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Name:
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1. Crop resear chers are interested in the productivity of a new variety of com. They plant 25
plots with r mdomly-selected seeds of the new variety, record the yield in bushels per acre,
and find the t a 99% confidence intervall for the true mean yield is 118 to 130 bushels per
acre.
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.
(a) What is the point estimate from this sample?
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(b) What is the margin of error?
(c) Interpre t the 99% confidence interval 118 to 130 in the context of the problem.
(d) Interpre: the confidence level of 99% in the context of the problem.
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2011 BFW Pub] ishers
The Practice
or
I
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3.61
2. A universit y health services physician is concerned about how much sleep freshman are
getting in t ie first few months of school. She asks a simple random sample of 20 students
how much sleep they got the previous night and constructs a 95% confidence interval for the
mean amount of sleep in hours.
(a) Discus: whether this study meets the necessary conditions for constructing a confidence
interva l. If you think one of the conditions has not been met, what additional information
would )e required or what change in the study would you recommend?
(b) If, inst ead of constructing a 95% confidence interval, the physician constructed a 90%
confid ence interval, would the 90% interval be wider, narrower, or the same width as the
95% iiuerval? Explain.
(c) How, vould the width of confidence interval change if the physician took a larger
sampl e? Explain.
362
© 2011 BFW Publishers
Quiz 8.1~
AP Statistics
Name:
(
1. Suppose yc u know that the distribution of finishing times for a certain crossword puzzle has
a mean of ~5 minutes, a standard deviation of 8 minutes, and is moderately skewed left. You
take an SR:; of 45 finish times from this distribution and calculate the mean finish time, x.
(a) Describe the shape, center, and spread ofthe sampling distribution of
x.
(b) Find a r umber, k, such that 95% of the values in the sampling distribution will lie within
k minut ~sof the mean of the distribution.
(c) If you t~ke repeated samples of size 45 from this population, what proportion of the time
will the interval. :X- ± k contain the number 25? Explain.
© 2011 BFW Pub] ishers
The Practice of Statistics, 4/e- Chapter 8
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363
2. The confid ence level is sometimes called the "capture rate." Explain why this is an
appropriate term.
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3. An insect scologist reports a 95% confidence interval for the mean length of full-grown
aquatic lat vae of the Phantom Midge Chaoborus albatus to be 6.9 to 8.5 mm, based on a
sample of 9 individual larvae.
(a) What; ue the point estimate and margin of error associated with this confidence interval?
(b) The t:< ologist stated that "all necessary conditions for constructing this confidence
interval were met." What does this tell you about his methods and about the population
of ins. :ct larvae?
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(c) If the ecologist had reported a 99% confidence interval instead ofa 95% interval, how
woulc. it have been different? Explain.
(d) The t cologist was unhappy with how wide this interval was. What should he do to
prodice a narrower interval with the same level of confidence? Explain.
364
The Practice of Statistics, 4/e- Chapter 8
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Quiz 9.1C
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AP Statistics
--------------------------------------------------------------
1. For each of he following settings, define the parameter of interest and write the appropriate
null and alte mative hypotheses for the test that is described.
(a) You suspect that a certain six-sided die is not correctly balanced, so that the probability
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(b) Statistic: can help decide the authorship of literary works. Sonnets by an Elizabethan poet
are known to contain an average of f1 = 6.9 new words (words not used in the poet's
other we rks) and the number of new words is approximately Normally distributed. Now a
new mar uscript has come to light with many new sonnets, and scholars are debating
whether it is the poet's work They take a simple random sample of five sonnets from the
new mar uscript and count the number of new words in each one. We expect poems by
another: uthor to contain more new words than found in the Elizabethan poet's poems.
2. Consider the test of an Elizabethan poet's sonnets form question l(b). Scholar's have
determined t hat the number of new words in works by this poet Normally distributed with a
mean of 6.9 words and a standard deviation of (J = 2.7 words. When you examine the five.
new works, :TOU find that the mean number of new words is x = 9.2. Below is a dot plot
showing the results of simulating 200 samples of size 5 from aN ormal distribution with a
mean of6.9 and a standard deviation of2.7, and calculating the meanfor each sample. Use it
to estimate the P-value ofthis test, and draw an appropriate conclusion for a significance
level of a. = 1).05.
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3. A certain cgarette brand advertises that the mean nicotine content of their cigarettes is 1.5
mg, but you are suspicious and plan to investigate the advertised claim by testing the
hypotheses Ho: f.1 = 1.5 versus H; : f.1 > 1.5 at the a = 0.05 significance level. You will do
so by meas uring the nicotine content of 30 randomly selected cigarettes of this brand.
(a) Descril ,e what a Type I error would be in this context.
(b) Descri re what a Type II error would be in this context.
(c) From' he perspective of public health, which error-Type
Explan.
I or Type II-is
more serious?
(d) Expla in why it might be a good idea to increase the significance level to O.I 0 for this test.
(e) You 'lave determined that at the a = 0.05 significance level, the power of the test against
the atemativezz = 1.75 is 0.88. Explain what the power of the test means in the context
of th.~problem.
