DAREN STARNES: Hello, AP Stat students.
And welcome to AP Daily practice sessions.
I'm Mr. Starnes coming to you from Hilton Head, South
Carolina.
And in this session, we'll be focusing
on multiple choice questions from units 6 through 9
on statistical inference.
If you haven't done so already, please take a moment
and download the questions and try the questions,
actually work them.
All right, here we go.
First question.
Now I'm going to read the question,
I'm also going to annotate some key words as I go along.
I encourage you to do the same as you're taking the exam.
Helps you focus on some of these key words and phrases.
An agricultural engineer selected a random sample
of 30 farms in the United States to construct a 95% confidence
interval for the mean size in acres
of farms in the United States.
The resulting interval was 367 to 558.
Which of the following is an appropriate interpretation
of the 95% confidence level?
So one strategy you can use is to anticipate the answer
before you even look at the answer choices on the exam.
If you think about interpreting a confidence level, what
comes to mind, repeating the sampling process
many, many times and thinking about
what percent of the intervals would capture
the true population parameter?
So if you're thinking that way, you
might be able to pick out the correct answer choice
or at least to eliminate some of the incorrect ones.
Here are the choices.
So the first thing to notice is that answer choice A is focused
on 95% of the farm sizes.
That's about individual farms.
But the confidence interval is about the population mean
or the average farm size.
So it's not A.
Answer choice B also focuses on farm sizes,
not about the mean farm size.
So that's not correct.
Now C does talk about 95% of all random samples of size 30
from the population having a mean farm size.
And it shouldn't say that falls within the interval.
But not this interval, 367 to 558.
It's within the interval you generate each time,
95% of all those intervals would capture the truth.
So it's not C.
Well, it's down to D or E. So in D, we have approximately 95%
of all random samples of size 30 from the population
will produce intervals that contain the mean size of farms
in the US.
And E, we have approximately 95% of all random samples of size
30 from the population will produce intervals
that contain the sample mean, while all of the intervals
contain the sample mean because the sample mean is
right in the middle of them.
So it's definitely not E. And the answer is D.
Let's move to question two.
The manager of a car company will select a random sample
of its customers to create a 90% confidence
interval to estimate the proportion of its customers
who have children.
Of the following, which is the smallest sample size
that will result in a margin of error of no more
than six percentage points?
On this one, there's really no anticipating the answer
because it's a calculation.
So we'll look at the answer choices now.
We're going to need a formula for the confidence
interval for a proportion.
Now on your formula sheet that you
can use during the exam on both sections,
you've got this general formula for a confidence interval,
statistic plus or minus critical value times the standard error
of your statistic.
Where do you get the standard error of the statistic?
Well, it's on the next page of the formula sheet.
So we look for that for a sample proportion.
And there's our formula.
Now we're going to need to fill in the parts of the margin
of error, which is this part after the plus
or minus, the whole thing.
So the critical value is going to have to do
with the 90% confidence level.
And you might know that from working several questions
in your class off the top of your head,
or you might have to look it up in the table that's
provided on the exam, or you could use your calculator.
But however you do it, it's a number
that you should recognize as 1.645.
So that goes in for critical value.
Here's the formula for the standard error
of the proportion.
And we want that to be less than or equal to the six percentage
points or 0.06.
The rest is just some algebra of solving the inequality.
So we're going to first of all multiply both sides
by the square root of n.
And then we're going to square both sides to get rid
of that square root.
And now we have n greater than or equal to something.
The key here is to realize that you'll
get the largest sample size possible
if p hat is equal to 0.5.
That will make the sample size as large as it possibly
needs to be to get your margin of error
within six percentage points.
So we plug in 0.5, and we reach for a calculator and we type
those in, and we get something like 187.9ish.
So 188 would be the smallest sample size we could take.
Looking at the answer choices, we got to go higher than that.
So we would need at least 200 out of those options.
It's the most computational one we'll have.
All right, question three.
The mayor of a large city will run for governor
if he believes that more than 30% of the voters in the state
already support him.
He'll have a survey firm ask a random sample of n voters,
whether or not they support him.
And he'll use a large sample test for proportions
to test the null hypothesis that the proportion of all voters
who support him is 30% or less against the alternative
hypothesis that the percentage is higher than 30%.
Suppose that 35% of all voters in the state
actually support him.
In which of the following situations
will the power for this test be the highest?
Remember that the power is the probability
that the test will correctly reject the null hypothesis when
the alternative hypothesis is true.
In this case, the alternative hypothesis
is true because the true population proportion
is 0.35, 35%.
So to get the highest power for the test, we have some choices.
What will give us the highest power?
Well, will it be a larger sample or a smaller sample?
Which gives you better information
about the population parameter, the proportion?
Larger sample, right?
Larger sample gives better information.
So we can rule out the smaller sample size options--
250, and 500.
We can also then take a look at the significance level
because that has an effect on the power.
A lower significance level like 0.01
makes it harder to reject the null, whether the null is true
or not.
It requires stronger evidence to reject the null.
So in this case, the null is wrong
and the alternative is true because 35% of the voters
support him.
