LUKE WILCOX: Hello, AP stats students,
and welcome to AP Daily practice sessions.
My name is Luke Wilcox.
I am an AP Stats teacher at East Kentwood High
School in Kentwood, Michigan.
And in this session, I'm going to take you
through two free-response questions, both of which
are multi focus.
Now, I highly recommend that you download these questions
and try them on your own first before watching the video.
So you can click the link on the video
or down below in the description,
download those questions, give them a try,
and then come back here to check your work.
So let's jump right in.
Here's your first free-response question.
Studies have shown that foods rich in compounds
known as flavonoids help lower blood pressure.
Researchers conducted a study to investigate whether there
was a greater reduction in blood pressure for people who consume
dark chocolate, which contains flavonoids,
than people who consumed white chocolate, which does not
contain flavonoids.
25 healthy adults agreed to participate in the study
and add 3.5 ounces of chocolate to their daily diets.
Of the 25 participants, 13 were randomly
assigned to the dark chocolate group,
and the rest were assigned to the white chocolate group.
All participants had their blood pressure
recorded in millimeters of mercury
before adding chocolate to their daily diets
and again 30 days after adding chocolate to their diets.
The reduction in blood pressure before minus after for each
of the participants in the two groups
is shown in the dot plots below.
Part A, determine and compare the medians
of the reduction in blood pressure for the two groups.
There are two things that this question is asking for.
The first is to determine the medians of each of the groups.
So let's start with the dark chocolate
group where we know there are 13 subjects or 13 data values.
Because that is an odd number of values,
the median is, in fact, the middle number
if you put them in order.
So you'll notice there's six dots to the left
and six to the right.
So we can see here that the median for dark chocolate
is equal to seven.
Now, moving to the other group, the white chocolate group,
there are 12 subjects in that group.
So we don't actually have a middle value
there because there is an even number of values.
So we have to look at the two middle values, which
are negative two and two and then find
the average of those, which here, is going to be zero.
Now, the other part of the question
is to compare the median.
So you want to make an explicit comparison.
So I'm going to say that the median for dark chocolate
is greater than the median for white chocolate.
The researchers found the mean reduction in blood pressure
for those who consume dark chocolate is
x bar dark equal to 6.08, and the mean reduction in blood
pressure for those who consume white chocolate, x bar white,
is 0.42.
So you can see a bigger reduction
in blood pressure for the dark chocolate group
compared to the white chocolate group.
In part B, one researcher indicated
that because the difference in sample means of 5.66
is greater than 0, there is convincing statistical evidence
to conclude that the population mean blood pressure
reduction for those who consume dark chocolate is greater
than for those who consume white chocolate.
Why might the researchers conclusion based only
on the difference in sample means of 5.66 not necessarily
be true?
Well, let's first just acknowledge
that there is a difference between those two means
where the dark chocolate seemed to have performed better
than the white chocolate.
But the question is whether this is convincing
statistical evidence.
And I think I want to make the distinction
between some evidence and convincing evidence.
And I'm going to say this, while the difference
of means being 5.66 does provide us
some evidence that dark chocolate is better
because it's greater than zero, it
is unclear if this is convincing evidence.
Now, here's why.
Remember, in the setup of the experiment,
we used random assignment to place 13 individuals in one
group and 12 into the other.
This idea of random assignment here is important
because it is possible that dark chocolate is not
more effective, and this result happened
purely by chance just because of the random assignment.
Another way to think about this is
if we were to do another random assignment,
we would get slightly different results,
and another random assignment would
get slightly different results.
In other words, it's likely to produce different sample means,
thus introducing variability into the sampling
distribution of the difference of sample means.
So what do we need in order to be able to say for sure
we have convincing evidence?
Well, this is why we need statistical inference.
Specifically, something like a significance test.
A significance test would allow us
to determine if we, in fact, have convincing evidence
that dark chocolate is better for reducing blood pressure.
A simulation was conducted to investigate whether there
is a greater reduction of blood pressure for those who
consume dark chocolate than for those
who consume white chocolate.
The simulation was conducted under the assumption
that no difference exists.
The results of 120 trials of the simulation
are shown in the following dot plot.
Now, very important, in order to create this simulated dot
plot here, we had to assume that no difference exists.
Now, to me, that sounds like the null hypothesis
for a significance test.
So here's what the null hypothesis
would look like, that the difference in means
is equal to zero.
There is no difference.
So part C says, use the results of the simulation
to determine whether the results from the 25
participants in this study provide convincing
statistical evidence at a 5% level of significance
that adding dark chocolate to a daily diet
will result in a greater reduction in blood
pressure on average than adding white chocolate
to a daily diet.
