Download AP Statistics Mixed Review Free Response

AP Statistics Mixed Review
Free Response
(2009B #5)
A bottle-filling machine is set to dispense 12.1 fluid ounces into juice bottles. To ensure that the machine is filling
accurately, every hour a worker randomly selects four bottles filled by the machine during the past hour and
measures the contents. If there is convincing evidence that the mean amount of juice dispensed is different from 12.1
ounces or if there is convincing evidence that the standard deviation is greater than 0.05 ounce, the machine is shut
down for recalibration. It can be assumed that the amount of juice that is dispensed into bottles is normally
distributed.
During one hour, the mean number of fluid ounces of four randomly selected bottles was 12.05 and the standard
deviation was 0.085 ounce.
a)
Perform a test of significance to determine whether the mean amount of juice dispensed is different from
12.1 fluid ounces. Assume the conditions for inference are met.
b) To determine whether this sample of four bottles provides convincing evidence that the standard deviation
of the amount of juice dispensed is greater than 0.05 ounce, a simulation study was performed. In the
simulation study, 300 samples, each of size 4, were randomly generated from a normal population with a
mean of 12.1 and a standard deviation of 0.05. The sample standard deviation was computed for each of the
300 samples. The dotplot below displays the values of the sample standard deviations.
Use the results of this simulation study to explain why you think the sample provides or does not provide evidence
that the standard deviation of the juice dispensed exceeds 0.05 fluid ounce.
(2010B #5)
An advertising agency in a large city is conducting a survey of adults to investigate whether there is an association
between highest level of educational achievement and primary source for news. The company takes a random
sample of 2,500 adults in the city. The results are shown in the table below.
a)
If an adult is to be selected at random from this sample, what is the probability that the selected adult is a
college graduate or obtains news primarily from the internet?
b) If an adult who is a college graduate is to be selected at random from this sample, what is the probability
that the selected adult obtains news primarily from the internet?
c)
When selecting an adult at random from the sample of 2,500 adults, are the events “is a college graduate”
and “obtains news primarily from the internet” independent? Justify your answer.
d) The company wants to conduct a statistical test to investigate whether there is an association between
educational achievement and primary source for news for adults in the city. What is the name of the
statistical test that should be used?
What are the appropriate degrees of freedom for this test?
(2010 #1)
Agricultural experts are trying to develop a bird deterrent to reduce costly damage to crops in the United States. An
experiment is to be conducted using garlic oil to study its effectiveness as a nontoxic, environmentally safe bird
repellant. The experiment will use European starlings, a bird species that causes considerable damage annually to
the corn crop in the United States. Food granules made from corn are to be infused with garlic oil in each of five
concentrations of garlic —0 percent, 2 percent, 10 percent, 25 percent, and 50 percent. The researchers will
determine the adverse reaction of the birds to the repellant by measuring the number of food granules consumed
during a two-hour period following overnight food deprivation. There are forty birds available for the experiment,
and the researchers will use eight birds for each concentration of garlic. Each bird will be kept in a separate cage and
provided with the same number of food granules.
a) For the experiment, identify
i. the treatments
ii. the experimental units
iii. the response that will be measured
b) After performing the experiment, the researchers recorded the data shown in the table below.
i. Construct a graph of the data that could be used to investigate the appropriateness of a linear regression
model for analyzing the results of the experiment.
ii. Based on your graph, do you think a linear regression model is appropriate? Explain.
(2010 #4)
An automobile company wants to learn about customer satisfaction among the owners of five specific car models.
Large sales volumes have been recorded for three of the models, but the other two models were recently introduced
so their sales volumes are smaller. The number of new cars sold in the last six months for each of the models is
shown in the table below.
The company can obtain a list of all individuals who purchased new cars in the last six months for each of the five
models shown in the table. The company wants to sample 2,000 of these owners.
a)
For simple random samples of 2,000 new car owners, what is the expected number of owners of model E
and the standard deviation of the number of owners of model E?
b) When selecting a simple random sample of 2,000 new car owners, how likely is it that fewer than 12
owners of model E would be included in the sample? Justify your answer.
c)
The company is concerned that a simple random sample of 2,000 owners would include fewer than 12
owners of model D or fewer than 12 owners of model E. Briefly describe a sampling method for randomly
selecting 2,000 owners that will ensure at least 12 owners will be selected for each of the 5 car models.
