I need help with Statistics exam
Name: BME 530 Midterm Exam (2021) Q1 (3pts) A manufacturer of a flu vaccine is concerned about the quality of its flu serum. Batches of serum are processed by three different departments having rejection rates of 0.10, 0.08, and 0.12, respectively. The inspections by the three departments are sequential and independent. 1) What is the probability that a batch of serum survives the first departmental inspection but is rejected by the second department? 2) What is the probability that a batch of serum is rejected by the third department? 3) What is the probability that a batch of serum survives inspections of all three departments? Q2 (3pts) There is a 50-50 chance that the queen carries the gene of hemophilia. If she is a carrier, then each prince has a 50-50 chance of having hemophilia independently. If the queen is not a carrier, the prince will not have the disease. Suppose the queen has had three princes without the disease. What is the probability that the queen is a carrier? Q3 (4pts) The proportion of people who respond to a certain mail-order solicitation is a continuous random variable X that has the density function ?(?) = { 2(? + 2) 5 0 if 0 <>< 1="" elsewhere.="" 1)="" show="" that="" (0=""><>< 1).="" 2)="" find="" probability="" (="" 1="" 4=""><>< 1 2 ). 3) find e(x). 4) find var(x). q4. (4pts) a medical research team wishes to assess the usefulness of a certain symptom (call it s) in the diagnosis of a particular disease. in a random sample of 775 patients with the disease, 744 reported having the symptom. in an independent random sample of 1380 subjects without the disease, 21 reported that they had the symptom. 1) in this exercise, what is a false positive? what is a false negative? 2) compute the sensitivity of the symptom and the specificity of the symptom. 3) compute the positive predictive value. 4) compute the negative predictive value. q5 (2pts) a pharmaceutical company knows that approximately 5% of its birth-control pills have an ingredient that is below the minimum strength, thus rendering the pill ineffective. 1) let x denote the number of ineffective pills in a sample of 200 pills. what probability distribution that x follows? please write the distribution function. 2) what is the probability that fewer than 10 in the sample of 200 pills will be ineffective? q6. (6pts) toxins. an investigation on toxins produced by molds that infect corn crops was performed. a biochemist prepared extracts of the mold culture with organic solvents and then measured the amount of toxic substance per gram of solution. from 11 preparations of the mold culture the following measurements of the toxic substance (in milligrams) were obtained: 3, 2, 5, 3, 2, 6, 5, 4.5, 3, 3, and 4. the sample mean and standard deviation are 3.682 and 1.714 respectively. 1) compute a 95% confidence interval for the mean weight of toxic substance per gram of mold culture. state the assumption you make about the population and justify the sampling distribution used in the calculation. 2) based on the data, can you draw the conclusion that the toxic substance in this type of the corn crops is smaller than 4? using the significant level of α=0.05. please provide the details on hypothesis, test statistic and its distribution, and the basis of your conclusion. 3) test the null hypothesis that the variance of the population is 4 at α=0.05. how do you compute the p-value for the test statistic? shown the area in the probability distribution curve. q7 (6pts) a research team used telephone interviews of randomly selected respondents in hong kong to obtain information regarding individuals' perceptions of health and smoking history. among 1222 current male smokers, 72 reported that they had 'poor' or 'very poor' health, while 30 among 282 former male smokers reported that they had 'poor' or 'very poor' health. can we conclude that among hong kong men there is a difference between current and former smokers with respect to the proportion who perceive themselves as having 'poor' or 'very poor' health? let ?