LUKE WILCOX: Hello, AP stat students.
And welcome to AP Daily--
Practice Sessions.
In this series of videos, we are going
to focus in on practice multiple choice
and practice free response in the hopes of getting
you prepared for the AP Statistics exam.
Now, I can tell you that each one of the questions that we've
chosen to go into these videos are very carefully chosen
because we know they cover content that you absolutely
have to know for the AP exam.
So if you can stick with us through all these videos,
you can be sure that you can walk into the AP exam feeling
confident.
So let's get started.
My name is Luke Wilcox.
I am an AP Stats teacher at East Kentwood High
School in Kentwood, Michigan, go Falcons,
where I've been an AP Stats teacher for about 20 years now.
I've also been in an AP exam grader for many years.
And so I'll do my best to bring all of those experiences
into the discussions that we're going
to have about these practice questions within these videos.
Now, in this first session, I'm going
to take you through six multiple choice questions all
covering content at the beginning half of the AP Stats
course.
Now, I highly recommend that you download these questions first
and try them on your own, then come back to the video
to look through the solutions.
So the link to those questions is on the video
or down below in the description.
Download those, give those a try,
and then come back here to see how you did.
So here is question number one.
A company determines the mean and standard deviation
of the number of sick days taken by its employees in one year.
Which of the following is the best description
of the standard deviation?
And here are the five answer choices.
Now, this question is about the interpretation
of this standard deviation.
And in order to arrive at that, let's
start by looking at the formula for standard deviation.
Now, this formula is provided for you
on the formula sheet for the AP exam.
And if we break this down by starting on the inside,
you can see that the first calculation that's done here
is to take each one of the individual values in a list
and figure out how far it is away from the mean.
In other words, we're finding the distance
from each individual value to the mean.
Now some of those are negative and some are positive,
so we end up squaring those in order
to get the squared distances.
And then in the next part of the formula,
you can see that essentially what we're doing here
is finding an average, adding them all up dividing
by the number of numbers.
Now, this is going to give us the average squared distance
from the mean.
The final step in the calculation
is to take the square root so that standard deviation is
the average distance from each individual value to the mean.
So now that we know that, let's go back to our answer choices
here and see which one we like.
I'm going to start with the answer choice A. Approximately
the mean distance between the number of sick days taken
by individual employees and the mean number of sick days taken
by all employees.
Now, that sounds exactly like what we were just
talking about in the formula.
I like this answer choice A, but I
want to make sure that the other answer choices are incorrect.
So let's look at each one of those individually.
Answer choice B you can see is referring
to the median distance, and we don't want the median distance.
We want the mean distance, or the average distance.
So answer choice B is out.
Answer choice C is talking about the distance
between the greatest number of sick days in the mean.
And it really should be each individual value in the mean.
So answer choice C is out.
Answer choice D is looking for the distance
between the fewest sick days and the most sick days.
Now, that's the distance between the minimum and the maximum.
And we know that distance is called the range of the data.
That's a different measure of variability, not the same
as the standard deviation.
So that one is out.
And then finally, answer choice E,
you can see in this one we're looking for the middle 50%
of the distribution.
This is what's called the InterQuartile Range or the IQR.
And that's a different measure of variability
besides standard deviation.
So that answer choice is out.
Which means that our gut instinct
to begin with answer choice A, is in fact, the correct answer
here for number one.
The researcher collected data on the latitude
in degrees North of the equator and the average low temperature
in degrees Fahrenheit for a random sample
of cities in Europe.
The data were used to create the following equation
for the least squares regression line.
And you can see the equation given.
Which of the following is the best interpretation
of the slope of the line.
Now, this question is all about interpreting slope,
and you can see all of our answer choices here.
But before we look at those, I want to talk about the best way
to interpret slope for AP Statistics.
Now, in order to do that, we first have to identify
the slope in the equation, which you can see here is negative
0.70.
Now, I'm Going to take you back to your first algebra class
ever where you probably learned that slope was the change in y
over the change in x.
Now, this slope of negative 0.70,
we could think of that as being a fraction over 1.
And in the context of the problem
that is the change in average low temperature divided
by the change in latitude.
