statistics 641 - final exams - 1998 through 2002longneck/fn641_98,02.pdf · statistics 641 - final...

Statistics 641 - FINAL EXAMS - 1998 through 2002

December 11, 1998

I. (50 points) Consider a set of observations Yi, i = 1, 2, . . . 400, which are assumed to be independent andidentically distributed with a mean µ and variance σ2. NOTE that due to the large sample size involvedhere, you may use normal-distribution tables for any probability calculations or decision rules requiredbelow.

a. Consider the null hypothesis H0 : µ = 12 and the two-sided alternative hypothesis H1 : µ 6= 12. Usethe following steps to present the customary t-test of this null hypothesis based on Y1, . . . , Y400 andα = 0.05.

i. Write down a general formula for the t test statistic commonly used for this hypothesis test.ii. Write down the decision rule for this hypothesis test. Use α = 0.05.

b. In the context of the hypothesis test presented in (a), give clear, explicit definitions of the followingterms.

i. Type I errorii. Type II erroriii. Power of the test.

c. For parts (c) and (d) of this question, you may assume that σ = 1.

i. Calculate the power of your test for the following six values of the true parameter µ: 11.9, 11.95,11.975, 12.025, 12.05, 12.1.ii. Use your results from (c.i) to sketch a power curve for your test. Be sure to label your axesclearly.

d. An agonomist reviews your work from (a) through (c) and objects, “The power you have for µ = 12.025is lower than I was hoping to get. How can I increase it?” Answer your agronomist’s question, payingcareful attention to: (i) your specific recommendation on how to increase the power; and (ii) explanation(based on the ideas from parts (a) through (c)) of why your recommendation will result in an increasein power.

e. The 400 observations considered above represent the weight (in grams) of pecans (a type of nut).However, (unknown to you) the 400 pecans were actually collected from 10 trees, with 40 pecanspicked from each tree. Also, your agronomist admits that within a given tree, pecan weights cannotbe considered independent, and will have a strong positive correlation, due to common genetic andenvironmental factors. Given this additional information, answer the following questions withoutcarrying out additional calculations.

i. How will this positive correlation within trees affect the expectation of the variance estimatoryou used in part (a.i)?ii. Suppose you ignored the positive correlation within trees and proceeded to use the t-testyou proposed in part (a). Will this increase or decrease the numerical values of test power youcalculated in part (c)? Explain.

f. In light of your answer to (e), your agronomist says, “OK, I see that it’s wrong to use the t-test from(a) to test our null hypothesis. What should I do instead?” Answer your agronomist’s question bypresenting a standard testing method that will account appropriately for the sampling design describedin (e). Be sure to give clear, explicit statements of both your test statistic formula and your decisionrule.

II. (2 points each) Place the letter of the best answer in the blank to the left of each question.

(1) The most crucial of the conditions imposed on the sampled data and the populations in order for thepooled t-test to be valid is

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A. normality.B. equal variance.C. independence.D. all three conditions are equally important.E. none of the conditions are crucial

(2) In a hypotheses test of Ho : µ ≥ 5 vs H1 : µ < 5, with σ known, if the sample size remains constant,but the level α is increased from .01 to .05, then the power of the test at µ=4,

A. increases.B. decreases.C. remains the same.D. may increase or decrease depending on the sample size.E. cannot be determined with the given information.

(3) A 95/99 tolerance interval for a normal population

A. has a higher degree of confidence than a 99% confidence interval on the population mean.B. is a 95% confident estimate of µ and a 99% confident estimate of σ.C. is an estimate of a region of values which will contain between 95% and 99% of the population

values.D. is a region of values for which we are 99% confident that the region contains 95% of the population

values.E. has a lower degree of confidence than a 99% confidence interval on the population mean

(4) An experimenter wants to test Ho : F = Fo, where F is the process cdf and Fo is a proposed cdf.Which one of the following statements is TRUE?

A. The Chi-squared GOF test is the preferred test statistic.B. The most powerful test statistic depends on the shape of Fo.C. The Anderson-Darling test has greater power than any other test.D. The Shapiro-Wilk test has greater power than the Chi-squared test.E. The Chi-squared GOF test can only be used when Fo is discrete.

(5) An unbiased estimator θ of the parameter θ

A. is never wrong.B. has a sampling distribution which is approximately normal for large n.C. has a sampling distribution in which the average value is θ.D. has a smaller MSE than most biased estimators.E. is based on random sampling principles.

(6) If f(y; θ) is a pdf which is symmetric about θ, then, amongest the three test statistics discussed inclass, the test statistic having greatest power

A. is the Wilcoxon Signed Rank test.B. is the sign test if f(y; θ) is a Cauchy pdf.C. is the t-test if f(y; θ) is a double exponential pdf.D. is the sign test if f(y; θ) is heavy-tailed.E. none of the above

(7) The reason that experimental units are paired in a study to compare the average responses of twotreatments

A. is to reduce the degrees of freedom of the t-test.B. is to reduce the variance of the difference in the two sample means.C. is to increase the degrees of freedom of the t-test.

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D. is to make the difference in the two sample means normally distributed.E. none of the above

(8) An administrator is studying the quality of high-school curriculums in Michigan. She randomly selects50 high schools out of the 357 high schools in Michigan for the study. A careful examiniation of theircurriculum is performed. Let X be the number of high schools in which the curriculum was found tobe unsatisfactory. The distribution of X is

A. binomialB. geometricC. hypergeometricD. normalE. can not be determined with the given information

(9) Let σ be the standard deviation of a population having a population which is highly skewed to the right .Suppose the experimenter wants to test hypotheses about σ but she can only run 10 experiments. (Ourstandard methods for drawing inferences about σ require the population distribution to be normallydistributed or the sample size to be relatively large.) The most appropriate advice for the experimenteris to

A. use a nonparametric approach.B. tell the experimenter to collect more data.C. use the Box-Cox transformation.D. apply the normal based procedures since they are usually robust to departures from normality.E. none of the above

(10) The Wilcoxon rank sum statistic is called a distribution-free test statistic since

A. it can take on only a finite number of values.B. its distribution under both the null hypothesis and alternative hypothesis does not depend on the

population distribution.C. its distribution is free of the assumption of equal variance.D. its distribution is free of the assumption of normality.E. its distribution under the null hypothesis does not depend on the population distribution.