= 0.01. justify the test statistic used in your test. indicate the p-value using the distribution of the test statistic. q8 (6pts) two groups of elementary school students are taught mathematics by two different methods: traditional (group 1) and small group interactive teaching by discovery based on piagetian theory (group 2). the results of a learning test are analyzed to test the difference in mean scores using the two methods. group 1 had 16 students while group 2 had 14 students and the scores are given below. test the hypothesis that the methods have no influence on test scores against the alternative that the students in group 2 have significantly higher scores. take α=0.05. justify the test statistic being used. indicate the p-value using the distribution of the test statistic. show your steps and do not use the r function (t.test) for hypothesis testing. q9 (6pts) a researcher is claimed that a new diet will reduce a person’s weight by 4.5 kilograms on average in a period of 2 weeks. the weights of 7 women who followed this diet were recorded before and after the 2-week period. woman weight before weight after 1 58.5 60.0 2 60.3 54.9 3 61.7 58.1 4 69.0 62.1 5 64.0 58.5 6 62.6 59.9 7 56.7 54.4 test the claim that the new diet reduced a person’s weight in a period of 2 weeks at α=0.05. justify the test statistic being used. indicate the p-value using the distribution of the test statistic. 1="" 2="" ).="" 3)="" find="" e(x).="" 4)="" find="" var(x).="" q4.="" (4pts)="" a="" medical="" research="" team="" wishes="" to="" assess="" the="" usefulness="" of="" a="" certain="" symptom="" (call="" it="" s)="" in="" the="" diagnosis="" of="" a="" particular="" disease.="" in="" a="" random="" sample="" of="" 775="" patients="" with="" the="" disease,="" 744="" reported="" having="" the="" symptom.="" in="" an="" independent="" random="" sample="" of="" 1380="" subjects="" without="" the="" disease,="" 21="" reported="" that="" they="" had="" the="" symptom.="" 1)="" in="" this="" exercise,="" what="" is="" a="" false="" positive?="" what="" is="" a="" false="" negative?="" 2)="" compute="" the="" sensitivity="" of="" the="" symptom="" and="" the="" specificity="" of="" the="" symptom.="" 3)="" compute="" the="" positive="" predictive="" value.="" 4)="" compute="" the="" negative="" predictive="" value.="" q5="" (2pts)="" a="" pharmaceutical="" company="" knows="" that="" approximately="" 5%="" of="" its="" birth-control="" pills="" have="" an="" ingredient="" that="" is="" below="" the="" minimum="" strength,="" thus="" rendering="" the="" pill="" ineffective.="" 1)="" let="" x="" denote="" the="" number="" of="" ineffective="" pills="" in="" a="" sample="" of="" 200="" pills.="" what="" probability="" distribution="" that="" x="" follows?="" please="" write="" the="" distribution="" function.="" 2)="" what="" is="" the="" probability="" that="" fewer="" than="" 10="" in="" the="" sample="" of="" 200="" pills="" will="" be="" ineffective?="" q6.="" (6pts)="" toxins.="" an="" investigation="" on="" toxins="" produced="" by="" molds="" that="" infect="" corn="" crops="" was="" performed.="" a="" biochemist="" prepared="" extracts="" of="" the="" mold="" culture="" with="" organic="" solvents="" and="" then="" measured="" the="" amount="" of="" toxic="" substance="" per="" gram="" of="" solution.="" from="" 11="" preparations="" of="" the="" mold="" culture="" the="" following="" measurements="" of="" the="" toxic="" substance="" (in="" milligrams)="" were="" obtained:="" 3,="" 2,="" 5,="" 3,="" 2,="" 6,="" 5,="" 4.5,="" 3,="" 3,="" and="" 4.="" the="" sample="" mean="" and="" standard="" deviation="" are="" 3.682="" and="" 1.714="" respectively.