Justify your answer.
It says specifically here that we
need to use the results from the 25 participants.
Well, we saw those results in the previous part.
We saw that the difference between the two means was 5.66,
but the question is, could that difference
have happened purely by chance just because
of the random assignment?
Well, this is where we're going to use the simulated dot
plot to see is it possible we would
get a difference that large purely by chance.
Well, you can see, here's 5.66 on the dot plot,
and we did get a difference that large or larger but only three
times out of the 120.
That's going to give us a proportion of 0.025.
Now, what this is really telling us
is an estimate of the p value for the significance test.
The Holy Grail of AP Statistics, this is the p value.
So in our solution here, let's start
by interpreting that p value.
We're going to say, assuming the null,
assuming there is no difference in mean reduction of blood
pressure for those who consume dark chocolate
and those who consume white chocolate,
there is a 0.025 probability of getting
a difference of sample means of 5.66
or greater purely by chance.
Now, that is a small value, less than 5%.
So we do have convincing evidence
that there is a greater reduction of blood pressure
for those who consume dark chocolate than for those
who consume white chocolate.
Let's move on to our second free-response question, which
says, a Research Center conducted a national survey
about teenage behavior.
Teens were asked whether they had consumed
a soft drank in the past week.
The following table shows the accounts
for three independent random samples from major cities.
Part A, suppose one team is randomly
selected from each city sample.
A researcher claims that the likelihood
of selecting a teen from Baltimore
who consumed a soft drank in the past week
is less than the likelihood of selecting
a teen from either one of the other cities who consumed
a soft drank in the past week.
Because Baltimore has the least number
of teens who consumed a soft drink.
Is the researchers claim correct?
Explain your answer.
Now, this claim is being made about Baltimore.
So let's go to the table and look at the values
that we have for Baltimore.
And you can see the number of yeses at 727
is, in fact, the lowest of the three cities,
but also note that the sample size is much lower in Baltimore
than the other two cities.
Here, instead of looking at the count of the number of teens
that have had a soft drink, it makes more sense
to look at the proportion of teens
who have had a soft drink.
So I'm going to calculate the proportion
for Baltimore is about 0.804.
And now, let's compare that to the other two cities.
In Detroit, you can see about 0.741 of the teens
had consumed a soft drink.
And in San Diego, it looks like we have about 0.65.
So let's go ahead and answer our question now.
We're going to say, no, the proportion of Baltimore teens
who consumed a soft drank in the past week
is 0.804, which is certainly greater
than the proportion for those other two cities
that you see there.
Part B, construct a segmented bar chart
of relative frequencies based on the information in the table.
Now, relative frequencies.
The frequency is the number.
The relative frequency is the percent or the proportion.
Now, we actually just calculated those in part A.
We have the proportion for each one of the cities.
So we're going to take those three proportions
and show those in the segmented bar chart as a dark gray,
and then we're going to use light
gray to represent those teens that did not have a soft drink.
Now, very important, any time you have a segmented bar chart,
you need to make sure that you clearly
indicate in a key what each one of those sections represents.
Now, part B II says, which city had the smallest proportion
of teens who consumed a soft drink in the previous week.
Determine the value of the proportion.
Well, we've already done the work there,
and we can clearly see that San Diego has the smallest
proportion of teens at 0.65 compared to the other two
cities.
Part C says, consider the inference procedure that
is appropriate for investigating whether there
is a difference among the three cities in the proportion
of all teens who consumed a soft drank in the past week.
Part I, identify the appropriate inference procedure.
Looking at our data table here, we
can see that there is one variable that
is being measured here, and this is a categorical variable about
whether or not a soft drink has been drank in the last week.
And you'll also notice that we have
three different populations.
So we've selected three independent random samples
from three different populations.
So any time you have one categorical variable
that you're measuring across multiple populations,
the correct significance test here
would be a chi-square test for homogeneity.
Now, part II says to identify the hypotheses of the test.
Well, let's think about the null hypothesis, which
typically says something like, there is no difference.
And in this case, we're looking at the proportion
of teens that have had a soft drink between the three cities.
So for the null hypothesis, we're
going to say that there is no difference in the proportion
of all teens who consumed a soft drank in the past week
across the three cities.
Now, this means that the alternative
is going to say that there is, in fact, a difference.
And we're going to say there is at least one difference
in the proportion of all teens who
consumed a soft drank in the past week
across the three cities.
So I hope that this video was helpful for you,
and I hope that you will also join us in the next video
where you will meet my colleague and good friend, Daren
Starnes, who is going to lead you through session five, where
he takes you through some multiple choice
questions on the second half of the content from AP Stats.
Hope that you'll join Daren in the next video.