Now, for AP Stats here is the best way to interpret slope.
We say for a change of latitude of one,
we predict that the change in the average low temperature
would go down by 0.7 degrees.
Now looking at all of our answer choices here,
let's examine the first part of the interpretation,
which should be talking about a change in latitude of one.
Now answer choices and B say exactly that.
It says for each one degree North of the equator.
And the other three answer choices
are all incorrect interpretation.
So let's get rid of those three answer choices,
leaving us with just the answer choice A and B. Now
the difference between those two answer choices
you can see an answer choice A it's
talking about an increase in temperature.
Well, we know that the slope is a negative value, which
means answer choice A is incorrect
leaving us with the correct interpretation for slope answer
choice B.
Moving on to number three, which of the following
is not a characteristic of stratified random sampling.
And we have our five answer choices here.
Now, notice, we are looking for the statement
that is not a characteristic.
In other words, we're looking for the false statement
out of these five.
Now, before we look at those answer choices,
I want to make sure you understand the idea
of a stratified random sample.
And I'm going to do that with a quick example here.
So let's suppose that we had a population of High School
students, and we wanted to take a stratified random sample
of 100 students.
Now, what we would do first is we would take that population
and split it into groups or strata based on some variable.
In this case, the variable is the grade level.
So we have 9th graders, 10th graders, 11th graders,
and 12th graders.
So we have split the population into strata.
Next from each one of those strata
we're going to select a sample random sample.
So we would take a sample random sample
of 25 freshmen, 25 sophomores, 25 juniors, and 25 seniors
in order to get our sample of size 100.
Now, keeping this example in mind,
let's go back to our answer choices here for number 3.
And remember, we're looking for the statement that is false.
So answer choice A says random sampling is
part of the sampling procedure.
Well, yeah that's true.
We took a random sample from each one of the grade levels.
Now, because that's a true statement.
This is not the one that we're looking for.
Answer choice B says the population
is divided into groups of units that are
similar on some characteristic.
Well, this is true in our example
as well because we divided the population of High School
students into grade levels where each grade
level, all the freshmen were ninth graders.
So this is a true statement, which
means we're not looking for that answer.
Answer choice C it says the strata are based on facts known
before the sample is selected.
Well, we definitely know the grade level of students
before we selected this sample.
So this is a true statement.
That one's out.
And then D, each individual unit in the population
belongs to one and only one of the strata.
Well, students only belong to one grade level at a time.
So this is a true statement, which means
that answer choice is out.
Now, our final answer choice says,
every possible subset of the population
of the desired sample size has an equal chance
of being selected.
Now, remember, we were selecting a sample of size 100,
and let's consider the special case that the sample of 100
was all seniors.
Is it possible that we could have a sample of 100 seniors
if we're using a stratified random sample?
No, that's not possible because we're taking 25
from each grade level.
So it's actually impossible that the sample
would be all seniors.
Which means this statement, every possible subset
of the population has an equal chance
of being selected is not true.
And remember, we were looking for the statement
that was false.
So answer choice E is the correct answer here.
Question four, an experiment has three mutually
exclusive outcomes--
A, B, and C. If the probability of A is 0.12, probability of B
is 0.61, and the probability of C
is 0.27, which of the following must be true?
And then we're given three statements to consider here.
Now, there's two very important ideas
that I want to review in this question,
and that is what does it mean for events to be mutually
exclusive, and what does it mean for events to be independent.
So let's start there.
And I'm going to start with mutually exclusive.
Now, what it means for two events to be mutually exclusive
is that they cannot occur at the same time.
Now, if they cannot occur at the same time,
when we find the probability of one event and the other event,
well, because they can't occur at the same time,
that probability has to be equal to 0.
Now, the other equation that's true for mutually exclusive
events is that the probability of A or B
is just equal to the probability of A plus the probability of B.
We don't have to worry about subtracting out the overlap
because there isn't any overlap for mutually exclusive events.
Now, let's think about the idea of independent events.
And what that means is knowing whether one event has occurred
does not change the probability that the other event
will occur.
And if you have two events that are independent,
then the probability of A and B can
be found by simply multiplying those two probabilities
together, the probability of A times the probability of B.
