Why do we take random samples?
What conclusion can be made for a study that uses random sampling?
Define random assignment:
Why do we do random assignment?
What conclusion can be made for a study that uses random assignment?
30
unemployment by randomly dialing phone
numbers until it had gathered responses from
1000 adults in the state. In the survey, 19% of
those who responded said they were not
currently employed. In reality, only 6% of the
adults in the state were not currently employed
at the time of the survey. Which of the following
best explains the difference in the two
percentages?
We shouldn’t expect the results of a random
sample to match the truth about the
population every time.
that they are unemployed.
survey included only adults and did not
include teenagers who are eligible to work.
d. The difference is due to nonresponse. Adults
who are employed are less likely to be
available for the sample than adults who are
unemployed.
2. A study of treatments for angina (pain due to low
blood supply to the heart) compared bypass
surgery, angioplasty, and use of drugs. The study
looked at the medical records of thousands of
angina patients whose doctors had chosen one of
these treatments. It found that the average
survival time of patients given drugs was the
highest. What do you conclude?
a. This study provides convincing evidence
that drugs prolong life and should be the
treatment of choice.
because the patients were volunteers.
c. We can’t conclude that drugs prolong life
because this was an observational study.
d. We can’t conclude that drugs prolong life
because no placebo was used.
3. Some studies of the relationship between car
color and frequency of accidents have found that
red cars are more likely to be in accidents than
black cars, despite how visible they are. Some
experts warn that we should not conclude red
cars are less safe than black cars because of
possible confounding. Which of the following
best describes what this means?
a. There are too many variables involved in
traffic accidents to isolate the effect of car
color.
b. The accidents used in the study might not be
a random sample of all accidents.
c. It is not possible to separate the effect of car
color from the type of people who choose to
drive red cars.
d. It can be hard to determine if red cars or black
cars are more visible on modern highways.
4. A new headache remedy was given to a group of
25 subjects who had headaches. Four hours after
taking the new remedy, 20 of the subjects
reported that their headaches had disappeared.
From this information, you conclude
a. that the remedy is effective for the treatment
of headaches.
comparison.
5. One hundred volunteers who suffer from
attention deficit hyperactivity disorder (ADHD)
are available for a study. Fifty are randomly
assigned to receive a new drug that is thought to
be particularly effective in treating ADHD. The
other 50 are given a commonly used drug. A
psychiatrist evaluates the symptoms of all
volunteers after four weeks to determine if there
has been substantial improvement in symptoms.
The study would be double-blind if
a. neither drug had any identifying marks on it.
b. none of the volunteers were allowed to see
the psychiatrist, and the psychiatrist is also
not allowed to see the volunteers during the
evaluation session.
knew which treatment any person had
received.
31
effectiveness of different insecticides in
controlling pests and their impact on the
productivity of tomato plants. What is the best
reason for randomly assigning treatments
(spraying or not spraying) to the farms?
a. Random assignment allows researchers to
generalize conclusions about the
b. Random assignment will tend to average out
all other variables, such as soil fertility, so
that they are not confounded with the
treatment effects.
other variables, like soil fertility.
d. Random assignment eliminates chance
variation in the responses.
7. Which of the following is not a benefit of keeping
other variables the same in an experiment?
a. Keeping other variables the same helps
prevent confounding.
variability in the response variable.
c. Keeping other variables the same makes it
easier to get statistically significant results if
one treatment is more effective than the
other.
the need for random assignment.
8. Every hour, the quality control manager at a
factory making tortilla chips selects a random
sample of 20 bags of chips and weighs the
contents. If the manager is convinced that the
mean weight of all bags produced that hour
differs from the target of 16 ounces, the
production line will be stopped so the problem
can be identified. In the current sample of 20
bags, the mean weight is 15.94 ounces, and this
estimate has a margin of error of 0.12 ounces.
Which of the following conclusions is best?
a. The manager should stop the production line
because the mean of 15.94 is less than 16.
b. The manager should stop the production line
because the margin of error is greater than 0.
c. The manager should not stop the production
line because the difference between 15.94 and
16 is less than the margin of error.
d. The manager should not stop the production
line because the margin of error is very small.