(11) The Wilcoxon rank sum statistic is preferred to the pooled t-test

A. if the population distributions are normally distributed.B. for all continuous distributions.C. if the population distributions are symmetric.D. if the population distributions have equal variance.E. if the population distributions have heavy tails.

(12) A nonparametric density estimator has four components which must be selected prior to computingthe estimator. The component having the greatest impact on the shape of the resulting estimate is

A. the band width.B. the sample size.C. the number of plotting points.D. the kernel function.E. in fact all four components are equally important.

(13) In testing the hypotheses H0 : p ≤ .7 vs HA : p > .7, where p is a population proportion, α = .05 andn=15, the probability of a Type II error at p = .8

A. is .050.B. is .950.C. is .1788.

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D. is .8212.E. can not be determined using the given information

(14) An experiment was to be designed to study the the average compressive strength of steel beams.What sample size would be sufficient to ensure that the average compressive strength is estimated toa precision of 500 psi with a reliability of 0.99? The compressive strength of these types of beams hasapproximately a normal distribution with values ranging from 1,000 to 10,000 psi.

A. 200B. 60C. 246D. 538E. can not be determined using the given information

(15) In the estimation of the population quantile function, Q(u), the reason X(i) is used as an estimator ofQ((i-.5)/n) and not as an estimator of Q(i/n) is

A. there are only n plotting points and we need n+1 estimators.B. the resulting curve is much smoother.C. Q(1) is the largest value in the population and X(n) will nearly always underestimate it.D. all the aboveE. none of the above

(16 Which one of the following statements is FALSE?

A. The proportion of the data greater than or equal to the median is at least 50%.B. The sample mean is preferred to sample median for estimating the center of a distribution when

the data is from a Normal distribution.C. The probability of an outlier in a box plot depends on the population distribution.D. The distances from the median line to the two edges of the box in a box plot are equal if we are

sampling from a normal distribution.E. If the distribution is skewed to the right, then the median has a smaller value than the mean.

(17) Let X1, ..., X25 be iid N(µ, σ2) random variables. In testing H0 : σ ≥ 5 vs H1 : σ < 5, if the level ofsignificance was α=.01, the probability of Type II error at σ = 3.41 is

A. .01.B. .25.C. .50.D. .99.E. computed using the non-central chi-squared distribution and hence requires a computer program

in order to make the computation.

(18) An experimenter wants to test Ho : F = Fo, where F is the process cdf and Fo is a normal cdf. Thebest test statistic for testing this hypothesis

A. is the Anderson-Darling statistic.B. is the Kolmogorov statistic.C. is the Shapiro-Wilks statistic.D. is the Chi-squared GOF statistic.E. is any one of the statistics in A., B., C. or D. since they are equally powerful.

(19) A new type of transistor is in development. Using the data from an accelerated life test of the transistor,the failure rate function is found to be approximately a quadratic function. Let T be the time to failureof a randomly selected transistor. The distribution of T is modelled by a

A. gamma distribution.B. normal distribution.

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C. lognormal distribution.D. Weibull distribution.E. exponential distribution.

(20) Given that the population proportion, P, is known to be greater than 0.8, then in order to be 99%confident that the difference between the sample estimator P and the true value P is at most 0.1, thesample size n must be at least

A. 107.B. 166.C. 100.D. 207.E. cannot be determined without further information

(21) The reason for taking a stratified random sample is

A. to make all inferences correct.B. to increase the chance of reaching a desired conclusion.C. to reduce the risk of a oversampling a particular group.D. to guarantee that certain groups in the population will be included in the sample.E. all of the above

(22) Let N be the number of alpha particle emissions of carbon-14 that are counted per second by a Geigercounter. Suppose that N has a Poisson distribution with mean rate λ. Let T be the time in secondsuntil the third particle is recorded. The distribution of T is

A. normal.B. Weibull.C. hypergeometric.D. negative binomial.E. gamma.

(23) In testing H0 : µ ≥ 5 vs H1 : µ < 5, the P-value of the test statistic was 0.03. If the level of significancewas α=.01, and the true value of µ was µ=7, then the decision based on the data

A. was a Type I errorB. was a Type II errorC. was correctD. cannot be determinedE. all of the above

(24) A relative frequency histogram was used as an estimator of a continuous population pdf. The relativefrequency was plotted versus class intervals of greatly different widths. The plot will result in agraphical distortion since

A. the plotted rectangles will be too discrete.B. the population pdf is continuous and the relative frequency histogram is a step function.C. the relative frequency histogram will be highly skewed.D. there will be too much area under the curve for the widest class intervals.E. none of the above

(25) The P-value of the computed value of a test statistic is

A. the probability of observing a value of the test statistic more extreme to H1

B. the weight of evidence in favor of H1

C. the smallest value of α for which the observed data will reject Ho

D. the largest value of α for which the observed data will reject Ho

E. none of the above

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December 14, 1999

I. (50 points) The tensile strength of a material is the ability that the material possesses to resist deformationwhen a force or a load is applied to it. A metallurgist conducts a study to evaluate the tensile strength ofductile iron strengthened at two different temperatures. She thinks that the lower temperature will yield thehigher mean tensile strength. At each of the two temperatures, 800◦C and 1000◦C, 300 specimens of ductile ironwere heat treated. The data consists of the tensile strengths from 300 specimens heated to 800◦C: X1, . . . , X300

which are iid with mean µ1 and standard deviation σ1 and the tensile strengths from 300 specimens heated to1000◦C: Y1, . . . , Y300 which are iid with mean µ2 and standard deviation σ2. Furthermore, the X ′s and Y ′s areindependent.

a. The metallurgist is interested in the null hypothesis H0 : µ1 ≤ µ2 versus the alternative hypothesis H1 :µ1 > µ2 Use the following steps to present the customary t-test of this null hypothesis based on X1, . . . , X300

and Y1, . . . , Y300.

i. Write down a general formula for the t test statistic commonly used for this hypothesis test.

ii. Write down the decision rule for this hypothesis test. Use α = 0.05.

iii. State the necessary conditions needed for your procedure to be valid and how you would verify whetherthe conditions in are satisfied in this experimental setting.

b. In the context of the hypothesis test presented in (a), give clear, explicit definitions of the following terms,Make Sure to Frame Your Definitions in Terms of This Specific Problem

i. Type I error

ii. Type II error

iii. Power of the test.