="" 1)="" compute="" a="" 95%="" confidence="" interval="" for="" the="" mean="" weight="" of="" toxic="" substance="" per="" gram="" of="" mold="" culture.="" state="" the="" assumption="" you="" make="" about="" the="" population="" and="" justify="" the="" sampling="" distribution="" used="" in="" the="" calculation.="" 2)="" based="" on="" the="" data,="" can="" you="" draw="" the="" conclusion="" that="" the="" toxic="" substance="" in="" this="" type="" of="" the="" corn="" crops="" is="" smaller="" than="" 4?="" using="" the="" significant="" level="" of="" α="0.05." please="" provide="" the="" details="" on="" hypothesis,="" test="" statistic="" and="" its="" distribution,="" and="" the="" basis="" of="" your="" conclusion.="" 3)="" test="" the="" null="" hypothesis="" that="" the="" variance="" of="" the="" population="" is="" 4="" at="" α="0.05." how="" do="" you="" compute="" the="" p-value="" for="" the="" test="" statistic?="" shown="" the="" area="" in="" the="" probability="" distribution="" curve.="" q7="" (6pts)="" a="" research="" team="" used="" telephone="" interviews="" of="" randomly="" selected="" respondents="" in="" hong="" kong="" to="" obtain="" information="" regarding="" individuals'="" perceptions="" of="" health="" and="" smoking="" history.="" among="" 1222="" current="" male="" smokers,="" 72="" reported="" that="" they="" had="" 'poor'="" or="" 'very="" poor'="" health,="" while="" 30="" among="" 282="" former="" male="" smokers="" reported="" that="" they="" had="" 'poor'="" or="" 'very="" poor'="" health.="" can="" we="" conclude="" that="" among="" hong="" kong="" men="" there="" is="" a="" difference="" between="" current="" and="" former="" smokers="" with="" respect="" to="" the="" proportion="" who="" perceive="" themselves="" as="" having="" 'poor'="" or="" 'very="" poor'="" health?="" let="" =="" 0.01.="" justify="" the="" test="" statistic="" used="" in="" your="" test.="" indicate="" the="" p-value="" using="" the="" distribution="" of="" the="" test="" statistic.="" q8="" (6pts)="" two="" groups="" of="" elementary="" school="" students="" are="" taught="" mathematics="" by="" two="" different="" methods:="" traditional="" (group="" 1)="" and="" small="" group="" interactive="" teaching="" by="" discovery="" based="" on="" piagetian="" theory="" (group="" 2).="" the="" results="" of="" a="" learning="" test="" are="" analyzed="" to="" test="" the="" difference="" in="" mean="" scores="" using="" the="" two="" methods.="" group="" 1="" had="" 16="" students="" while="" group="" 2="" had="" 14="" students="" and="" the="" scores="" are="" given="" below.="" test="" the="" hypothesis="" that="" the="" methods="" have="" no="" influence="" on="" test="" scores="" against="" the="" alternative="" that="" the="" students="" in="" group="" 2="" have="" significantly="" higher="" scores.="" take="" α="0.05." justify="" the="" test="" statistic="" being="" used.="" indicate="" the="" p-value="" using="" the="" distribution="" of="" the="" test="" statistic.="" show="" your="" steps="" and="" do="" not="" use="" the="" r="" function="" (t.test)="" for="" hypothesis="" testing.="" q9="" (6pts)="" a="" researcher="" is="" claimed="" that="" a="" new="" diet="" will="" reduce="" a="" person’s="" weight="" by="" 4.5="" kilograms="" on="" average="" in="" a="" period="" of="" 2="" weeks.="" the="" weights="" of="" 7="" women="" who="" followed="" this="" diet="" were="" recorded="" before="" and="" after="" the="" 2-week="" period.="" woman="" weight="" before="" weight="" after="" 1="" 58.5="" 60.0="" 2="" 60.3="" 54.9="" 3="" 61.7="" 58.1="" 4="" 69.0="" 62.1="" 5="" 64.0="" 58.5="" 6="" 62.6="" 59.9="" 7="" 56.7="" 54.4="" test="" the="" claim="" that="" the="" new="" diet="" reduced="" a="" person’s="" weight="" in="" a="" period="" of="" 2="" weeks="" at="" α="0.05." justify="" the="" test="" statistic="" being="" used.="" indicate="" the="" p-value="" using="" the="" distribution="" of="" the="" test="">