(f) Wha t impact will reducing the significance level to 0.01 have on the power of the test?
418
The Practice of Statistics, 4/e- Chapter 9
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Test 9B
AP Statistics
Name:
Part l:: Multiple Choice. Circle the letter corresponding to the best answer.
1. A significance test was performed to test the null hypothesis Ho: P = 0.5 versus the
alternative
Ha: p > 0,,:i. The test statistic is z = 1.40. Which of the following is closest to the P-value
for this test ~
(a) 0.0808
(b) 0.1492
(c) 0.1616
(d) 0.2984
(e) 0.9192
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2. The mean t me it takes for a person to experience pain relief from aspirin is 25 minutes. A
new ingred ent is addedto help speed uprelief Let fl. denote the mean time to obtain pain
relief with 1 he new product. An .experiment is conducted to verify if the new prod uct works
more quick y.What are the null and alternative hypotheses for the appropriate test of
significanct :?
(a) Ho : ti= 25 vs. R'a: fl. :;t:25
(b)Ho : u= 25 ve.H; :./1 < 25
(c) H«: fl. < 25 vs.H; :fl. = 25
(d) Ho : fl. -c ·25 vs. Ha: fl.. > 25
(e) Ho : fl.:= 25 vs. Ha: fl. > 25
3. A test of H 0 : f-l
= 60 versus
H; : f-l
:;t:
60 produces a sample mean of
x = 58 and a P-value
of
0.04. At ar IX = 0.05 level, which of the following is an appropriate conclusion?
(a) There ill sufficient evidence to conclude that fl. < 60.
(b) There ill sufficient evidence to conclude that fl. = 60.
(c) There ill, insufficient evidence to conclude that p: = 60.
(d) There i~,
insufficient evidence to conclude that J1 60.
(e) There i~,
sufficient evidence to conclude that J1 60.
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4. Because t procedures are robust, the most important condition for their use is
(a) the popi ilation standard deviation is known.
(b) the population distribution is approximately Normal.
(c) the data can be regarded as a random from the population.
(d) np and i 1(1 - p) are both at least 10.
(e) all values in the sample are within two standard deviations of the mean.
I
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The Practice of Statistics, 4/e- Chapter 9
© 2011 BFW Publishers
12. When the m mufacturing process is working properly, NeverReady batteries have lifetimes
that
follow a slig htly right-skewed distribution with Ii = 7 hours. A quality control supervisor
selects a srit ple random sample of n batteries every hour and measures the lifetime of each.
If she is CO]] ~mce(nhat the mean lifetime £f all batteries produced that hour is lesn than 7
hours at the 5~ significance level, then ill those batteries are discarded.
(a) Define t ie parameter of interest ~:6.dlstate appropriate hypotheses for the quality control
supervis Drto test,
(b) Since te sting the lifewne of a battery requires draining the battery completely, the
supervi: or wants to sfa.mpleas few batteries as possible from each hour's production. She
is consi iering a sa}tipk size of n = 4. Explain why this sample size may lead to problems
in carrying out t~~ sign\i.ficancetest from (a).
(c) Descril e a Type I and a Typ~ II error in this situation and the consequences of each.
(d) The qu ality control officer is considering changing the significance level of the test to
1%.. Discuss the impact this might have on error probabilities and the power of the test,
and de scribe the practical consequences of this change.
© 2011 BFW Pu blishers
The Practice of Statistics, 4/e- Chapter 9
435
S. We want to t estH« fJ. = 1.5 vs. H; : fJ.
"* 1.5 at a
= 0.05 . A 95% confidence interval for fJ.
calculated fn nn a given random sample is (1.4, 3.6). Based on this finding we
(a) fail to rej ect Ho .
(b) reject Ho .
(c) cannot make any decision at all because the value of the test statistic is not available.
(d) cannot make any decision at all because the distribution of the population is unknown.
(e) cannot make any decision at all because (1.4, 3.6) is only a 95% confidence interval for
fJ..
6. Which of the following statements is/are: correct?
L 'I'ae powerofa significance test depends on the alternative value of the parameter.
II. T re probability of a Type II error is equal to the significance level of the test.
IILE Tor probabilities can be expressed only when a significance level has been
speci led.
(a) I and.Il cnly
(b) .I and III.Imly
(c) II and III only
(d) I, II, and ill
(e) None of he above gives the complete set of correct responses.
Use the followin g for questions 7 and 8:
'-
The. water diet n quires one to drink two cups of water every half hour from the time one gets up
until one goes to bed, but otherwise allows one to eat whatever one likes. Four adult volunteers
agree to test the Iiet. They are weighed prior to beginning the diet and after six weeks on the
diet. The weigh s (in pounds) are
Subject
B
C
D
A
Weight before diet
240
150
180
125
Weight after 6 weeks
130
215
152
170
7. Which of the following conditions must be met in order to use a t-procedure on these paired
data?
.
(a) Only the distribution of pre-diet weights must be approximately Normal.
(b) Only the distribution of differences (after 6 weeks - before) must be approximately
Normal.