c. For parts (c) and (d) of this question, you may assume that σ1 = σ2 = 1 and that the sample sizes are largeenough to invoke the central limit theorem if necessary.

i. Calculate the power of your test for the following six values of the parameter:

∆ =µ1 − µ2√

1/300 + 1/300= .5, 1.0, 1.5, 2.0, 2.5, 3

ii. Use your results from (c.i) to sketch a power curve for your test. Be sure to label your axes clearly.

d. The metallurgist in discussing your results from (a) through (c) states, “The power of the test when ∆ = 1.0is not large enough to meet industry standards. What needs to be done to increase it?” Answer themetallurgist’s question, paying careful attention to: (i) your specific recommendation on how to increasethe power; and (ii) explanation (based on the ideas from parts (a) through (c)) of why your recommendationwill result in an increase in power.

e. The 600 observations considered above represent the tensile strength obtained from the two levels of heattreatment. However, after the experiments were conducted, the metallurgist informs you that the heattreatment for the 300 specimens for each heat level were conducted in the following manner. The furnaceused to heat treat the specimens could hold only 5 specimens at a time. Thus, a tray containing 5 randomlyselected specimens was heated to the specified temperature for the prescribed length of time and thenthe tensile strength measurements were taken on the 5 specimens. The metallurgist states that there issome variation in the temperature from one experimental run to the next. Thus, there may be a strongpositive correlation between tensile strength readings for specimens on the same tray. Given this additionalinformation, answer the following questions without carrying out any additional calculations.

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i. How will this positive correlation within specimens affect the expectation of the variance estimatoryou used in part (a.i)?

ii. Suppose you did not adjust for the positive correlation within specimens and proceeded to use theordinary t-test you proposed in part (a). Will the positive correlation in the data increase or decreasethe numerical values of power you calculated for the test statistic in part (c)? Explain.

f. In light of your answer to (e), the metallurgist states, “Using the t-test from (a) to test the researchhypothesis is obviously flawed. What is an alternative approach to testing the research hypothesis?” Answerthe metallurgist’s question by presenting a standard testing method that will account appropriately for thesampling design described in (e). Be sure to give clear, explicit statements of both your test statistic formulaand your decision rule.


(1) Which one of the following statements is FALSE?

A. The proportion of the data greater than the median is at least 50%.

B. The standard deviation is preferred to MAD as an estimator of the population dispersion when thedata is from a Gamma distribution.

C. The sample median is preferred to the sample mean as an estimator of the population level when thedata contains extreme values.

D. The semi-interquartile range is the average of the difference between median and the first quartile andthe difference between the third quartile and the median.

E. If the distribution is skewed to the right, then the median has a smaller value than the mean.

(2) Let X1, ..., X25 be iid N(µ, σ2) random variables. In testing H0 : σ ≥ 5 vs H1 : σ < 5, if the level ofsignificance was α=.01, the probability of Type II error at σ = 3.1 is

A. .01

B. .25

C. .75

D. .99

E. computed using the non-central chi-squared distribution

(3) Let the random variable X have an Exponential distribution with cdf F (x) = 1− e−.25x for x > 0. The 20thpercentile of X is

A. .8926

B. 6.438

C. .0558

D. .4024

E. .20

(4) Nearly all (say 99.73%) of the units in a population have values between 10 and 900. Assume the populationis approximately normal. Rough estimates of the mean and standard deviation are

A. 455 and 150, respectively.

B. 450 and 222.5, respectively.

C. 455 and 225, respectively.

D. 455 and 148.3, respectively.

E. no estimates can be determined

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(5) Given that the population proportion, π, is known to greater than 0.8, then in order to be 99% confidentthat the difference between the sample estimator π and the true value π is at most 0.1, the sample size nmust be at least

A. 166

B. 136

C. 107

D. 150

E. cannot be determined without further information

(6) Unbiased estimators with small variances are desirable since

A. they have smaller mean square error than biased estimators

B. all their values are nearly equal to the parameters being estimated

C. their sampling distributions are highly concentrated about the parameter being estimated

D. they have known distributions whereas biased estimators do not


(7) The reason that experimental units are paired in a study to compare the average responses of two treatments

A. is to reduce the degrees of freedom of the t-test

B. is to reduce the variance of the difference in the two sample means

C. is to increase the degrees of freedom of the t-test

D. to make the difference in the two sample means normally distributed


(8) In a α = 0.05 test of Ho : µ ≥ 5 vs H1 : µ < 5, where the population distribution is approximately normalwith σ =4 and n=25, what is the power of the test at µ = 3.

A. 0.1963

B. 0.0500

C. 0.9500

D. 0.8037

E. cannot be determined with the given information

(9) The power of a test of hypotheses is

A. the probability that the test rejects Ho at specified points in the parameter space.

B. 1-α

C. the ability of the test to determine when the null hypothesis is false.

D. 1-β


(10) Suppose a plot of log(-log(1-ui)) vs log(Y(i)), where ui=(i-.5)/n and Y(i) is the ith order statistic, is nearlya straight line. The population distribution is likely a

A. normal distribution

B. Weibull distribution

C. Cauchy distribution

D. LogNormal distribution

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E. cannot be determined

(11) The Wilcoxon signed rank sum statistic is preferred to the paired t-test if

A. the population distribution of the differences is normally distributed

B. the Wilcoxon signed rank test is never preferred to the t-test

C. the population distribution of the differences is symmetric

D. the population distributions have unequal variances

E. the population distribution of the differences has extremely heavy tails

(12) The purpose of randomization in experimentation is to

A. point out the effects of extraneous factors.

B. eliminate governmental complaints.

C. eliminate the effects of extraneous factors.

D. estimate the effects of extraneous factors.

E. validate the reference distribution for inference purposes.

(13) In testing the hypotheses H0 : π ≤ .3 vs Ha : π > .3 using the Z-statistic, where π is a populationproportion, α = .05 and n=50, the probability of a Type II error at π = .4 is

A. .5398

B. .95

C. .4602

D. .1867


(14) An experiment was to be designed to study the the average compressive strength of concrete slabs. Whatsample size would be sufficient to ensure that the sample mean estimated the average compressive strengthto within 5 units with a reliability of 0.99? Compressive strength has approximately a normal distributionwith a standard deviation of approximately 15.