Section II: Free Response
survey results, Sam and Joshan randomly
selected 60 students from their school’s student
list and divided them at random into two groups
of 30. By email, they asked the students the
following questions about a nearby convenience
store, the Quick Shop:
think the prices at the Quick Shop are
fair, given that the typical markup is
25%?
of the Quick Shop, do you think the
prices are fair, given that the typical
markup is 25%?
Question Wording
Total 30 30 60
study? Justify your answer.
responded “fair” in each of the two treatment
groups and the difference in proportions
(Convenience – Markup).
performed to see what differences in
proportions would happen due only to
chance variation in the random assignment,
assuming that the wording of the question
didn’t matter. Use the results of the
simulation below to determine if the
difference in proportions from part (b) is
statistically significant. Explain your
conclusions can we draw? Explain.
32
university wants to evaluate the satisfaction level
of students living in one of the dormitories on
campus. She plans to interview a random sample
of 50 of the 816 residents of the building.
a. Explain how you could use a random
number generator to select a simple random
sample of 50 residents of the dormitory.
b. During each interview, the director asks, “If
you could make the choice again, would you
still choose this dorm?” In the sample of 50,
66% of the residents answered “Yes.” If the
director took a second random sample of 50
students, would this percent be the same?
Explain.
might make the estimate from part (b)
unreliable. Would increasing the sample size
solve this problem? Explain.
turns out that elephants dislike bees. They
recognize beehives and avoid them. Can this
information be used to keep elephants away from
trees? Will elephant damage be less in trees with
hives? Will empty hives even keep elephants
away? Researchers want to design an experiment
to answer these questions using 72 acacia
trees.106
and the response variable.
a completely randomized design for this
experiment. Include a description of how the
treatments should be assigned.
experiment in which subjects were told they were
“teachers” and were instructed to give electric
shocks to other subjects (“students”) when the
latter made mistakes on a word-matching test.
What the “teachers” didn’t know was that the
“students” were Milgram’s assistants, who faked
their discomfort at receiving nonexistent shocks.
Milgram showed that people were willing to
administer what they thought were serious
shocks rather than disobey someone in authority.
In what way did this experiment violate the
principles of ethical research?
.
What is the probability of getting an odd product from two dice?
We’re going to play a game to answer this question. You and your partner must decide who will be “Odds” and
who will be “Evens”. Then you will roll two dice and multiply the numbers. If the product is odd, the odd
person wins and vice versa for evens. Play 20 times, keeping track of how many wins each person has.
1. How many times did the odds win? Write this as a fraction out of 20 and turn it to a percentage.
Maybe the odds just had a run of bad luck. Let’s see how the rest of the class did with odds. Write the number of
odds wins for your group in the table on the board.
2. Find the total percent of rolls that were odd products for the whole class. How does this compare to your
group’s results?
3. To determine the true probability of rolling an odd product, we should list out all possible products that
we could get. Complete the table below to show all possible products.
4. Use your table to find the probability of rolling an odd product.
5. Which was closer to the percentage you found in #4, your group data
or the classroom data? Why do you think that is?
6. Explain what the probability of rolling odds means in this setting.
1 2 3 4 5 6
1
2
3
4
5
6
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New Jersey Transit claims that its 8:00 a.m. train from Princeton to New York has probability 0.9 of arriving on
time. Assume for now that this claim is true.
7. Explain what probability 0.9 means in this setting.
8. The 8:00 a.m. train has arrived on time 5 days in a row. What’s the probability that it will arrive on time
tomorrow? Explain.
9. A businessman takes the 8:00 a.m. train to work on 20 days in a month. He is surprised when the train arrives
late in New York on 3 of the 20 days. Should he be surprised? Describe how you would carry out a simulation
to estimate the probability that the train would arrive late on 3 or more of 20 days if New Jersey Transit’s
claim is true. Do not perform the simulation.
10. The dotplot below shows the number of days on which the train arrived late in 100 repetitions of the
simulation. What is the resulting estimate of the probability described in Question 3? Should the businessman
be surprised?
Probability
.
What is the probability of being a male student with blue eyes?
To answer today’s question, we will randomly select 10 students to come up front. Use those students’
information to answer the following questions.