(2011 #6)
Every year, each student in a nationally representative sample is given tests in various subjects. Recently, a random
sample of 9,600 twelfth-grade students from the United States were administered a multiple-choice United States
history exam. One of the multiple-choice questions is below. (The correct answer is C.)
Of the 9,600 students, 28 percent answered the multiple-choice question correctly.
a) Let p be the proportion of all United States twelfth-grade students who would answer the question
correctly. Construct and interpret a 99 percent confidence interval for p.
Assume that students who actually know the correct answer have a 100 percent chance of answering the question
correctly, and students who do not know the correct answer to the question guess completely at random from among
the four options.
Let k represent the proportion of all United States twelfth-grade students who actually know the correct answer to
the question.
b) A tree diagram of the possible outcomes for a randomly selected twelfth-grade student is provided below.
Write the correct probability in each of the five empty boxes. Some of the probabilities may be expressions
in terms of k.
c)
Based on the completed tree diagram, express the probability, in terms of k, that a randomly selected
twelfth-grade student would correctly answer the history question.
d) Using your interval from part (a) and your answer to part (c), calculate and interpret a 99 percent
confidence interval for k, the proportion of all United States twelfth-grade students who actually know the
answer to the history question. You may assume that the conditions for inference for the confidence
interval have been checked and verified.
(2011B #6)
Grass buffer strips are grassy areas that are planted between bodies of water and agricultural fields. These strips are
designed to filter out sediment, organic material, nutrients, and chemicals carried in runoff water. The figure below
shows a cross-sectional view of a grass buffer strip that has been planted along the side of a stream.
A study in Nebraska investigated the use of buffer strips of several widths between 5 feet and 15 feet. The study
results indicated a linear relationship between the width of the grass strip (x), in feet, and the amount of nitrogen
removed from the runoff water (y), in parts per hundred. The following model was estimated.
yˆ = 33.8 + 3.6x
a)
Interpret the slope of the regression line in the context of this question.
b) Would you be willing to use this model to predict the amount of nitrogen removed for grass buffer strips
with widths between 0 feet and 30 feet? Explain why or why not.
A scientist in California wants to know if there is a similar relationship in her area. To investigate this, she will place
a grass buffer strip between a field and a nearby stream at each of eight different locations and measure the amount
of nitrogen that the grass buffer strip removes, in parts per hundred, from runoff water at each location. Each of the
eight locations can accommodate a buffer strip between 6 feet and 13 feet in width. The scientist wants to
investigate which combination of widths will provide the best estimate of the slope of the regression line.
Suppose the scientist decides to use buffer strips of width 6 feet at each of four locations and buffer strips of width
13 feet at each of the other four locations. Assume the model, yˆ = 33.8 + 3.6x , estimated from the Nebraska study
is the true regression line in California and the observations at the different locations are normally distributed with
standard deviation of 5 parts per hundred.
c) Describe the sampling distribution of the sample mean of the observations on the amount of nitrogen
removed by the four buffer strips with widths of 6 feet.
d) Using your result from part (c), show how to construct an interval that has probability 0.95 of containing
the sample mean of the observations from four buffer strips with widths of 6 feet.
For the study plan being implemented by the scientist in California, the graph on the left below displays intervals
that each have probability 0.95 of containing the sample mean of the four observations for buffer strips of width 6
feet and for buffer strips of width 13 feet. A second possible study plan would use buffer strips of width 8 feet at
four of the eight locations and buffer strips of width 10 feet at the other four locations. Intervals that each have
probability 0.95 of containing the mean of the four observations for buffer strips of width 8 feet and for buffer strips
of width 10 feet, respectively, are shown in the graph on the right below.
If data are collected for the first study plan, a sample mean will be computed for the four observations from buffer
strips of width 6 feet and a second sample mean will be computed for the four observations from buffer strips of
width 13 feet. The estimated regression line for those eight observations will pass through the two sample means. If
data are collected for the second study plan, a similar method will be used.
e) Use the plots above to determine which study plan, the first or the second, would provide a better estimator
of the slope of the regression line. Explain your reasoning.
f)
The previous parts of this question used the assumption of a straight-line relationship between the width of
the buffer strip and the amount of nitrogen that is removed, in parts per hundred. Although this assumption
was motivated by prior experience, it may not be correct. Describe another way of choosing the widths of
the buffer strips at eight locations that would enable the researchers to check the assumption of a
straightline relationship.