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G. Umphrey STAT*2120: Probability and Statistics for Engineers Winter 2021 Assignment #4 Your solutions will be submitted via Crowdmark. Further instructions and some hints will be provided in class. Be neat! 1. (a). Plot a density histogram for the data set “Aluminum Strips Under Stress - Lifetime (in 1000s of cycles)”. Calculate the mean and standard deviation for this sample, and use these values to estimate the corresponding population parameters for a normal distribution. Overlay the normal density over the histogram. How good would you say the fit is, based simply on a visual inspection of the graph? Hand this graph in. [Notes: Be sure to modify the limits of your axes if necessary. Also remember to label your axes and provide a title.] (b). Referring to the example in (a), let X be the lifetime of an aluminum strip under stress (in 1000's of cycles). Estimate P(X > 1500) by (i) making a direct estimate from the empirical data and (ii) using the normal distribution you fit to the data. Comment on the closeness of the two probabilities. 2. Go to Random.Org, the “True Random Number Service”. Under “Numbers” click on “Decimal Fraction Generator”. Generate six values in the [0,1] interval, each to five decimal places. Record these. Now go to R and enter these values as a vector. For example, here’s a set of values I generated: Next, create a vector named “URS1” (or whatever) with the command: URS1 <- c(0.18165,="" 0.31232,="" 0.76549,="" 0.96598,="" 0.79596)="" now="" treat="" this="" vector="" as="" a="" set="" of="" six="" random="" cumulative="" probabilities="" that="" will="" be="" used="" to="" generate="" a="" random="" sample="" of="" six="" normal="" values,="" each="" drawn="" at="" random="" from="" a="" normal="" distribution="" with="" mean="" ì="60" and="" standard="" deviation="" ó="10." use="" the="" qnorm="" function="" to="" do="" so,="" like="" so:="" normrs1=""><- qnorm(urs1,="" 60,="" 10)="" you="" can="" replace="" urs1="" and="" normrs1="" by="" your="" preferred="" names.="" take="" a="" look="" at="" the="" values="" you="" generated.="" this="" is="" your="" own="" personal="" random="" sample="" of="" six="" “sheep="" weights”.="" [note:="" do="" not="" use="" rnorm()="" to="" generate="" your="" sample="" here.]="" obtain="" the="" mean="" (xbar)="" and="" standard="" deviation="" (s)="" on="" your="" random="" sample="" of="" six="" sheep="" weights.="" calculate="" the="" standard="" error="" (se)="" of="" the="" sample="" mean;="" this="" is="" the="" sample="" standard="" deviation="" divided="" by="" the="" square="" root="" of="" the="" sample="" size.="" calculate="" the="" following="" intervals="" for="" your="" sample="" data:="" (a)="" (xbar="" !="" z0.025se,="" xbar="" +="" z0.025se)="" (b)="" (xbar="" !="" t0.025;="" 5se,="" xbar="" +="" t0.025;="" 5se)="" for="" each="" interval="" check="" to="" see="" if="" the="" population="" mean="" ì="60" is="" included="" in="" the="" interval.="" notes:="" (1)="" the="" first="" interval="" is="" a="" “faulty”="" 95%="" confidence="" interval,="" calculated="" using="" the="" standard="" normal="" value="" of="" 1.96.="" the="" problem="" is="" that="" the="" sample="" size="" is="" small="" and="" the="" sample="" standard="" deviation="" is="" estimating="" the="" population="" standard="" deviation.="" for="" this="" size="" of="" sample="" (n="6)" the="" correct="" multiplier="" of="" the="" se="" is="" a="" t="" value="" with="" 5="" degrees="" of="" freedom,="" which="" we="" can="" get="" from="" the="" t="" table:="" t5;0.025="2.571." (make="" sure="" you="" know="" how="" these="" values="" were="" obtained.)="" (2)="" on="" “random.org”="" you="" might="" find="" “introduction="" to="" randomness="" and="" random="" numbers”="" of="" interest.="" 3.="" you="" can="" generate="" random="" samples="" (actually="" pseudorandom="" samples)="" very="" quickly="" in="" r="" with="" the="" rnorm()="" function.="" to="" get="" a="" better="" sense="" of="" how="" estimation="" of="" the="" population="" mean="" is="" influenced="" by="" sample="" size,="" obtain="" random="" samples="" of="" sizes="" n="100," 10,000,="" and="" 1,000,000="" from="" a="" normal="" distribution="" with="" mean="" ì="60" and="" standard="" deviation="" ó="10." for="" each="" data="" set="" obtain="" the="" mean,="" standard="" deviation,="" and="" calculate="" the="" 95%="" confidence="" interval="" for="" the="" population="" mean.="" the="" 95%="" confidence="" intervals="" can="" be="" calculated="" as="" (xbar="" !="" 1.96se,="" xbar="" +="" 1.96se)="" since="" the="" sample="" sizes="" are="" all="" quite="" large.="" did="" your="" confidence="" intervals="" capture="" ì="60?" (most="" of="" the="" time="" the="" answer="" will="" be="" yes,="" but="" sometimes="" the="" answer="" will="" be="" no!)="" 4.