(c) The distribution of both pre-diet weights and six-week weights must be approximately
Normal.
(d) The distr ibution of pre-diet weights and the distribution of differences (after 6 weeks before)
must be :ipproximately Normal.
(e) All three distributions-before
diet, after 6 weeks, and the difference-must he
approximate ly
Normal.
©12011 BFW Publi-hers
The Practice of Statistics, 4!e- Chapter 9
437
8. What wou ld a Type IIerror be in this setting?
(a) Conch ding that the diet leads to weight loss when it doesn't.
(b) Conch iding that the diet leads to weight loss when it really does.
(c) Not cc ncluding that the diet leads to weight loss when it does.
(d) Not cc ncluding that the diet leads to weight loss when it really doesn't.
(e) Drawi Ilg a conclusion from this test when the Normality condition has not been satisfied.
9. A researcl ier wishes to determine if people are able to complete a certain pencil and paper
maze mor ~quickly while listening to classical music. Suppose previous research has
establishe i that the mean time needed for people to complete a certain maze (without music)
is 40 seco ads. The researcher, therefore, decides to test.the hypotheses
Ho : f.J = L. 0 versus Ha :p < 40, where u = the time in seconds needed to complete the maze
while lisu ning to classical music.
To do so, the researcher has 10,000 people complete the maze with classical music playing.
The mean time for these people is x = 39.92 seconds, and the P-value of his significance test
is 0.0002. Which statement below best describes the appropriate conclusion to draw from this
study?
(a) The n searcher has proved that listening to classical music substantially improves the
time i:takes to complete the maze,
(b) The n searcher has strong evidence that listening to classical music substantially
impro ves the time it takes to complete the maze.
(c) The n searcher has moderate evidence that listening to classical music subst antially
impro ves the time it takes to complete the maze.
(d) Altho rgh the researcher has obtained a statistically significant result, it appears to have
little] iractical significance.
(e) Since the P-vailue is greater than the reciprocal of the sample size, this is not a significant
result.
10. The recor amended daily Calcium intake for women over 21 (and under 50) is 1000mg per
day. The health services at a college are concerned that women at the college get less
Calcium han that, so they take a random sample of female students in order to test the
hypotheses Ho : f.J = 1000 versus H; : f.J < 1000. Prior to the study they estimate that the
power of their test against the alternative H; : f.J = 900 is 0.85. Which of the following is the
best inter oretation of this value?
(a) The probability of making a Type IIerror.
(b) The probability of rejecting the null hypothesis when
(c) The j robability of rejecting the null hypothesis when
(d) The r robability of failing to reject the null hypothesis
(e) The Irobability of failing to reject the null hypothesis
438
the parameter value
the parameter value
when the parameter
when the parameter
The Practice of Statistics, 4/e- Chapter 9
if: 1000.
ia 900.
value is 1000.
value is 900.
© 2011 BFW Publishers
Part 2: .Free Response
Show all your WI Irk. Indicate clearly the methods you use, because you will be graded on the
correctness of y( ur methods as well as on the accuracy and completeness of your results and
explanations.
11. Publishing s( .ientific papers online is fast, and the papers can be long. Publishing in a paper
journal meat s that the:paper wi11live forever in libraries. The British Medical Journal
combines the two: it prints short and readable versions, with longer versions available online.
Is this OK w .th authors?
The journal, sked a random sample of 104 of its recent authors several questions. One
question was "Should the journal continue using this system?" In the sample, 72 said "Yes."
(a) Do the data give good evidence that more than two-thirds (67%) of authors support
continuir g this system? Carry out an appropriate test to help answer this question,
(b) Interpret the P-value from your test in the context of the problem.
c 2011 BFW Publishers
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.
The Practice of Statistics, 4/e- Chapter 9
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439
12. "Red tide" is a bloom of poison-producing algae-. a few different species of a class of
plankton c illed dinoflagellates. When weather and water condition cause these blooms,
shellfish S11Chas clams living in the area develop dangerous levels of a paralysis-inducing
toxin. In 1tiassachusetts, the Division of Marine Fisheries (DMF) monitors leve ls of the toxin
in shellfisl l by regular sampling of shellfish along the coastline. If the mean level of toxin in
clams exceeds 800l1g (micrograms) of toxin per kg of clam meat in any area at a 5% level of
significan .e, clam harvesting is banned there until the bloom is over and levels of toxin in
clams sub ride. During a bloom, the distribution of toxin levels in clams on a single mudflat
is distinct y non-Normal.
(a) Defino the parameter of interest and state appropriate hypotheses for the DMF to test.
(b) Because of budget constraints and the large number of coastal areas that must be tested,
the D VIFwould like to sample no more than 10 clams from any single area. Explain why
this s unple size may lead to problems in carrying out the significance test from (a}.
(c) Desc ribe a Type I and a Type II error in this situation and the consequence s of each.
(d) The DMF is considering changing the significance level ofthe test to 10%. Discuss the
imp act this might have on error probabilities and the power of the test, and describe the
prai :tical consequences of this change.
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The Practice of Statistics, 4/e- Chapter 9
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