LUKE WILCOX: Hello, AP stat 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 video session 2, we are
going to go through two free response questions related
to probability.
Now I highly recommend that you download these questions first
and try them on your own before watching the video.
So there's a link on the video here or down
below in the description where you
can download, give those questions a try
and then come back here to check your work.
So let's jump right into the first free response question.
A machine at a manufacturing company
is programmed to fill shampoo bottles such
that the amount of shampoo in each bottle
is normally distributed with mean 0.6 and standard deviation
0.04.
Let the random variable A represent the amount
of shampoo in liters that is inserted into a bottle
by the filling machine.
Part A, a bottle is considered to be under filled
if it has less than 0.5 liter of shampoo.
Determine the probability that a randomly selected
bottle of shampoo will be under filled.
Show your work.
So any time you have some sort of random process here
and a random variable that you're going to be collecting,
it's good practice to start out by defining
what your random variable is.
Now even though they already said this in the question,
I'm going to put this down as part of my solution.
I'm going to let random variable A equal the amount of shampoo
in a randomly selected bottle.
Now the next thing to think about is the distribution
of that random variable A.
And you can see in the question, we're
told that it's normally distributed,
and we know what the mean and the standard deviation is.
So we want to make sure that we include that
as part of our solution here.
And there are several ways that you can do that
but my recommendation would be to start with a picture.
So here's my normal distribution.
And then I'm going to go ahead and add the mean at the center
of the distribution at 0.6.
And then I'm going to show the standard deviation
value by calculating one, two, and three standard deviations
away from the mean.
Now there are other ways that you can indicate all of this.
There is the shorthand notation of using the capital
N with the mean value of 0.6 and then the standard deviation
value of 0.04.
Now the next thing we want to make
sure we include in our solution here
is what's called the boundary value and then
the shaded region.
So in the question, it says that we're
looking for less than 0.5 liter of shampoo.
So I'm going to go ahead and mark
that boundary value in the picture
as well as the shaded region to the left.
So now we're looking for the area of that shaded region
to the left.
In other words, we're looking for the probability that
our random variable A is less than 0.50.
So the next thing that we need to do
is we need to standardize that boundary
value by calculating a z-score.
So you can see the calculation for the z-score
here, where we arrive at a value of negative 2.5.
Now what this is telling me is that boundary value of 0.50 is
2.5 standard deviations below the mean.
So we can then go to table A in order
to find the area of the shaded region, which is about 0.0062.
So I'm going to summarize this in a final statement
here where I say that the probability our randomly
selected bottle of shampoo will be under filled is about 0.062.
Now after the bottles are filled,
they are placed in boxes of 10 bottles per box.
After the bottles are placed in the boxes,
several boxes are placed in a crate
for shipping to a beauty supply warehouse.
The manufacturing company's contract with the beauty supply
warehouse states that one box will be randomly selected
from a crate.
If two or more bottles in the selected box are under filled,
the entire crate will be rejected and sent
back to the manufacturing company.
Part B, the beauty supply warehouse manager
is interested in the probability that a crate shipped
to the warehouse will be rejected.
Assume that the amounts of shampoo in the bottles
are independent of each other.
So part i says, define the random variable of interest
for the warehouse manager and state how the random variable
is distributed.
OK, this warehouse manager is going
to be taking one random sample of a box that
has 10 bottles in it.
And that warehouse manager is going
to be interested in how many of those 10
bottles are under filled?
So we're going to define the random variable of interest
as X which is the number of under filled bottles
in a box of 10 bottles.
Now the other part of this question that we have to answer
is to say how this random variable X is distributed.
Now if we're looking for the number of unfilled bottles
in a box of 10, there's four things we need to think about.
First, that each one of the bottles
has two possible outcomes, under filled or not under filled.
Each one of the bottles is independent,
that's stated in the question.
There is a fixed number of trials
or a fixed number of bottles, that's 10,
and the probability of being under filled
for each one of them is the same.
And actually that's what we calculated in part
A was the probability that each individual bottle
would be under filled.
Now because those four conditions are satisfied,
we know that X is a binomial random variable with a fixed
number of trials n equal to 10.
And then the probability of each bottle being under filled
is 0.062.
That's what we calculated in part A.
Now ii says determine the probability that a crate will
be rejected by the warehouse manager.
Show your work.
Now in order to be rejected, there
would have to be two or more bottles that were under filled.
So what we're looking for here is the probability
that X is greater than or equal to 2.
Now if you think about X being greater than or equal to 2,
that could be 2 bottles under filled or 3
or 4 or 5, 6, 7, 8,9,10.
That's a whole lot of values to calculate.
It's actually much easier here if we use the complement rule.
So I'm going to do 1 minus the probability that X is less than
or equal to 1.
And when I say X is less than or equal to 1,
that only includes two possibilities,
the possibility that X is 1 and the possibility that X is 2.
So we just subtract those two probabilities from 1
and we should arrive at our answer.
Now in order to calculate those probabilities,
we need the binomial formula here.
Now you can grab this formula from the formula
sheet on the AP exam.
And then I'm going to go ahead and substitute the values where
the probability that a bottle is under filled is 0.062
and the probability that it's not under filled
is the complement of that 0.9938.
