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1. Crop resear chers are interested in the productivity of a new variety of com. They plant 25plots with r mdomly-selected seeds of the new variety, record the yield in bushels per acre,and find thet a 99% confidence intervall for the true mean yield is 118 to 130 bushels peracre.

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(a) What is the point estimate from this sample? 9l.i"A-p\e. ~~d.Jv-

(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.

@ 2011 BFW Pub] ishers The Practice or Statistics, 4Je- Chapter 8

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3.61

2. A universit y health services physician is concerned about how much sleep freshman aregetting in t ie first few months of school. She asks a simple random sample of 20 studentshow much sleep they got the previous night and constructs a 95% confidence interval for themean amount of sleep in hours.

(a) Discus: whether this study meets the necessary conditions for constructing a confidenceinterva l. If you think one of the conditions has not been met, what additional informationwould )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 the95% iiuerval? Explain.

(c) How, vould the width of confidence interval change if the physician took a largersampl e? Explain.

362 © 2011 BFW Publishers

Quiz 8.1~ AP Statistics Name:(

1. Suppose ycu know that the distribution of finishing times for a certain crossword puzzle hasa mean of ~5 minutes, a standard deviation of 8 minutes, and is moderately skewed left. Youtake 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 withink minut ~sof the mean of the distribution.

(c) If you t~ke repeated samples of size 45 from this population, what proportion of the timewill the interval. :X- ± k contain the number 25? Explain.

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2. The confid ence level is sometimes called the "capture rate." Explain why this is anappropriate term.

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3. An insect scologist reports a 95% confidence interval for the mean length of full-grownaquatic lat vae of the Phantom Midge Chaoborus albatus to be 6.9 to 8.5 mm, based on asample 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 confidenceinterval were met." What does this tell you about his methods and about the populationof ins. :ct larvae?

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(c) If the ecologist had reported a 99% confidence interval instead ofa 95% interval, howwoulc. it have been different? Explain.

(d) The t cologist was unhappy with how wide this interval was. What should he do toprodice a narrower interval with the same level of confidence? Explain.

364 The Practice of Statistics, 4/e- Chapter 8 e 2011 BFW Pubiishers

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Quiz 9.1C --------------------------------------------------------------AP Statistics N2Lme: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.

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(a) You suspect that a certain six-sided die is not correctly balanced, so that the probability

of rollin: ~a 5 is something other than.!.. You plan to roll the die many times to test6

whether it's correctly balanced.

(b) Statistic: can help decide the authorship of literary works. Sonnets by an Elizabethan poetare known to contain an average of f1 = 6.9 new words (words not used in the poet'sother we rks) and the number of new words is approximately Normally distributed. Now anew mar uscript has come to light with many new sonnets, and scholars are debatingwhether it is the poet's work They take a simple random sample of five sonnets from thenew mar uscript and count the number of new words in each one. We expect poems byanother: 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 havedetermined t hat the number of new words in works by this poet Normally distributed with amean 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 plotshowing the results of simulating 200 samples of size 5 from aN ormal distribution with amean of6.9 and a standard deviation of2.7, and calculating the meanfor each sample. Use itto estimate the P-value ofthis test, and draw an appropriate conclusion for a significancelevel of a.= 1).05.

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Ie 2011 BFW Pub1i shers The Practice of Statistics, 4/e- Chapter 9 417

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3. A certain cgarette brand advertises that the mean nicotine content of their cigarettes is 1.5mg, but you are suspicious and plan to investigate the advertised claim by testing thehypotheses Ho: f.1 = 1.5 versus H; : f.1 > 1.5 at the a = 0.05 significance level. You will doso by meas uring the nicotine content of 30 randomly selected cigarettes of this brand.

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(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 I or Type II-is more serious?Explan.

(d) Expla in why it might be a good idea to increase the significance level to O.I0 for this test.

(e) You 'lave determined that at the a = 0.05 significance level, the power of the test againstthe atemativezz = 1.75 is 0.88. Explain what the power of the test means in the contextof 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 © 2011 BFW Publishers

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

XI alternative_x 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. Anew ingred ent is addedto help speed uprelief Let fl. denote the mean time to obtain painrelief with 1he new product. An .experiment is conducted to verify if the new prod uct worksmore quick y.What are the null and alternative hypotheses for the appropriate test ofsignificanct :?(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 of0.04. 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.

