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起止时间:2018-01-30到2018-01-30
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作业第一周 Chapter1 Introduction homework1
1、 A coin is flipped twice. Let Y =number of heads obtained, when the probability of a head for a flip equals π. a. Assuming π =0.50 , specify the probabilities for the possible values for Y, and find the distribution’s mean and standard deviation. b. Find the binomial probabilities for Y when π equals (i) 0.60, (ii) 0.40. c. Suppose you observe y =1 and do not know π. Calculate and sketch the likelihood function. d. Using the plotted likelihood function from (c), show that the ML estimate of π equals 0.50.
评分规则: （a）10分Y=0，1，2时的概率各2分，均值和标准差各2分，标准差算成方差得1分，结果算错公式写对得1分，结果算错没有公式不得分。
（b）6分Y=0，1，2时，针对2个π值都有三个概率，每个概率计算1分。
（c）4分likelihood function 2分，描绘 likelihood function 的图2分
（d）5分有求导过程得3分，结果写对得2分。
2、 Genotypes AA, Aa, and aa occur with probabilities (π1,π2,π3). For n=3 independent observations, the observed frequencies are (n1,n2,n3). a. Explain how you can determine n3 from knowing n1 and n2. Thus, the multinomial distribution of (n1,n2,n3) is actually two-dimensional. b. Show the set of all possible observations, (n1,n2,n3) with n=3. c. Suppose (π1,π2,π3)=(0.25,0.50,0.25). Find the multinomial probability that (n1,n2,n3)=(1,2,0). d. Refer to (c).What probability distribution does n1 alone have? Specify the values of the sample size index and parameter for that distribution.
评分规则: （a）3分
写出公式得3分
（b）10分三个参数的10个取值可能，每个1分
（c）6分公式4分，结果2分
（d）6分写出分布及分布的参数得满分，参数写错一个扣2分。若没有分布形式，但写出概率公式和概率也可得分。
3、 To collect data in an introductory statistics course, recently I gave the students a questionnaire. One question asked whether the student was a vegetarian. Of 25 students, 0 answered “yes.”They were not a random sample, but let us use these data to illustrate inference for a proportion. (You may wish to refer to Section1.4.1onmethodsofinference.) Let π denote the population proportion who would say “yes.” Consider H0:π =0.50 and Ha:π =0.50.a. What happens when you try to conduct the “Wald test,” for which
uses the estimated standard error?b. Find the 95% “Wald confidence interval” for π. Is it believable? (When the observation falls at the boundary of the sample space, often Wald methods do not provide sensible answers.) c. Conduct the “score test,” for which 
uses the null standard error. Report the P-value. d. Verify that the 95% score confidence interval (i.e., the set of π0 for which |z|< 1.96 in the score test) equals (0.0,0.133). (Hint: What do the z test statistic and P-value equal when you test H0:π =0.133 against Ha:π = 0.133.)
评分规则: （a）6分得出计算z的公式中分母为0的结果得满分。
（b）6分计算出IC区间得4分，写出unbelievable得2分。
（c）6分z值3分，p值3分
z值公式3分，z的不等式写出来2分，求得结果2分
作业第三周 Chapter2 Contingency Tables（2）+程序课 homework2
1、 A newspaper article preceding the 1994 World Cup semifinal match between Italy and Bulgaria stated that “Italy is favored 10–11 to beat Bulgaria, which is rated at 10–3 to reach the final.” Suppose this means that the odds that Italy wins are 11/10 and the odds that Bulgaria wins are 3/10. Find the probability that each team wins, and comment.
评分规则: 算出两队各自赢的概率，各10分说明哪一队更容易赢，5分
2、 Data posted at the FBI website (www.fbi.gov) stated that of all blacks slain in 2005, 91% were slain by blacks, and of all whites slain in 2005, 83% were slain by whites. Let Y denote race of victim and X denote race of murderer.a. Which conditional distribution do these statistics refer to, Y given X, orX given Y?b. Calculate and interpret the odds ratio between X and Y.c. Given that a murderer was white, can you estimate the probability that the victimwaswhite?What additiona linformation would you need to do this? (Hint: How could you use Bayes’s Theorem?)