="" generate="" a="" random="" sample="" of="" size="" 900="" from="" a="" normal="" distribution="" with="" mean="" ì="60" and="" standard="" deviation="" ó="10" with="" the="" r="" command="" rnorm(900,="" 60,="" 10).="" make="" sure="" to="" assign="" your="" generated="" values="" to="" a="" variable="" name.="" obtain="" the="" mean="" and="" standard="" deviation="" for="" your="" sample.="" next="" obtain="" a="" density="" histogram="" of="" your="" sample="" data="" and="" then="" overlay="" the="" density="" function="" for="" a="" normal="" random="" variable="" with="" the="" mean="" and="" standard="" deviation="" parameters="" estimated="" by="" the="" sample="" mean="" and="" standard="" deviation.="" the="" method="" for="" producing="" this="" graph="" is="" very="" close="" to="" the="" example="" in="" question="" 1.="" hand="" this="" graph="" in.="" 5.="" (a).="" plot="" a="" density="" histogram="" for="" the="" “airplane="" air="" conditioning="" failure="" data”.="" this="" data="" consists="" of="" time="" intervals="" (in="" hours)="" between="" successive="" failures="" of="" the="" air="" conditioning="" systems="" of="" 13="" boeing="" 720's="" (data="" from="" derman,="" gleser="" and="" olkin,="" 1973).="" you="" will="" use="" the="" data="" set="" with="" the="" nas="" removed.="" calculate="" the="" mean="" this="" sample,="" and="" use="" this="" value="" to="" estimate="" the="" appropriate="" parameter="" for="" an="" exponential="" distribution.="" you="" need="" to="" be="" careful="" here;="" sometimes="" the="" exponential="" distribution="" is="" defined="" with="" parameter="" â="ì," but="" sometimes="" with="" parameter="" ë="1/ì." what="" does="" r="" use?="" fit="" the="" exponential="" density="" over="" the="" histogram.="" how="" good="" would="" you="" say="" the="" fit="" is,="" based="" simply="" on="" a="" visual="" inspection="" of="" the="" graph?="" hand="" this="" graph="" in.="" [notes:="" be="" sure="" to="" modify="" the="" limits="" of="" your="" axes="" if="" necessary.="" also="" remember="" to="" label="" your="" axes="" and="" provide="" a="" title.]="" (b).="" referring="" to="" the="" example="" in="" (a),="" let="" x="" be="" the="" time="" between="" successive="" failures="" (in="" 1000's="" of="" cycles).="" estimate="" p(x=""> 150) by (i) making a direct estimate from the empirical data and (ii) using the exponential distribution you fit to the data. Comment on the closeness of the values obtained. 6. In a certain communications system, there is an average of 1.3 transmission errors per 10 seconds. Let the distribution of transmission errors be Poisson. (a). What is the probability of more than 2 errors in a communication 10 seconds in duration? (b). What is the probability of more than 2 errors in a communication 30 seconds in duration? 7. There are 24 computers in a computer lab. Three of them are infected with a computer virus. If four students randomly select computers to work at (all at the same time, one student per computer), give an expression for p(x), the probability that X of the students will select a computer with the virus. 8. On Assignment #2, question 10 (c), you obtained a probability mass function for a discrete random variable. Your function was (or should have been) an example of which special probability distribution? 9. The geometric and the exponential distributions both have "memoryless" properties. Prove the "memoryless" properties for each of these distributions; that is, prove that if the random variable X has a geometric distribution, P[X = a + b * X > a] = P[X = b], where a and b are positive integer constants, and if the random variable Y has an exponential distribution, P[Y > c + d * Y > c] = P[Y > d], where c and d are positive real constants. 10. Use the Poisson approximation to determine the following binomial probabilities. (a) If the probability that a person has a certain heart defect is 0.0012, what is the probability that among two thousand people (randomly chosen) at least two have this heart defect? (b) On average, 99.4% of cars have safe brakes when new. What are the probabilities that among a group of 600 new cars (assumed to be “randomly selected”): (i) all have safe brakes; (ii) no more than 598 cars have safe brakes. 11. The manufacturer of “1 kg” jars of jam wants to fill the jars with a mean weight of 1.045 kg and a standard deviation of 0.063 kg. (a) If these standards are being met, what is the probability that 4 randomly selected jars will have a mean fill weight of less than 1 kg? Assume fill weights are normally distributed. (b) As in (a), but calculate for (i) n = 10 and (ii) n = 25. (c) The line is stopped and the equipment checked if the mean fill weight of 10 randomly selected jars is less than 1.00 kg or if it exceeds 1.075 kg. What is the probability the line will be stopped if specifications are being met? (d) What is the probability the line will be stopped if the mean weight of 1.045 kg is maintained but variability increases such that the population standard deviation is now 0.134 kg?