A. 150

B. 60

C. 49

D. 538

E. cannot be determined using the given information

(15) The life length, L, in thousands of hours of a new type of electronic control is to be determined. Theengineer finds that the distribution of L is not normal, but she finds that a plot of log(L) yields nearly astraight line on a normal probability plot. The distribution of L is

A. normal with µ 6= 0 and σ > 1.

B. Weibull.

C. exponential.

D. lognormal.

E. gamma

(16) Of the three conditions imposed on the experiment in order for the pooled t-test to be valid, the one mostaffecting the power of the test is

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A. normality

B. equal variance

C. independence

D. all three conditions are equally important

E. none of the conditions are crucial

(17) In a hypotheses test of Ho : µ ≥ 5 vs H1 : µ < 5, with σ known, if the sample size remains constant, butthe level α is increased from .01 to .05, then the power of the test at µ=4,

A. increases

B. decreases

C. remains the same

D. may increase or decrease depending on the sample size


(18) An engineer wants to determine the number of miles, W, such that 5% of all cars produced by his companyin 1999 will have a transmission fail at a mileage less than W. The engineer evaluates 25 cars on a testtrack and determines the number of miles until transmission failure. These 25 values yield y = 55, 000 ands = 1, 000 with a p-value of .237 for the Shapiro-Wilks test. A 99% lower confidence bound on W is

A. 52,367

B. 54,485

C. 57,633

D. 53,355

E. cannot be determined from this data

(19) An experimenter wants to test Ho : F = Fo, where F is the process cdf and Fo is a specified discrete cdf.Which one of the following statements is TRUE?

A. The Chi-squared GOF test is only for testing hypotheses about pmf’s.

B. The Shapiro-Wilk test has greater power than the Chi-squared test.

C. The Anderson-Darling test has greater power than any other test.

D. The Chi-squared GOF test is the preferred test statistic.

E. The Shapiro-Wilk and Anderson-Darling are equally preferred.


A. the probability of observing a value of the test statistic more extreme to H1


C. the smallest value of α for which the observed data will reject Ho

D. the largest value of α for which the observed data will reject Ho


(21) In an experiment to study the effects of vibration on the strength of tempered steel, the tensile strengthof steel specimens was measured. A statistician suggested making 4 separate measurements of the tensilestrength of each specimen and recording the average of the 4 measurements. This will help to

A. decrease the bias in measuring strength.

B. increase the validity of the measurement.

C. increase the reliability of the measurement.

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D. decrease the significance of the measurement.

E. increase the unbiasedness of the measurement.

(22) In a Box Plot, the probability that a data point is designated as an extreme outlier

A. depends on the sample size

B. depends on the population distribution

C. depends on the median of the population distribution

D. is the same for all population distributions


(23) In testing H0 : µ ≤ 5 vs H1 : µ > 5, the P-value of the test statistic was computed to be 0.003. If the levelof significance was α=.01, and the true value of µ was µ=4, then the decision based on the data

A. was a Type I error

B. was a Type II error

C. was a Type III error

D. was correct


(24) As the sample size n increases, the sample relative frequency histogram will tend towards the shape of

A. the population probability density function(pdf)

B. the population cumulative distribution function(cdf)

C. the population quantile function

D. the pdf of a normal distribution


(25) An unbiased estimator of a parameter θ

A. is never wrong

B. has a symmetric distribution

C. is the best possible estimator

D. is a method of moments estimator

E. has average value equal to θ

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December 12, 2000

I. (28 points) Weight gain in the first 3 months after birth is important for new born infants. A pediatricianwishes to test a new feeding formula to determine if it will cause greater weight gain in new born infants thanthe standard formula.From her records she finds that the first 3 months weight gains of single birth infants on the standard formulahave the following characteristics:

µS = 15oz. and σS = 6oz.On the other hand, the first 3 months weight gains of identical twins on the standard formula have the

characteristics:µT = 12oz., σT = 6oz. variation between sets of twins, andρ = .8 (correlation in weight gain of identical twins)

She wants to run a 3 month experiment on a group of infants to test if the new formula provides a greater increasein weight than the old formula. She has decided to use a 5% probability of Type I error, and wishes to be ableto detect an increase in weight gain of at least 3 oz. with 90% probability.

(a.) What sample size must she use if she does the experiment with a random sample of n single birth infants?State any assumptions you are making in this calculation.

(b.) What sample size must she use if she does the experiment with n pairs of identical infants? State anyassumptions you are making in this calculation.

(c.) Discuss the relative merits of the two experiments in terms of practicality and of her basic goal.

II. (3 points each) For each of the following statements, state whether the given statement is TRUE or FALSE.If the statement is FALSE, explain VERY BRIEFLY why the statement is FALSE or CORRECT thestatement by changing a few words or numbers.

(1) The sample standard deviation is prefered to MAD as an estimator of population dispersion when thepopulation distribution has very heavy tails.

(2) Given that the population proportion, π, is known to less than 0.8, then in order to be 99% confident thatthe difference between the sample estimator π and the true value π is at most 0.1, the sample size n mustbe at least 107.

(3) The reason that experimental units are paired in a study to compare the average responses of two treatmentsis to reduce the degrees of freedom of the t-test.

(4) The power of a test of hypotheses is the probability that the test rejects Ho at specified points in theparameter space.

(5) A level α = .10 test of Ho : π ≥ .20 vs Ha : π < .20, where π is a population proportion, is conducted basedon a random sample of n=20 units from the population. The probability of a Type II error of this test ifπ=.1 is .002.

(6) The Wilcoxon signed rank sum statistic has greater power than the paired t-test when the population distri-bution of the differences is symmetric but has extremely heavy tails since the t-test has smaller probabilitiesof Type I errors.

(7) Of the three conditions imposed on the experiment in order for the pooled t-test to be valid, the one mostaffecting the power of the test is equal variance.

(8) In a hypotheses test of Ho : σ ≤ 20 vs H1 : σ > 20, from a population having a normal distribution , ifthe sample size remains constant, but the level α is increased from .01 to .05, then the power of the test atσ=24 increases.

(9) An experimenter wants to test Ho : F = Fo, where F is the process cdf and Fo is a specified discrete cdf.The Anderson-Darling test has greater power than any other test.