So now that we've reviewed those two important concepts,
let's go back to our three statements
and try and decide which of those are true.
And I'm going to start with the easy ones here.
Statement 2 says that the probability of A and B is zero.
Now, remember we know that A, B, and C are mutually exclusive.
They can't occur at the same time.
So, of course, this is a true statement.
Now, also because we know that all these events are mutually
exclusive, if I want to find the probability of B or C,
I can just simply add the two probabilities together.
So that's also a true statement.
Now, I think statement one is the trickiest here.
It says A and C are independent.
Now, let's just suppose to start that this is true,
that A and C are independent.
Now, if they are, then the probability of A and C
can be found by simply multiplying those two
probabilities together.
So we'd take those two probabilities,
multiply them together to get a value of 0.0324.
But remember, it says that events A and C are mutually
exclusive and cannot occur at the same time.
So the probability of A and C should be equal to zero.
It should not be equal to 0.0324.
Now, this means that statement one is an incorrect
or a false statement.
And that therefore means that the correct answer for number 4
is answer choice D, II and III.
Question number 5 says random variable
x is normally distributed with a mean 10 and standard deviation
3, and random variable y is normally
distributed with mean 9 and standard deviation 4.
If x and y are independent, which of the following
describes the distribution of y minus x.
So we are combining two random variables here
by subtracting y minus x, and we'd
like to describe the distribution.
Now, that usually means we're going to talk about
shape, center, and variability.
And the shape on all five of the answer choices
is the same, that's normal.
So we're good there.
So let's move to the center.
And you can see here we have to calculate
the mean of this random variable y minus x,
which is very easy to do.
It's just simply the mean of y minus the mean of x.
So we plug-in the values given in the problem.
And we see that means should be equal to negative 1.
Now, this allows us to eliminate a couple of answer choices
over here because they have the incorrect mean.
Now, the last thing that we have to do here
is to calculate the standard deviation.
And what students often want to do
is they want to subtract the standard deviations
or maybe even add the standard deviations,
but you got to remember, one does not simply
add standard deviations.
What we can add is the variances.
Now, that means we have to calculate the variance
for each, which is easy to do.
We just take the standard deviations there
and we square them and add those two variances together.
Now this is going to give us the variance of y minus x.
But remember, we want the standard deviation
of y minus x.
So you have to take the square root of the 25, which
will give you a standard deviation of 5,
leaving us with the correct answer
choice here of answer choice C.
Question number 6, in one region of the country,
the mean length of stay in hospitals
is 5.5 days with standard deviation 2.6 days.
Because many patients stay in the hospital for considerably
more days, the distribution of length of stay
is strongly skewed to the right.
Consider random samples of size 100 taken from the distribution
with the mean length of stay x bar recorded for each sample.
Which of the following is the best description
of the sampling distribution of x bar.
So this question is asking us for a description
of the sampling distribution.
And once again, I want to talk about
shape, center, and variability.
So let's start by thinking about the shape of the sampling
distribution of x bar.
Now in the problem, we are told that the population
distribution is strongly skewed to the right.
But we're also told that we're taking samples of size
100, a very large sample size.
In fact, larger than 30, which means
that we can use the central limit theorem here,
which would say that the sampling distribution of x bar
is approximately normal.
So even though the population distribution
is skewed to the right because of the central limit theorem,
the sampling distribution is approximately normal.
Now, that means I can get rid of these first three
answer choices there--
A, B, and C. And we are now just left with the answer choices D
and E.
Now, notice that the mean is 5.5 for each of those.
So it's all going to come down to the standard deviation here.
And remember, we want the standard deviation
of the sampling distribution of x bar.
Now, we can get that formula from the formula
sheet for the AP exam.
It's just sigma over the square root of n.
And then we plug-in our numbers from the problem
and we get a standard deviation of 0.26,
which means the correct answer for this one is answer choice
E.
So I hope that this video has been helpful for you.
I also hope that you will join me in the next session, session
2, where we will go through two free response questions on one
of the most challenging topics in all of AP Statistics
that have probability.
Hope to see you soon.