So if we want to be able to figure that
out the easiest using a significance level of 0.05
means we need less strong evidence
to be able to reject the null in general.
So that means it's answer choice E.
Right.
Next up.
A sports physician conducted a study
to investigate whether there's an association between running
experience and the occurrence of a certain sport
injury for marathon runners while training for a marathon.
You'll notice data were collected
for a random sample of 51 marathon runners,
and they categorized each of the runners from the sample as low,
medium, or high, running experience,
and also whether or not the runner experienced a sport
injury while training for the marathon.
So we have two variables, both categorical--
running experience, low, medium, high, and yes
or no on the sports injury.
The conditions for inference were met.
We like that.
And a chi-square test statistic of approximately 8.12
was calculated.
Which of the following describes the p-value of the test?
So we don't really have any advantage
to think about the answers till we see them.
There's a couple of ways to get to the answer,
but we need to think first, we have three possible values
for one variable, two possible values for another.
If you picture that in a two-way table,
you might have three rows and two columns,
or three columns and two rows.
But either way, your degrees of freedom
is going to be the number of rows minus 1 times the number
of columns minus 1.
So you're going to get two.
And that helps you be able to calculate
the p-value either using the table that comes with the exam,
if you prefer to do that.
You just look on the df equal to 2 row
and try to find your statistic of 8.12,
which is over here somewhere.
And that tells you that your probability
is between 0.01 and 0.02.
Or if you're not a table person, you could use your technology.
And here's the command for chi-square CDF
for the area to the right of that 8.12,
and we get a p-value of 0.017, which
is in between 0.01 and 0.02.
Whichever way you do it, you should
get answer choice D. So this is one where the technology is
really your friend.
It helps you not have to look on the table
if you're not a big fan of those tables.
Question five.
This one's a little longer.
A research study indicated a negative linear relationship
between two variables.
Number of hours per week spent exercising, exercise time,
and number of seconds it takes to run one lap around a track,
running time.
Computer output from the study is shown below.
So this is a regression analysis on the relationship
between these two quantitative variables.
Assuming that all conditions for inference are met,
we like that again.
Which of the following is an appropriate test statistic
for testing the null hypothesis that the slope
of the population regression line equals 0?
That's our null hypothesis.
All right.
You might be able to do this without looking at the answer
choices.
Always see if you can formulate a response to the question
without having to look because then you might not
be distracted by some of the tricky answer choices
that they deliberately put in there.
So you might know the general formula
for the standardized test statistic that
comes from the formula sheet.
You can also look on the first page of the formula sheet.
It's right here.
Standardized test statistic is statistic
minus parameter over standard error of the statistic.
So now what we need to do is figure out where on the output
is the statistic we're interested in for testing
the slope of the regression line.
Well, it's on this line.
It's the coefficient of exercise time
is the slope of the sample regression line.
That's the statistic we use.
And we're supposed to subtract statistic
minus the null value of the parameter, which
is given a zero.
And then divide that by the standard error,
which is given right here in the table.
So we can rule out all the ones with like 88 and 74.
Those are coming from elsewhere on the printout.
And we're down to the last two.
Well, we're just supposed to take the statistic minus 0
divided by the standard error.
It's just D. E is to trick you because, they've
put a square root of 11 down there in the denominator
as if this is an inference about a mean that might look familiar
from an inference about a mean formula.
Last question.
A team of psychologists studied the effect
of multitasking on the completion of cognitive tasks.
A group of 40 women participated in the study.
Each woman owned a smartphone equipped
with the same type of keyboard.
The women typed a text passage on the phone
twice once while sitting in a quiet room, that's
a single task, and the other time while walking,
that's their multitask.
The order of the single task in the multitask
was randomly determined for each woman.
The psychologist recorded the time
it took each woman to type the text for both tasks.
If the conditions of inference are met, yes,
which of the following tests is most appropriate to analyze
the data?
We have a menu of choices.
First thing to note.
What variable did the researchers record?
Time.
Is time a categorical variable or a quantitative variable?
It's quantitative.
So if we're analyzing a quantitative variable,
we should not be thinking about proportions.
We were thinking about means.
So C and D are out.
E is a test for categorical data as well.
Chi-square test for independence.
So we're down to A or B.
What helps us decide?
Well, the method of data collection
is what helps us decide.
So in this case, we need to know,
did the researchers randomly assign one group of women
to do the single task, sitting in the room,
and one group, a different group to do the multitask?
Because if so, that would be a two-sample t-test.
No.
What they did was they randomly assigned each woman
in a random order to do both things,
sitting in the quiet room and also doing the task while
walking.
So this is a matched pairs t-test for a mean difference,
because you would take the difference for each
of the women and the times that it took them to do that task,
and you would analyze the list of differences
and do a t-test on those differences.
Well, I hope you enjoyed the variety of questions there.
If you feel like you need a little bit more practice,
go into AP Classroom, click on the Review tab,
and you can look at some of on-demand review sessions
that come with completely different questions, a handout
that you can download and try the questions before you
look at the solutions.
But our next session will be session six,
free-response questions about insurance.
Hope to see you in the next session.