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(10) The P-value of the computed value of a test statistic is the largest value of α for which the observed datawill reject Ho

(11) In an experiment to study the effects of vibration on the strength of tempered steel, the researcher wasgreatly restricted in the number of replications that could be conducted. Thus, she made 10 measurementson each of the 5 steel specimens and constructed a 95% confidence interval on the average strength, µ usingn=50. In fact, the multiple measurements on a given specimen are very positively correlated. This willresult in a confidence interval having a higher level of confidence than the stated 95% confidence.

(12) In a Box Plot, the probability that a data point is designated as an extreme outlier is the same for allpopulation distributions since the Box plot is a distribution-free procedure.

(13) In testing H0 : π ≤ .3 vs Ha : π > .3, the P-value of the test statistic was computed to be 0.3. If the levelof significance was α=.01, and the true value of π was .4, then the decision based on the data was a TypeI error.

(14) The skewness and kurtosis parameters for a given cdf are generally thought to represent the heaviness ofthe tails of the cdf and how much the cdf differs from a normal cdf.

(15) A 95/99 lower tolerance interval for a Weibull population is an estimate of a region of values which willcontain between 95% and 99% of the population values.

(16) If f(y; θ) is a pdf which is symmetric about θ, then, amongest the three test statistics discussed in class,the test statistic having greatest power is the Wilcoxon Signed Rank test.

(17) Let σ be the standard deviation of a population having a distribution which is highly skewed to the right.Suppose the experimenter wants to estimate σ using a 90% confidence interval but she can only run 10experiments. The most appropriate advice for the experimenter is to use a Chi-squared based confidenceinterval for σ but with a higher level of confidence, say 99item[(18)] In a test of the difference in the means of two population means, where both populations havea normal distribution, the researcher designs the study so that the sample sizes are the same. The mainreason for having equal sample sizes is to simplify the calculations involved in using the pooled-variancet-test.

(19) A nonparametric density estimator has four components which must be selected prior to computing theestimator. The component having the least impact on the shape of the resulting estimate is the number ofplotting points.

(20) In the estimation of the population quantile function, Q(u), the reason X(i) is used as an estimator of Q((i-.5)/n) and not as an estimator of Q(i/n) is there are only n plotting points and we need n+1 estimators.

(21) The reason for taking a stratified random sample is to guarantee that certain groups in the population willbe included in the sample.

(22) In a level α = .01 t-test of Ho : µ ≤ 5 vs Ha : µ > 5, where µ is the mean of a normally distributedpopulation, a random sample of n=7 observations was selected. The power of the t-test when µ is onestandard deviation greater than 5 is .3085.

(23) A relative frequency histogram was used as an estimator of a continuous population pdf. The relativefrequency was plotted versus class intervals of greatly different widths. The plot will result in a graphicaldistortion since the plotted rectangles will be too discrete.

(24) The GOF statistics, Kolmogorov-Smirnov, Cramer von Mises, and Anderson-Darling are referred to asdistribution-free statistics when Fo is completely specified since the distributions of all three statistics onlydepend on location-scale parameters.

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December 10, 2001

I. (50 points) A company designs a study to evaluate two methods (M1,M2) for converting recycled automobiletires into surfaces for tennis courts. The company wants to compare the average surface traction of the materialproduced by the two processes. Method M1 is the conventional method of conversion and Method M2 is a newmethod which is more expensive in its conversion of the tires. The company wants to determine if M2 producesa surface having a higher average traction rating than M1. Since M2 is more expensive, the mean traction of M2

must be at least 5 units larger than the mean for M1 in order for it to be considered economically feasible. Thecompany decides to take a random sample of material on 50 consecutive days of production from each of the twomethods. The data consists of the surface traction measurements of the 50 samples from M1 : X1, . . . , X50 withprocess mean µ1 and process standard deviation σ1 and the surface traction measurements from 50 specimensfrom M2: Y1, . . . , Y50 with process mean µ2 and process standard deviation σ2.

(a) The company is interested in the research hypothesis H1 : µ1 + 5 < µ2.

i. Write down a general formula for the t test statistic for this hypothesis test (It is presumed that M2

will produce a product having a more consistent surface traction than M1.).

ii. Write down the decision rule for this hypothesis test. Use α = 0.05.

iii. State the necessary conditions needed for your procedure to be valid and how you would verify whetherthe conditions are satisfied in this experimental setting.

(b) In the context of the hypothesis test presented in (a), give clear, explicit definitions of the following terms,Make Sure to Frame Your Definitions in Terms of This Specific Problem

i. Type I error

ii. Type II error


c. For parts (c) and (d) of this question, you may assume that σ1 = 3 and σ2 = 1 and that the sample sizesare large enough to invoke the central limit theorem if necessary.

i. Calculate the power of your test for the following six values of the parameter:

µ2 − µ1 = 4.5, 5.0, 5.5, 6.0, 6.5, 7

ii. Use your results from (c.i) to sketch a power curve for your test. Be sure to label your axes clearly.

d. The company’s engineer examines your results from (a) through (c) states, “The power of the test whenµ2 − µ1 = 5.5 is not large enough. Determine the minimum sample size necessary to achieve a power of atleast 0.90 when µ2 − µ1 ≥ 5.5.

e. The 50 observations considered above represent the surface traction obtained from 50 consecutive days ofproduction. Thus, there may be a strong positive correlation between the surface traction measurements.Given this additional information, answer the following questions.

i. If the correlations between pairs of daily measurements from method M1 are equal to ρ > 0, demon-strate with a mathematical calculation how this positive correlation between the daily measurementswill affect the estimated standard error of µ2?

ii. Suppose you did not adjust for the positive correlation between the daily measurements and proceededto use the ordinary t-test you proposed in part (a). Will the positive correlation in the data increaseor decrease the numerical values of power you calculated for the test statistic in part (c)? Explain.

iii. Suppose you did not adjust for the positive correlation between the daily measurements and proceededto obtain a 95% C.I. for µ2 using procedures for independent random samples. What is the effect ofthe positive correlation in the data on the level of confidence of your C.I.? What is the effect of thepositive correlation in the data on the width of your C.I.?

