STA 032 Spring 2016 Homework 5 – Book Portion

STA 032 Spring 2016Homework 5 – Book Portion – Due Friday, May 13thReminder: Book homework and R homework are to beturned in to separate piles(a) Find the probability the first item sold for more thanthe second item.(b) Find the probability that both items sold for over400 gold pieces total.(c) Find the probability that the average of the two itemssold for between 220 and 250 gold pieces.Book Homework(d) Find the 30th percentile for the average price of thetwo items.1. A doctor knows from experience that the probability apatient shows side effects on a particular drug is 0.10. Assume that probability does not change, and that patientsare independent.(e) Find the 60th percentile for the total of the twoitems.(f) Find the value of the total price for both where only10% of the time, both items will sell for more thanthat value.(a) How many patients on average should he expect togive the drug before one of them has a side effect?(b) What is the probability that the 4th patient he givesthe drug to is the first to have a side effect?(c) What is the probability that the 2nd patient to havea side effect occurs between the 4th, 5th, and 6thpatient to receive the drug?(d) What is the probability that the 3nd patient to have aside effect occurs exactly on the 6th patient to receivethe drug?5. The average amount of time until a car accident on aparticular 60 mile stretch of road is 60 minutes. Assume(unreasonably) that car accidents are independent, andthat two accidents cannot occur at the same time.(a) What is the probability of a car accident occurringin the two hours?(b) What is the probability of a car accident occurringbetween 45 and 75 minutes?2. At a particular manufacturing plant, the probability thata created item does not meet specifications is 0.04. Theassembly will stop for recalibration if it detects 2 itemsthat do not meet specifications. Assume items are independent, and the probability of not meeting specificationsdoes not change.(c) What is the variance of the time until a car accidentoccurs?(d) If a car accident has not happened in 3 hours, what isthe probability it will happen in the next two hours?(a) Find the expected number of items produced beforethe machine needs recalibration, and the standarddeviation.(b) What is the probability the machine stops on exactlythe 8th item?(c) What is the probability the machine stops after the4th item?(d) What is the probability the first item that does notmeet specifications is the 4th item?6. In a particular state, a major flood happens once every 3years on average. If you can assume floods are independent, and that two floods cannot happen simultaneously,find the following:(a) The probability that a flood happens within the first5 years.(b) The probability that a flood happens after the first2 years.(c) If there has been no flood for 4 years, what is theprobability there is a flood in the next 2 years?3. Assume that IQ is normally distributed, with mean 100and standard deviation 15.(d) The expected time until a flood happens, and thestandard deviation.(a) What is the probability that a randomly selected persons IQ is over 120?(b) Find the values of Q1 , Q2 , and Q3 for IQ.(c) Find the values of lower = Q1 − 1.5(Q3 − Q1 ) andupper = Q3 + 1.5(Q3 − Q1 ) for IQ. Recall that thesewere cutoffs for outliers.(d) Find the probability of an outlier for IQ for a singleperson based on your values for (c).(e) If we randomly selected 10 people, what is the probability their average IQ is over 105?(e) Find the 50th percentile of the time until a flood.7. The wait time at a popular fast food restaurant is uniformly distributed between 1 and 15 minutes on average.(a) Find the probability that you wait less than 6 minutes.(b) Find the probability that you wait between 5 and 10minutes.(c) What is the expected wait time, and the variance ofwait time?4. Assume that the selling price of a particular item (callit I1 ) is normally distributed, with mean 200 gold coins,and standard deviation 50 coins. Assume the selling priceof another item is also normally distributed, with mean230 gold coins, and standard deviation 10 gold coins (callit I2 ). Assume these distributions are independent.(d) If you have waited 5 minutes already, what is theprobability you wait an additional 5 minutes?8. The total amount of tips a waitress receives is uniformlydistributed between $1 and $20 in an hour on average.1(a) Find the probability the waitress receives between $5and $10.III. Consider the function rnorm. This simulates a normalrandom variable, for example the following code will generate 1000 normal random variables with mean 5, standard deviation 6:(b) What is the expected tip amount the waitress willreceive, and the standard deviation?(c) Find the following probabilities: P r(X < µX ),P r(X > µX ) where X =the amount of tips a waitress receives.X = rnorm(1000, mean = 5, sd = 6)(a) Generate 10000 values of a normal random variablewith mean 2, standard deviation 1, and call this vector X. Generate 10000 values of a normal randomvariable with mean -2, standard deviation 5, andcall this vector Y . Report back the mean and standard deviation of both vectors (using the functionsin R).(d) Find the following probabilities: P r(X < µX − σX ),P r(X > µX + σX ) where X =the amount of tips awaitress receives.(e) What is the median amount of tips a waitress willreceive?(b) Create a vector W by adding the two vectors from(a) together, and find the mean and standard deviation W (using R).R HomeworkI. The goal of this problem is to simulate a Geometric random variable. Assume in a computer game Stardew Valley, there is a 10% chance it rains on any given day.(c) Find the probability that W is larger than 3.(d) We know what the answers to (b), (c) should beexactly. Calculate the error for all the values found(the error is what R found subtract by what we knowit should be).(a) Use R to simulate generating days up to and including the first time it rains. Return based on one (random) run of your simulation the number of days ittook up to and including the first rainy day.(b) Repeat (a) 20000 times, so that you should have20000 values of how many days it took up to andincluding the first rainy day. Do not print outthese values!!! Plot a histogram of these values.(c) What can you say about the distribution of thisparticular Geometric random variable based on (b)?Describe it.(d) Find the expected value and standard deviation ofyour vector in (b).(e) Find the probability that it took 15 days until thefirst rainy day.(f) Find the probability that if it took more than 10days, it would take an additional 5 days to observethe first rainy day.II. The goal of this problem is to simulate a Negative Binomial random variable. Consider that 40% of peoplewear glasses for near-sightedness.(a) Use R to simulate selecting random people until 5are selected wear glasses. Return one the number ofpeople it took until 5 wore glasses.(b) Repeat (a) 20000 times, so that you should have20000 values of how many people were were selecteduntil 5 wore glasses. Do not print out these values!!! Plot a histogram of these values.(c) Find the median of the vector from (b).(d) Find the mean and standard deviation of your vectorfrom (b).(e) Find the probability we had to sample more than 10people before we found 5 that needed glasses.(f) Find the probability that if it takes more than 10people, it will an additional 5 to find 5 that woreglasses.2