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 i1(1 - p) are both at least 10.(e) all values in the sample are within two standard deviations of the mean.

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436 The Practice of Statistics, 4/e- Chapter 9 © 2011 BFW Publishers

12. When the m mufacturing process is working properly, NeverReady batteries have lifetimesthat

follow a slig htly right-skewed distribution with Ii = 7 hours. A quality control supervisorselects 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 7hours 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 controlsupervis Drto test,

(b) Since te sting the lifewne of a battery requires draining the battery completely, thesupervi: or wants to sfa.mpleas few batteries as possible from each hour's production. Sheis consi iering a sa}tipk size of n = 4. Explain why this sample size may lead to problemsin 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 to1%.. 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 ectHo .(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 forfJ..

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 beenspeci 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 upuntil one goes to bed, but otherwise allows one to eat whatever one likes. Four adult volunteersagree to test the Iiet. They are weighed prior to beginning the diet and after six weeks on thediet. The weigh s (in pounds) are

Subject A B C DWeight before diet 180 125 240 150Weight after 6 weeks 170 130 215 152

7. Which of the following conditions must be met in order to use a t-procedure on these paireddata? .(a) Only the distribution of pre-diet weights must be approximately Normal.(b) Only the distribution of differences (after 6 weeks - before) must be approximatelyNormal.(c) The distribution of both pre-diet weights and six-week weights must be approximatelyNormal.(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 heapproximate ly

Normal.

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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 Ilga 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 papermaze mor ~quickly while listening to classical music. Suppose previous research hasestablishe 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 hypothesesHo : f.J = L. 0 versus Ha :p < 40, where u = the time in seconds needed to complete the mazewhile 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 testis 0.0002. Which statement below best describes the appropriate conclusion to draw from thisstudy?(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 significantresult.

10. The recor amended daily Calcium intake for women over 21 (and under 50) is 1000mg perday. The health services at a college are concerned that women at the college get lessCalcium han that, so they take a random sample of female students in order to test thehypotheses Ho : f.J = 1000 versus H; : f.J < 1000. Prior to the study they estimate that thepower of their test against the alternative H; : f.J = 900 is 0.85. Which of the following is thebest inter oretation of this value?(a) The probability of making a Type IIerror.(b) The probability of rejecting the null hypothesis when the parameter value if: 1000.(c) The j robability of rejecting the null hypothesis when the parameter value ia 900.(d) The r robability of failing to reject the null hypothesis when the parameter value is 1000.(e) The Irobability of failing to reject the null hypothesis when the parameter value is 900.

438 The Practice of Statistics, 4/e- Chapter 9 © 2011 BFW Publishers

Part 2: .Free ResponseShow all your WI Irk. Indicate clearly the methods you use, because you will be graded on thecorrectness of y( ur methods as well as on the accuracy and completeness of your results andexplanations.

11. Publishing s( .ientific papers online is fast, and the papers can be long. Publishing in a paperjournal meat s that the:paper wi11live forever in libraries. The British Medical Journalcombines 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. Onequestion 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 supportcontinuir 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.

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12. "Red tide" is a bloom of poison-producing algae-. a few different species of a class ofplankton c illed dinoflagellates. When weather and water condition cause these blooms,shellfish S11Chas clams living in the area develop dangerous levels of a paralysis-inducingtoxin. In 1tiassachusetts, the Division of Marine Fisheries (DMF) monitors leve ls of the toxinin shellfisl l by regular sampling of shellfish along the coastline. If the mean level of toxin inclams exceeds 800l1g (micrograms) of toxin per kg of clam meat in any area at a 5% level ofsignifican .e, clam harvesting is banned there until the bloom is over and levels of toxin inclams sub ride. During a bloom, the distribution of toxin levels in clams on a single mudflatis 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 whythis 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 theimp act this might have on error probabilities and the power of the test, and describe theprai :tical consequences of this change.

440 The Practice of Statistics, 4/e- Chapter 9 © 2011 BFW Publishers