1. In any given class, there are males and females who have blue, brown or green eyes. Create a table or
tree diagram that shows all possible combinations of gender with eye color.
2. Using the 10 students chosen, find each of the following probabilities:
P(Male) = P(Blue Eyes) =
P(Female) = P(Brown Eyes) =
P(Green Eyes) =
P(Other Eyes) =
3. Find each of the following probabilities and explain why your answer makes sense.
P(Male or Female) = P(Blue or Brown Eyes) =
4. Find each of the following probabilities and explain why your answer makes sense.
P(Male or Blue Eyes) = P(Female or Brown Eyes) =
5. Find each of the following probabilities and explain why your answer makes sense.
P(Not Green Eyes) = P(Not Male) =
37
Choose an American adult at random. Define two events:
A = the person has a cholesterol level of 240 milligrams per deciliter of blood
(mg/dl) or above (high cholesterol)
B = the person has a cholesterol level of 200 to <240 mg/dl (borderline high
cholesterol)
According to the American Heart Association, P(A) = 0.16 and P(B) = 0.29.
6. Explain why events A and B are mutually exclusive.
7. Say in plain language what the event “A or B” is.
8. Find P(A or B).
9. Let C be the event that the person chosen has normal cholesterol (below 200 mg/dl). Find P(C).
38
Sample Space
VS .
What is the probability that a randomly selected person from the class uses Facebook? Uses Twitter? Uses both?
Neither?
1. Collect class data to fill in the following two-way table.
Twitter No Twitter Total
Total
2. Suppose that we randomly choose a student from class. Find the following probabilities.
P(Facebook) = P(Twitter) =
P(no Facebook) = P(no Twitter) =
3. Now we will find some probabilities for two events occurring.
P(Facebook AND Twitter) = P(Twitter AND no Facebook) =
P(Facebook AND no Twitter) = P(no Facebook AND no Twitter) =
4. Is P(Facebook OR Twitter) = P(Facebook) + P(Twitter)? Why or why not?
5. Take the values from the two-way table and put them in the following Venn Diagram.
a. Where is “Twitter AND Facebook”
b. Where is “Twitter OR Facebook”
TwitterFacebook
40
What is the relationship between educational achievement and home ownership? A random sample of 500 U.S.
adults was selected. Each member of the sample was identified as a high school graduate (or not) and as a
homeowner (or not). The two-way table displays the data. Suppose we choose a member of the sample at
random. Define event G: is a high school graduate and event H: is a homeowner.
High School Graduate Not a High School
Graduate
Not a homeowner 89 71
6. Explain why P(G or H) ≠ P(G) + P(H). Then find P(G or H).
7. Make a Venn diagram to display the sample space of this chance process.
8. Write the event “is not a high school graduate but is a homeowner” in symbolic form.
41
Two-Way Tables
Venn Diagram
.
Definition: Two events are independent if knowing whether or not one event has occurred does not change the
probability that the other event will occur.
Are the events “Female” and “prefers English” independent?
1. Collect class data to fill in the following two-way table and Venn Diagram.
English Math Total
Female
Male
Total
2. Suppose that we randomly choose a student from class. Find the following probabilities.
P(Female) = P(English) =
P(Female AND no English) = P(no Female AND no English) =
3. Find P(Female OR English).
4. What is the probability that a student prefers English, given that they are a female? Write as a percent.
5. What is the probability that a student prefers English, given that they are a male? Write as a percent.
6. Are the events “Female” and “prefers English” independent? Explain.
Female English
43
Students at the University of New Hampshire received 10,000 course grades in a recent semester. The two-way
table below breaks down these grades by which school of the university taught the course. The schools are Liberal
Arts, Engineering and Physical Sciences (EPS), and Health and Human Services.
Grade Level
Engineering and Physical Sciences 368 432 800
Health and Human Services 882 630 588
Choose a University of New Hampshire course grade at random. Define a low grade as one that is below a B.
7. Find P(low grade | Engineering). Interpret this probability in context.
8. Find P(low grade | Liberal Arts). Interpret this probability in context.
9. Are events “low grade” and “engineering” independent? Justify your answer.
44
Probability of NO