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(1) The reason for taking a stratified random sample is to

A. reduce the required sample size.

B. increase the participation of persons in a survey.

C. increase the chance that certain subsets of the populations will be included in the sample.

D. reduce the operational costs of running a survey.


(2) Let X1, ..., X25 be iid N(µ, σ2) random variables. In testing H0 : σ ≥ 5 vs H1 : σ < 5, if the level ofsignificance was α=.01, the chance of rejecting Ho at σ = 2.6 is

A. .01

B. .98

C. .02

D. .99

E. computed using the non-central chi-squared distribution

(3) Let the random variable X have an Weibull distribution with cdf F (x) = 1 − e−.25x for x > 0. The 80thpercentile of X is

A. .8926

B. 6.438

C. .0558

D. .4024

E. mathematically intractable

(4) A relative frequency histogram having classes of greatly different class widths was used as an estimator ofa continuous population pdf. The relative frequency was plotted versus the class intervals. The plot willresult in a graphical distortion because

A. some of the classes will have too high a frequency

B. the sample size will be too small for the narrow class intervals

C. the area under the curve will not add to one

D. areas under the curve will not represent population proportions

E. in fact there will not be a distortion since it is an unbiased estimator of the pdf

(5) Suppose the population standard deviation is σ = 3. In order to be 95% confident that the differencebetween the sample estimator of the population mean µ and the true value µ is at most 0.5 units, thesample size n must be at least

A. 140

B. 100

C. 98

D. 35


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(6) The GOF statistics, such as K-S, A-D, and Cramer von Mises, for testing the goodness-of-fit of a continuouspdf are called distribution-free statistics because

A. their p-values do not depend on the exact form of the population pdf

B. the expected counts under Ho do not depend on the form of the pdf

C. the statistic must be adjusted for unspecified parameters

D. they have known distributions under Ho


(7) In a level α = .01 test of Ho : µ ≥ 12 versus H1 : µ < 12, where µ is the mean of a normally distributedpopulation, the sample size needed to have a probability of at least 0.95 of detecting that µ is half a standarddeviation less than 12 is (use the attached table to answer this question)

A. 75

B. 36

C. 63

D. 66

E. can not be answered with the given information

(8) The sample standard deviation is preferred to MAD as an estimator of population dispersion when thepopulation distribution

A. has absolutely no outliers

B. has a third central moment of 0 and a fourth central moment of 3σ4.

C. has a lognormal distribution

D. has a skewed but short-tailed distribution


(9) The power of a test of hypotheses

A. is greater than α for values of the parameter in Ho

B. is less than α for values of the parameter in Ho

C. is 1− β for values of the parameter in Ho

D. is β for values of the parameter in H1


(10) The reason we can use a plot of -log(-log(1-ui)) vs -log(Y(i)), where ui=(i-.5)/n and Y(i) is the ith orderstatistic to test the goodness of fit of a Weibull distribution to the population pdf is

A. the GOF statistics are distribution-free

B. the Weibull distribution has a tractable quantile function

C. the distribution of log(Yi) is a location-scale family

D. the quantile plot can always be used as a GOF evaluation


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(11) An experiment involving paired data (Xi, Yi) is conducted in order to test the hypothesis H1 : µ1 < µ2.Box plots of the original data and the differences Di = Xi − Yi reveals the following:

The box plot for Xi has many outliers

The box plot for Xi’s has much longer whiskers than the box plot for Yi’s

The box plot for Di’s has no outliers, whiskers of equal length, and median line falls in center of box

The preferred test statistic is

A. Wilcoxon Rank Sum testB. Wilcoxon signed rank testC. Paired t-testD. Separate variance t-testE. Sign test

(12) The reason that experimental units are paired in a study to compare the means of two processes is

A. to evaluate the effects of extraneous factors.B. to meet governmental requirementsC. to eliminate the effects of extraneous factors.D. to reduce the variance of the estimator of µ1 − µ2

E. to validate the reference distribution for inference purposes

(13) In testing the hypotheses H1 : σ1 > σ2, where σ1 and σ2 are the standard deviations of two normallydistributed populations, an α = .01 test was run using independent random samples of size n1 = 16 andn2 = 16. The probability of a Type II error when σ1 = 1.2σ2 is

A. .05B. .95C. .01D. .90E. need noncentral F-tables to compute power

(14) The skewness and kurtosis parameters for a given cdf are used to evaluate the following characteristics ofthe cdf

A. modality and heaviness of its tailsB. departure from normalityC. location and dispersionD. symmetryE. all the above

(15) The life length, L, in thousands of hours of a new type of electronic control is to be determined. Theengineer plots the sample failure rate function based on n iid observations from L. She finds that the plotis nearly a horizontal line. The distribution of L is

A. normal with µ > 0 and σ > 1B. Weibull with γ = .2C. exponential with β = .2D. lognormal µ 6= 0 and σ > 1E. gamma α = .2

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(16) Of the three conditions imposed on an experiment in order for the separate variance t-test to be valid, theone condition most affecting the level of the test is

A. normality

B. equal variance

C. independence

D. all three conditions are equally important

E. none of the conditions are crucial

(17) In a level α = .05 t-test of the hypothesis H1 : µ < 5, with a normal population and σ known, if the samplesize is increased from 10 to 30, then the level of significance of the test,

A. remains the same

B. decreases

C. increases

D. may increase or decrease depending on the value of σ


(18) General Electric wants to determine a warranty time for its top of the line dish washer. They want you, theirtop of the line statistician, to determine the number of hours, H, such that at most 5% of all dish washersproduced by the company in 2002 will need servicing before H hours of use. The company’s engineersevaluated 100 dish washers and recorded the number of hours until each of the machines needed servicing.The 100 times to service yield y = 5, 000 hours and s = 500 hours. The Shapiro-Wilks test produced ap-value of .367 for the 100 measurements. With 99% confidence, a lower bound for H is

A. 3972

B. 6028

C. 4178

D. 5822

E. cannot be determined from this data

(19) An experimenter wants to test Ho : F = Fo, where F is the process cdf and Fo is a specified cdf. Whichone of the following statements is FALSE?