评分规则: a
b.7分考察odds ratio的计算，过程7分，结果3分
c. 10分考察贝叶斯理论的应用，过程7分，结论3分
3、 A statistical analysis that combines information from several studies is called a meta analysis. A meta analysis compared aspirin with placebo on incidence of heart attack and of stroke, separately for men and for women (J. Am. Med. Assoc., 295: 306–313, 2006). For the Women’s Health Study, heart attacks were reported for 198 of 19,934 taking aspirin and for 193 of 19,942 taking placebo.a. Construct the 2×2 table that cross classifies the treatment (aspirin, placebo) with whether a heart attack was reported (yes, no).b. Estimate the odds ratio. Interpret.c. Find a 95% confidence interval for the population odds ratio for women. Interpret.(Asof2006,results suggested that for women, aspirin was helpful for reducing risk of stroke but not necessarily risk of heart attack.)
评分规则: a.6分表中6个数字每个2分
b.10分过程7分，结果3分
c. 9分计算置信区间的过程7分，结果2分
作业第五周 Chapter3 Generalized Linear Model（2） homework3
1、 
Refer to Table 2.7 on x =mother’s alcohol consumption and Y =whether a baby has sex organ malformation.WIth scores (0,0.5,1.5,4.0,7.0) for alcohol consumption, ML fitting of the linear probability model has the output:
a. State the prediction equation, and interpret the intercept and slope.b. Use the model fit to estimate the(i) probabilities of malformation for alcohol levels 0 and 7.0,(ii) relative risk comparing those levels.
评分规则: a. 写出模型公式3分；解释截距项和斜率各3分
b. (i)计算概率各3分，(ii)5分
2、 Refer to the previous exercise1 and the solution to (b).a. The sample proportion of malformations is much higher in the highest alcohol category than the others because,although it has only one malformation, its sample size is only 38. Is the result sensitive to this single malformation observation? Re-fit the model without it (using 0 malformations in 37 observations at that level), and re-evaluate estimated probabilities of malformation at alcohol levels 0 and 7 and the relative risk.b. Is the result sensitive to the choice of scores? Re-fit the model using scores (0,1,2,3,4), and re-evaluate estimated probabilities of malformation at the lowest and highest alcohol levels and the relative risk.c. Fit a logistic regression or probit model. Report the prediction equation. Interpret the sign of the estimated effect.
评分规则: a. 重新拟合模型5分，两个概率计算各3分， 相对风险3分， 最后的敏感度说明2分。
b.与a问得分原则相同。
c. 两个模型拟合结果各5分，参数解释2分。
作业第八周 Chapter4 Logistic Regression（2） homework4
小提示:本节包含奇怪的同名章节内容
1、 A study used logistic regression to determine characteristics associated with Y =whether a cancer patient achieved remission (1=yes). The most important explanatory variable was a labeling index (LI) that measures proliferative activity of cells after a patient receives an injection of tritiated thymidine. It represents the percentage of cells that are “labeled.” The first table shows the grouped data. Software reports the second table for a logistic regression model using LI to predict 
a. Conduct a Wald test for the LI effect. Interpret.b. Construct a Wald confidence interval for the odds ratio corresponding to a 1-unit increase in LI. Interpret.c. Conduct a likelihood-ratio test for the LI effect. Interpret.d. Construct the likelihood-ratio confidence interval for the odds ratio. Interpret.
评分规则: a.算出Z值，2分；结果拒绝原假设：1分具体分值分（各得分要求分值之和需为该题分值）全选01
b. 置信区间4分，解释2分
c. 统计量2分，结果1分
d.置信区间4分，解释2分
2、 For the horseshoe crab data , fit the logistic regression model for π =probability of a satellite, using weight as the predictor.a. Report the ML prediction equation.b. Find 
at the weight values 1.20, 2.44, and 5.20kg, which are the sample minimum, mean, and maximum.c. Find the weight at which 
.d. At the weight value found in (c), give a linear approximation for the estimated effect of(i) a 1kg increase in weight.This represents a relatively large increase, so convert this to the effect of(ii) a 0.10kg increase, and(iii) a standard deviation increase in weight (0.58kg).e. Construct a 95% confidence interval to describe the effect of weight on the odds of a satellite. Interpret.f. Conduct the Wald or likelihood-ratio test of the hypothesis that weight has no effect. Report the P-value, and interpret.Note: you can get the data in the R package “icda” which is named by “horseshoecrabs”.
