1) The first task is to review some information that might beuseful later:
a) Write a brief definition of the word "quartile" as we haveused it in previous weeks. Be sure to provide a citation:_____________________________.
b) Write a brief definition of the word "quantile" as it mightbe used in statistics. Be sure to provide a citation (do not cutand paste... use your own words to summarize what you discovered):________________________________.
c) From within interactive R, enter the command shown below(the command shows a help page for the pbinom command). Provide avery brief description of the arguments that are passed to thepbinom() command ("arguments" in computer programming are theoptions that you give to a function so that the function cancalculate what you want it to). Note that one of the arguments islower.tail = TRUE, and because there is a value assigned to it withthe equals sign, it means that if you do not enter a new value forlower.tail, it will be set to TRUE by default. Do not type the">" into R, it is the command prompt:
> ?pbinom
2) You can use the dbinom() command (function) in R todetermine the probability of getting 0 heads when you flip a faircoin four times (the probability of getting heads is 0.5):
dbinom(0, size=4, prob=0.5)
Find the equivalent values for getting 1, 2, 3, or 4 headswhen you flip the coin four times. TIP: after you run the firstdbinom() command, press the up arrow and make a small change andrun it again.
probability of getting exactly 1 head: _______
probability of getting exactly 2 heads: _______
probability of getting exactly 3 heads: _______
probability of getting exactly 4 heads: _______
3) Use the pbinom() function in R to show the cumulativeprobability of getting 0, 1, 2, 3, or 4 heads when you flip thecoin 4 times (this is the same as finding the probability than thevalue is less than or equal to 0, 1, 2, 3, or 4.)
probability of getting no more than 0 heads: ____
probability of getting no more than 1 head: _____
probability of getting no more than 2 heads:_____
probability of getting no more than 3 heads: ____
probability of getting no more than 4 heads: ____
4) The following R command will show the probability ofexactly 6 successes in an experiment that has 10 trials in whichthe probability of success for each trial is 0.5:
dbinom(6, size=10, prob=0.5)
(True/False)____________
5) What is the probability of getting fewer than 2 heads whenyou flip a fair coin 3 times (round to 2 decimal places) ?______
6) What is the probability of getting no more than 3 headswhen you flip a fair coin 5 times (be sure to understand thewording differences between this question and the previousone—round to 2 decimal places)? ________
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The exponential distribution is a continuous distribution. Themain R functions that we will use for the exponential distributionare pexp() and qexp(). Here is an example of the pexp() function.Leaves are falling from a tree at a rate of 10 leaves per minute.The rate is 10, or we can say that lambda = 10 (meaning 10 leavesfall per minute). The leaves do not fall like clockwork, so thetime between leaves falling varies. If the time between leavesfalling can be modeled with an exponential distribution, then theprobability that the time between leaves falling will be less than5 seconds (which is 5/60 of a minute) would be:
(note: this is an explanation of how pexp() works, you willanswer a different question below)
pexp(5/60, rate=10)
which is about 0.565 (meaning that the probability is a bithigher than 50% that the next time-span between leaves falling willbe less than 5 seconds).
For tasks 7-12, assume that the time interval betweencustomers entering your store can be modeled using an exponentialdistribution. You know that you have an average of 4 customers perminute, so the rate is 4, or you can say that lambda =4.
It is easiest to keep everything in the original units ofmeasurement (minutes), but you can also translate that to secondsbecause a rate of “4 customers per minute” is the same as “4customer per 60 seconds,” and you can divide each number by 4 toget a rate of “1 customer per 15 seconds” or a rate of “1/15customers per second.”
7) What is the expectation for the time interval betweencustomers entering the store? Be sure to specify the units ofmeasurement in your answer. Round to 3 decimal places:___________________
8) What is the variance of the the time interval? Be sure tospecify the units of measurement in your answer. Round to 3 decimalplaces:_________________
9) The pexp() function is introduced at the bottom of Yakir,2011, p. 79, and there are some tips above. What is the probabilitythat the time interval between customers entering the store will beless than 15.5 seconds. Be sure to enter values so that everythingis in the same unit of measurement. Be sure to specify the units ofmeasurement in your answer. Round your answer to 3 decimal places:_________________.
10) What is the probability that the time interval betweencustomers entering the store will be between 10.7 seconds and 40.2seconds?________
11) The qexp() function in R allows you to enter aprobability, and it will tell you the criterion value (“cutoffpoint”) that corresponds to that probability value (e.g., if youenter .05 it tells you the cutoff point below which 5% of thevalues in the distribution fall).
What value of x would be the criterion value (cut-off point)for the top 5% of time intervals (Round to 3 decimal places, andinclude the units of measurement)? _______
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12) Describe in your own words the meaning of the number thatthe following R command produces (you are asked to interpret theresulting number so that we understand what that numbermeans).
pexp(1.2, rate=3)
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You have now had practice with the binomial distribution andthe exponential distribution. The approach to solving problems forthe normal distribution is similar to that for the exponentialfunction, but obviously you use different R functions (usuallypnorm() or qnorm()).
For the following three exercises, assume that you have anormally distributed random variable, called A, with a mean of 7,and a population standard deviation of 3.
13) What is the probability that a randomly selected valuefrom variable A will be greater than 9?_______
14) What value of variable A would be the cutoff point(criterion value) for identifying the lowest 4% of values invariable A (use the qnorm function)?____________
15) What is the probability that a randomly selected valuefrom variable A will be more than one standard deviation above itsmean (there are couple ways to solve this, one way is to use thestandard normal distribution?________________