A. The Chi-squared GOF test is valid whether or not F is discrete.

B. The Shapiro-Wilk’s test is valid only if Fo is normal.

C. The power of Anderson-Darling test depends on the specific form of F.

D. The Kolmogorov-Smirnov test is invalid if F is discrete.

E. The Shapiro-Wilk test has greater power than the Chi-squared test.


A. the probability of rejecting Ho for specified values of the parameter


C. the largest value of α for which the observed data will reject Ho

D. the smallest value of α for which the observed data will reject Ho

E. the probability of accepting Ho for specified values of the parameter

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(21) A random sample of 100 data values was taken from the cdf F (·). A plot of Y(i) versus zui, where Y(i) is

the ith order statistic and zuiis the standard normal percentile at ui = i−.5

100 yields a curve having nearlyall of the plotted points above a straight line for the largest 30 values of zui and nearly all of the plottedpoints above a straight line for the smallest 30 values of zui

. The remaining 40 points fell very near the linefor middle size values of zui

. This would indicate that

A. F (·) has a normal distribution

B. F (·) has a Cauchy distribution

C. F (·) has a Uniform distribution

D. F (·) has a Weibull distribution


(22) In a Box Plot, the probability that a data point is designated as an outlier

A. is smaller for a normal distribution than for a Cauchy distribution

B. is distribution free

C. depends on the median of the population distribution

D. increases as the sample size increases

E. all the above

(23) Suppose we want to test H1 : θ > 5, where θ is the location parameter of a symmetric pdf f(·; θ). A randomsample of 31 data values yields a box plot having 11 extreme outliers. The most appropriate test statisticfor this situation would be

A. the separate variance t-test

B. the one sample t-test

C. the Wilcoxon rank sum test

D. the Wilcoxon signed rank test

E. the sign test

(24) As the sample size n increases, the sampling distribution of a test statistic will tend towards the shape of

A. a standard normal distribution

B. a normal distribution

C. a uniform on (0,1) distribution

D. Santa Claus


(25) In using a kernel density estimator to estimate a population pdf based on a random sample Y1, · · · , Yn, thedesign factor which is least crucial in determining the effectiveness of the estimator is

A. the sample size, n

B. the kernel k(·)C. the bandwidth, h

D. the number of plotting points, m

E. all four factors are equally crucial

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December 13, 2002

I. (40 points) A government health agency is concerned about the effects of exposure to certain metals, such aslead and cadmium, on the health of workers in the metal plating industry. From a large epidemiological study,it was found that 40% of such workers had extensive exposure to such metals. An innovative safety program wasdeveloped to reduce exposure to these metals. The program set a goal of at most 20% exposure. The program wasimplemented in a random sample of 100 small plating companies. From each company 5 workers were randomlyselected within the company to monitor the success of the program. One year after implementating the safetyprogram the workers were examined. The results of these examinations are given in the following table:

Number Exposed 0 1 2 3 4 5 TotalObserved Frequency 51 32 11 4 1 1 100

Use the above information to answer the following questions. You may assume that exposure to the hazard metalsis an independent event for each of the five workers at each of the 100 companies.

(A) Construct a 99% confidence interval for the proportion of workers exposed to the metals after the safetyprogram was implemented.

(B) Is there significant evidence at the α = .01 level that the safety program has achieved its goal of at most20% of workers still being exposed to the hazardous metals?

(C) In the context of the hypothesis test presented in (B), give clear, explicit definitions of the following terms,Make Sure to Frame Your Definitions in Terms of This Specific Problem

i. Type I error

ii. Type II error


(D) Compute the probability of a Type II error for your test in (B) if the true percentage of workers beingexposed after the safety program was implemented was 15%.

(E) A government official examines your results from (A) through (D) states, “The probability of a Type II ofthe test when the percentage equals 15% is too large. Determine the minimum sample size necessary tohave at most a 10% chance of a Type II error when the true percentage is 15% or smaller.

(F) A government statistician examines the results of the study and comments that the assumption exposure tothe hazard metals is an independent event for each of the five workers at each of the 100 companies is notvalid. However, she states if you can demonstrate that if the safety program was implemented industry-widethen more than 80% of the companies in the industry would have 20% or fewer workers exposed in a randomsample of 5 workers, then she would certify the new safety program. Does the data from the study provideyou with the evidence the government official seeks?

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II. (20 points) INSTRUCTIONS Write the ONE letter from the second column which BEST matches thestatement in the first column. Note, there may be multiple correct responses and there may be items in thesecond column which are unused. An item in the second column can be used only once.

.......1. Test for equality of population variances A. Kolmogorov-SmirnovB. Specificity

.......2. Test for difference in population medians C. Tolerance Boundwhen population means do not exist D. Confidence Interval

.......3. Test for equality of population proportions E. Regression Analysiswhen sample sizes are small F. Spearman’s Correlation

.......4. Probability of positive test result G. Pearson’s Product Correlationwhen disease is present H. Chi-squared GOF test

.......5. Test for difference in two normal distributions’ I. Anderson-Darling testmedians when variances are unequal J. Shapiro-Wilk test

.......6. Method of transforming data to approximate normality K. Normality of residualsL. Sensitivity

.......7. Method of comparing the probability of success for M. Central Limit theoem2 populations when the data is paired N. Box-Cox method

.......8. Test for normality O. McNemar’s testP. Pooled t-test

.......9. Test for determining whether a negative binomial distribution Q. Levine’s testprovides a adequate model for a statistical process R. Power of test

......10. A requirement needed to construct a S. Correlated Dataconfidence interval for the slope of a line T. Satterthwaite Approximation

U. Maximum LikelihoodV. Wilcoxon Signed Rank testW. Wilcoxon Rank Sum testX. Sign testY. Fisher’s Exact testZ. Simpson’s Paradox

III. (2 points each) Place the letter of the best answer in the blank to the left of each question.

(1) A researcher interviews 233 of the 200000 Florida voters whose ballots were not counted in the 2000 USApresidential election. The number X of voters in the sample of 233 who voted for Al Gore has a

A. Poisson distribution

B. Binomial distribution

C. Negative Binomial distribution

D. Hypergeometric distribution

E. Hanging-Chad distribution

(2) A stratified random sample of sizes n1, n2, n3, n4, n5, is taken from a population. The researcher estimatesthe population mean by just averaging the n = n1 + n2 + n3 + n4 + n5 observations. This will result in

A. an underestimation of the population mean.

B. an unbiased estimator of the population mean.

C. an estimator having a very impressive formula for its variance estimator.

D. a biased estimator of the population mean.

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E. all the above

(3) A kernel density estimator is an vast improvement over a plot of the relative frequency divided by classwidth versus the population classes as an estimator of the population pdf when the cdf of the population