评分规则: a.得出正确回归模型，5分
b.每个估计值2分，共6分
c.得出正确结果，2分
d. 前两问每问2分，第三问3分，共7分
e. 置信区间4分， 解释2分
f. 过程3分，p值2分，解释1分
3、 A study in Florida that stated that the death penalty was given in 19 out of 151 cases in which a white killed a white, in 0 out of 9 cases in which a white killed a black, in 11 out of 63 cases in which a black killed a white, and in 6 out of 103 cases in which a black killed a black. The table shows results of fitting a logit model for death penalty as the response (1=yes), with defendant’s race (1=white) and victims’ race (1=white) as indicator predictors.
a. Based on the parameter estimates, which group is most likely to have the “yes” response? Estimate the probability in that case.b. Interpret the parameter estimate for victim’s race.c. Using information shown, construct and interpret a 95% likelihood-ratio confidence interval for the conditional odds ratio between the death penalty verdict and victim’s race.d. Test the effect of victim’s race, controlling for defendant’s race, using a Wald test or likelihood-ratio test. Interpret.
评分规则: a. 4个概率计算各占2分，结果说明2分
b.解释合理可得3分
c.置信区间4分
d.解释及结果各两分
4、 For the horseshoe crab data , fit the logistic regression model for π =probability of a satellite, using weight as the predictor.a. Report the ML prediction equation.b. Find 
at the weight values 1.20, 2.44, and 5.20kg, which are the sample minimum, mean, and maximum.c. Find the weight at which 
.d. At the weight value found in (c), give a linear approximation for the estimated effect of(i) a 1kg increase in weight.This represents a relatively large increase, so convert this to the effect of(ii) a 0.10kg increase, and(iii) a standard deviation increase in weight (0.58kg).e. Construct a 95% confidence interval to describe the effect of weight on the odds of a satellite. Interpret.f. Conduct theWald or likelihood-ratio test of the hypothesis that weight has no effect. Report the P-value, and interpret.Note: you can get the data in the R package “icda” which is named by “horseshoecrabs”.
评分规则: a.得出正确回归模型，5分
b.每个估计值2分，共6分
c.得出正确结果，2分
d. 前两问每问2分，第三问3分，共7分
e. 置信区间4分， 解释2分
f. 过程3分，p值2分，解释1分
作业第九周 Chapter5 Building and Applying Logistic Regression Models homework5
小提示:本节包含奇怪的同名章节内容
1、 Table 4.13 shows the result of cross classifying a sample of people from the MBTI Step II National Sample (collected and compiled by CPP, Inc.) on whether they report drinking alcohol frequently (1=yes, 0=no) and on the four binary scales of the Myers–Briggs personality test: Extroversion/Introversion (E/I), Sensing/iNtuitive (S/N), Thinking/Feeling (T/F) and Judging/Perceiving (J/P). The 16 predictor combinations correspond to the 16 personality types: ESTJ, ESTP, ESFJ, ESFP, ENTJ, ENTP, ENFJ, ENFP, ISTJ, ISTP, ISFJ, ISFP, INTJ, INTP, INFJ, INFP.
Table 5.10 shows the result of fitting a model using the four scales as predictors of whether a subject drinks alcohol frequently.a. Conduct a model goodness-of-fit test, and interpret.b. If you were to simplify the model by removing a predictor, which would you remove?Why?c. When six interaction terms are added, the deviance decreases to 3.74. Show how to test the hypothesis that none of the interaction terms are needed, and interpret.
.
评分规则: a. 5分，统计量3分，解释2分
b. 变量2分，解释1分
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