Transcript AP Statistics - Somerset Independent Schools

AP Statistics
Chapter 7 Test
Review Answers
Concept Question #1
• A survey conducted by an unnamed polling firm
stated on October 29th, that Barack Obama was
predicted to receive 49.8% of the vote on
election day. On the actual day of the election,
he received 50.97% of the vote.
– Part a: Which of these numbers is a statistic?
• ANS: 49.8%
– Part b: Which of these number is a parameter?
• ANS: 50.97%
– Part c: after answering part a & b above, write each
of the given numbers using the correct ‘symbol’.
• ANS: p = .5097 & 𝒑 = .498
Concept Question #2
• When we talk about the ‘variability’ of a
statistic, then we are focusing on what aspect
of a distribution?
– ANSWER: The spread of the distribution.
Concept Question #3
• When we talk about the amount of “bias” in a
statistic, then what aspect of the distribution
are we talking about?
– ANSWER: The center of the distribution
Concept Question #4
• MULTIPLE CHOICE: Which of the following is the
preferred characteristic of an estimate of a
parameter
–
–
–
–
–
A. A statistic with high variability and low bias
B. A statistic with high variability and high bias
C. A statistic with low variability and low bias
D. A statistic with low variability and high bias
E. It doesn’t matter how much bias or variability a
statistic contains
• ANSWER: C…. The less bias and variability… the better.
Concept Question #5
• If you wish to reduce the amount of variability
in a statistic that we hope to use to estimate a
parameter, then what is the best way to
accomplish this?
– ANSWER: INCREASE the sample size.
Concept Question #6
• Two polling companies come to Pulaski County
(pop. 65,000) with the hopes of completing a
survey of its residents. One company plans to
survey 2500 residents, and the other plans to
survey 3100 residents. Which of the sampling
distributions would have the smaller 𝑠𝑥 ?
– ANSWER: The survey with 3100 residents… the
larger sample size, DECREASES variability/spread, so
it would produce a SMALLER standard deviation (𝒔𝒙 )
Concept Question #7
• A survey in 2006 of 500 randomly selected Kentuckians
found that 74% of them rooted for the Wildcats, 21% of
them supported the Cards, and 5% of them supported
neither. Suppose that the actual percentage of Wildcat
supporters was 63%.
– Part a: The mean of 𝜇𝑝 of p would be?
• ANSWER: 𝝁𝒑 = 𝟎. 𝟔𝟑… recall that we always use the ‘actual’ value
for this mean.
– Part b: Is 63% a parameter or a statistic?
• ANSWER: parameter
– Part c: Which of the following numbers is unknown: 74% or
63%?
• ANSWER: Technically, only God knows the value of a parameter, so
63% is unknown.
Concept Question #8
• A survey in 2006 of 500 randomly selected Kentuckians
found that 74% of them rooted for the Wildcats, 21%
of them supported the Cards, and 5% of them
supported neither. Suppose that the actual percentage
of Wildcat supporters was 63%.
– Part a: What would the standard deviation (𝜎𝑝 ) be?
• ANSWER:
.𝟔𝟑(.𝟑𝟕)
𝟓𝟎𝟎
= 𝟎. 𝟎𝟐𝟏𝟔
– Part b: What condition must be met in order to complete
the calculation stated in part a?
• ANSWER: the 10% condition, which states “the population MUST
be at least 10 times greater than the sample”
• This is often written symbolically as: n ≤ 0.10 N or n ≤ 𝟏 𝟏𝟎 𝑵
Concept Question #9
• A member of our boy’s basketball team makes 70% of
his free throw shots. Due to the fact that this player is
a starter and will play many minutes in each game, he
will make exactly 100 free throw attempts by season’s
end. Assuming that each free throw is independent,
what is the probability that his number of made free
throws exceeds 70?
– ANSWER: 0.500
• WORK SHOWN BELOW:
– 70 out of 100 is .70 (70%)
– P(z >
𝟎.𝟕−𝟎.𝟕
.𝟕(.𝟑)
𝟏𝟎𝟎
) = P(z > 0), which is 0.500
Concept Question #10
• Part a: In simplistic language, state the main
idea of the Central Limit Theorem.
– As ‘n’ increases, the sampling distribution
becomes more ‘normal’ in nature.
• Part b: In simplistic language, state the main
idea of the Law of Large Numbers.
– As ‘n’ increases, the value of 𝒙 gets closer and
closer to the value of 𝝁.
Concept Question #11
• What is the ‘magic number’ that ensures that
a sampling distribution (FOR MEANS) will be
approximately normal, even if the parent
population is highly skewed to the right?
– ANSWER: According to the Central Limit
Theorem, 30 is the ‘magic number’.