A. is continuous

B. is discrete

C. is discrete or continuous

D. depends on whether cdf is a member of location/scale family


(4) Suppose that X1, · · · , Xn are to be used to construct a 95% prediction interval for a normal population. Theresearcher notes that the data was collected by an automatic sampler which may result in X1, · · · , Xn havinga high positive correlation. If the prediction interval was computed using the formula: X± t.025,n−1S/

√n,

the resulting interval

A. will be too wide.

B. will have a level of confidence greater than 95%.

C. will have a level of confidence less than 95%.

D. will have a level of confidence equal to 95%.


(5) Suppose a normal population has a standard deviation of σ = 9. In order to be 95% confident that thedifference between the sample estimator of the population mean µ and the true value µ is at most 1.5 units,the sample size n must be at least

A. 140

B. 100

C. 98

D. 35


(6) The Anderson-Darling GOF statistic is prefered to the Cramer von Mises GOF statistic for testing thegoodness-of-fit of a continuous pdf because

A. it a more modern procedure.

B. it has a more accurate p-value.

C. it has a smaller probability of Type I error.

D. it is a more sensitive test statistic in the tails of the distribution.

E. it is easier to compute.

(7) In a level α = .05 test of Ho : µ ≤ 17 versus H1 : µ > 17, where µ is the mean of a normally distributedpopulation, the sample size needed to have a Type II error rate of at most 0.10 whenever µ > 17 + .5 ∗ σ is

A. 36

B. 22

C. 13

D. 70

E. need the non-central t cdf in order to determine sample size

(8) MAD is preferred to sample standard deviation as an estimator of population dispersion when the populationdistribution

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A. has only a few outliers.

B. has tails much heavier than the normal distribution.

C. has a normal distribution.

D. has a skewed but short-tailed distribution.

E. has a finite mean.

(9) The power of a test of the hypothesis: Ha : µ < µo

A. is not a function of the value of α

B. is the probability of a Type II error

C. is one minus the probability of a Type II error

D. varies depending on the value of µ


(10) An experiment is conducted to test the hypothesis H1 : µ1 < µ2. Box plots of the data reveals the following:

The box plot for Xi has many outliers

The box plot for Xi’s has much longer whiskers than the box plot for Yi’s

The preferred test statistic is

A. Wilcoxon Rank Sum test

B. Wilcoxon signed rank test

C. Pooled t-test

D. Separate variance t-test

E. Sign test

(11) When the experimental units are paired in a study to compare the means of two normal processes,

A. there is an increase in the degrees of freedom of the t-test.

B. the variance of X − Y is decreased if the correlation between X and Y is negative.

C. the variance of X − Y is decreased if the correlation between X and Y is positive.

D. the power of the t-test is increased over the power of the pooled t-test even if X and Y are uncorrelated.

E. the sign test should always be used.

(12) In testing the hypotheses Ho : σ ≤ 23.8 versus H1 : σ > 23.8, where σ is the standard deviation of a normallydistributed population, an α = .05 test was run using a independent random sample of size n = 10. Theprobability of a Type II error when σ = 47.9 is

A. .05

B. .95

C. .10

D. .90

E. need noncentral Chi-squared tables to compute power

(13) A process engineer wants to determine if the process cdf F (·) has remained unchanged after a new machinehas been installed in the process. Let Fo(·) be the process cdf prior to altering the process, where Fo(·) isan continuous cdf. Which one of the following statements is TRUE?

A. The distribution of the K-S statistic is a function of Fo(·).B. The most powerful test of Ho : F = Fo depends on the form o Fo(·).

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C. The Anderson Darling test is the most powerful test statistic.

D. The Shapiro-Wilk test has greater power than the Chi-square GOF test.

E. All of the above statements are true

(14) A random sample of 100 data values was taken from the cdf F (·). A graph was constructed with Y(i) onthe vertical axis and zui on the horizontal axis, where Y(i) is the ith order statistic and zui is the standardnormal percentile at ui = i−.5

100 . The scatterplot has most of the plotted points on a straight line but thelargest 5 values of Y(i) are above the line and the smallest 5 values of Y(i) are below the line. This wouldindicate that

A. F (·) has a normal distribution

B. F (·) has a Cauchy distribution

C. F (·) has a Uniform distribution

D. F (·) has a Double Exponential distribution

E. F (·) has a Weibull distribution

(15) The coefficient of Determination, R2, is highly affected by the appropriateness of using a straight-line tomodel the mean of Y and

A. the amount of correlation between the n pairs (Xi, Yi)

B. the amount of variability in the response variable, Y

C. the degrees of freedom in the model

D. the degrees of freedom used in estimating σ will reject Ho

E. the amount of variability in independent variable

(16) If n pairs of observations, (X1, Y1), · · · , (Xn, Yn) are perfectly related by the relationship: Y = X2 for0 < X < 2, what can you conclude about the correlation coefficient?

A. r = 0

B. r < 0

C. r > 0

D. r = 1

E. r cannot be used since the relationship is nonlinear

(17) A researcher wants to determine if there is an increase in the likelihood that people will purchase a productafter a redesign of the product. The current market share is 20%. Initially, the researcher was planning onusing a random sample of n=20 persons with an α = .05 test to evaluate the product. He wants you tocalculate the chance that the study will fail to detect that preference for the product has been increased ifin fact the preference for the new product is 40%. This chance is

A. .316

B. .596

C. .416

D. .950


(18) If the runs test statistic determines that the n observations are highly positively correlated, but an α = .05pooled t-test is still used

A. the true maximum probability of a type I error is greater than .05

B. the true maximum probability of a type I error is less than .05

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C. the true maximum probability of a type I error is still equal to .05

D. the true maximum probability of a type I error is completely unknown


(19) A random sample of n=15 from a normally distributed population is used to construct a level α = .01 testof Ha : µ ≤ 20 versus Ho : µ > 20, where µ is the mean of the population. The probability of a Type IIerror for µ > 20 + .8σ is at most

A. .05

B. .55

C. .22

D. .32

E. cannot be determined from the given information

(20) The bandwidth in a kernel density estimator is determined by

A. flipping a coin

B. flipping a statistician

C. a long period of deep thought

D. writing a letter to Santa Claus

E. any or all of the above

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