Type all that into your calculator
and you're going to get a final answer of 0.0017.
So it is very unlikely that the crate would be rejected
by the warehouse manager.
All right, to reduce the number of crates rejected
by the beauty supply warehouse manager,
the manufacturing company is considering
adjusting the program of the filling machine
so that the amount of shampoo in each bottle
is normally distributed with a mean 0.56
and a standard deviation of 0.03.
Would you recommend that the manufacturing company
use the original programming of the filling machine
or the adjusted programming of the filling machine?
Provide a statistical justification for your choice.
Well, let me remind you about the original programming, which
we calculated in part A that the probability that a bottle would
be under filled would be 0.062.
Now that's with the original programming.
In this part of the question they
have the adjusted programming, which
now has a new mean and a new standard deviation.
And so showing that same calculation here,
you can see now that the probability that a bottle would
be under filled is 0.02275.
Now let's make sure that we actually
answer the question that's being asked here.
And it says, would you recommend that the manufacturing company
use the original programming?
And the answer is yes, we should continue
to use the original programming.
But the other part of the question that's important
is to give a reason, a justification for our choice
and that's why we calculated this new probability here.
And the answer is that the probability
of an under filled bottle is greater
for this adjusted programming.
0.02275 is greater than 0.062.
So let's jump into free response question number two
which says a company manufactures model rockets that
require igniters to launch.
Once an igniter is used to launch a rocket,
the igniter cannot be reused.
Sometimes an igniter fails to operate correctly
and the rocket does not launch.
The company estimates that the overall failure rate defined
as the percent of all igniters that fail to operate correctly
is 15%.
A company engineer develops a new igniter
called the super igniter with the intent
of lowering the failure rate.
To test the performance of the super igniters,
the engineer uses the following process.
So here's the process.
Step one, one super igniter is selected at random
and used in a rocket.
Step two, if the rocket launches,
another super igniter is selected at random
and used in a rocket.
Step 2 is repeated until the process stops.
The process stops when a super igniter fails
to operate correctly or 32 super igniters
have successfully launched rockets, whichever comes first.
Assume the super igniter failures are independent.
Part A, if the failure rate of the super igniters
is 15%, what's the probability that the first 30
super igniters selected using the testing process
successfully launch rockets?
OK.
They are telling us in the problem
that the failure rate is 15%.
So we can then use that to calculate the probability
that the super igniter is successful by calculating
the complement of that, which is going to give me 0.85.
So now we want to try and calculate the probability
that the first 30 are successful.
Now put another way, that means that the first one was
successful and the second and the third,
and so on until we get to the 30th one.
Well, it does say that each one of the igniters
is independent, which means we can simply
multiply these probability of success
for each one of the igniters.
And because there are 30 of them,
we end up with 0.85 raised to the 30th power, which is going
to give us a value of 0.0076.
Part B says, given that the first 30 super igniters
successfully launch rockets, what
is the probability that the first failure occurs
on the 31st or the 32nd super igniter tested
if the failure rate of the super igniters is 15%?
Now very important in this question
is the thing that is given to us.
The condition that is given to us
is that the first 30 have launched successfully
and now we want to think about the 31st and the 32nd trial.
So let's start by calculating the probability
that the first failure is on the 31st.
Well, the first 30 were all good.
So when we're looking at the 31st rocket,
it's the same probability of failure
as any one of the individual rockets and that's 15%.
Next, let's think about the probability
that the first failure occurs on the second igniter.
Now what that means is that the 31st one was successful
followed by a failure on the 32nd.
Now the probability that the 31st one
was successful is 0.85 followed by a failure on the 32nd one
is 0.15.
Now the reason why we can multiply those
is because we're told that they're independent.
So we arrive at an answer of 0.1275.
Now overall this question is asking
us to find the probability that the first failure occurs
on the 31st or the 32nd.
Now because those two events are mutually exclusive,
I can simply add the two probabilities
that we've already calculated to arrive
at a final answer of 0.2775.
All right part C. Given that the first 30 super igniters
successfully launch rockets is it
reasonable to believe that the failure
rate of the super igniters is less than 15%?
Explain.
Now this question is asking if it's
reasonable to believe that the failure rate is less than 15%.
And that sounds similar to me to a question
that would ask us if we have convincing evidence
that the failure rate is less than 15%.
And that sounds like a significance test.
Now if this was a significance test,
we would want to think about what our hypotheses would
be for the significance test and they would look like this.
The null hypothesis would say that the failure rate is 15%.
The alternative that we're trying
to find convincing evidence for would
say that the failure rate is less than 15%.
Now if this was a significance test,
we would want to calculate a P value in order to come up
with a conclusion here.
So in part A remember we found the probability
that the first 30 are all successful was 0.0076 assuming
the null hypothesis is true, that 15% failure rate
is correct.
Now this is a very, very low p value,
a low chance of this happening purely by chance.
So because this probability is so small, less than 5%,
we do, in fact, have convincing evidence that the failure
rate is less than 15%.
So I hope this video was helpful for you.
I also hope that you will join me in session three
where we will cover a couple of free response questions related
to collecting data.
Hope to see you soon.