– Note: DO NOT use the phrase ‘magic number’ in
your answer to a FRQ on a test or on the AP
EXAM…
Concept Question #12
• What EXACTLY is the condition (FOR
PROPORTIONS) that must be met in order to
conclude that a ‘normal approximation’ can
be made (i.e.: distribution is normal)?
–ANSWER:
n·p ≥ 10 AND n·q ≥ 10
Concept Question #13
• What is the formula for standardizing (find a zscore) for a proportion?
Z=
𝑝−𝑝
𝑝𝑞
𝑛
Concept Question #14
• What is the formula for standardizing (find a zscore) for a sample mean?
Z=
𝑥−μ
𝑠𝑥
𝑛
Concept Question #15
• If we take many simple random samples
(SRS’s) from the exact same population what
do we expect to have happen?
• ANSWER: The values of the statistics that we
get from that population will vary from
sample to sample.
Concept Question #16
• The chipmunk population in a certain area is known to have a mean
weight of 84 grams and a standard deviation of 18 grams. A wildlife
biologist weighs 9 chipmunks that have been caught in live traps before
releasing them. Which of the following best describes what we know
about the sampling distribution of means for the biologist’s sample?
• NOTE: there was no mention of the word proportion on this problem for
good reason.. It has nothing to do with a proportion, but everything to
do with a mean…
• ANSWER:
• Center: μ𝒙 = 𝟖𝟒 𝒈
𝟏𝟖
• Spread: 𝒔𝒙 = = 𝟔 𝒈 (NOTE: We are allowed to make this calculation,
𝟗
because it is VERY safe to say that the population of chipmunks is almost
certainly greater than 10 times 9 (which would be 90 total chipmunks).
• SHAPE: Since the ‘n’ is rather small (n is only 9), then we CANNOT
assume normality, so the shape of the distribution is unknown and
cannot be assumed to be approximately normal. ‘n’ would have needed
to be greater than 30 for us to assume normality in this situation.
Concept Question #17
• Which of the following best describes a sampling
distribution?
• A. The distribution of all values of a statistic found in a
large number of simulated samples of size n.
• B. The set of all values of a variable in a sample of size n.
• C. The set of all values of a variable in a large number of
samples of size n.
• D. The distribution of parameter values in all possible
samples of size n.
• E. A probability distribution that describes the relative
likelihood of all possible values of a statistic.
• Answer: E
Concept Question #18
• In a large population of adults, the mean IQ is 112 with a
standard deviation of 20. Suppose 200 adults are randomly
selected for a market research campaign. The sampling
distribution of the sample mean IQ is
• A. exactly Normal, mean 112, standard deviation 20.
• B. approximately Normal, mean 112, standard deviation
0.1.
• C. approximately Normal, mean 112, standard deviation
1.414.
• D. approximately Normal, mean 112, standard deviation
20.
• E. exactly Normal, mean 112, standard deviation 1.414.
• ANSWER: C
Concept Question #19
• Suppose you are sampling from a distribution that is strongly skewed left.
Which of the following statements about the sampling distribution of the
sample mean is true?
• A. As the sample size increases, the shape of the sampling distribution
gets closer and closer to a Normal distribution.
• B. As the sample size increases, the shape of the sampling distribution
gets closer and closer to the shape of the population distribution.
• C. As the sample size increases, the mean of the sampling distribution
gets closer to the population mean.
• D. Regardless of the sample size, the shape of the sampling distribution
is similar to the shape of the population distribution.
• E.
Regardless of the sample size, the standard deviation of the sampling
distribution is approximately equal to the standard deviation of the
population.
• ANSWER: A
Concept Question #20
• Suppose we wish to estimate the percentage of students who smoke
cigarettes at each of several colleges and universities. Two of the colleges
are Wabash College (enrollment 900) and Purdue University (enrollment
36,000). What should the relative size of our samples be from each
school if we want the two sampling distributions to have approximately
the same standard deviation?
• A. Because the population sizes are so different, it’s impossible to make
the standard deviations equal by adjusting the two sample sizes.
• B. We should take a larger number of Purdue students, since there are
more of them.
• C. We should take a larger number of Wabash students, since there are
fewer of them.
• D. We should take the same size samples from each school.
• E.
We should take samples that are exactly 10% of each school’s
enrollment.
• ANSWER: D
Concept Question #21
• A statistic is said to be unbiased if
• A. the survey used to obtain the statistic was designed so
as to avoid even the hint of racial or sexual prejudice.
• B. the mean of its sampling distribution is equal to the
true value of the parameter being estimated.
• C. both the person who calculated the statistic and the
subjects whose responses make up the statistic were
truthful.
• D. the value from any sample is equal to the parameter
being estimated.
• E. it is used for honest purposes only.
• ANSWER: B