1) The first task is to review some information that might be useful later:
a) Write a brief definition of the word "quartile" as we have used it in previous weeks. Be sure to
provide a citation:
A quartile, which contains the root word “quarter,” does exactly as its root word suggests. It is the
numbers in a set of an organized data (from lowest to highest) that sorts them in four sections (The
Difference between Quantile and Quartile, n.d.).
b) Write a brief definition of the word "quantile" as it might be used in statistics. Be sure to provide a
citation (do not cut and paste... use your own words to summarize what you discovered):
A quantile is a less specified version of quartile. A quantile can a set of numbers in accordance to any
metric that the person specifies, which doesn’t necessarily have to be in quarters. (The Difference
between Quantile and Quartile, n.d.).
c) From within interactive R, enter the command shown below (the command shows a help page for
the pbinom command). Provide a very brief description of the arguments that are passed to the
pbinom() command ("arguments" in computer programming are the options that you give to a
function so that the function can calculate what you want it to). Note that one of the arguments is
lower.tail = TRUE, and because there is a value assigned to it with the equals sign, it means that if you
do not enter a new value for lower.tail, it will be set to TRUE by default. Do not type the ">" into R, it
is the command prompt:
> ?pbinom
Typing ?pbinom into R opens a website with the help page that specifies the following.
pbinom(q, size, prob, lower.tail = TRUE, log.p = FALSE)
The q refers to the quantile.
,The size refers to how many times the experiment is done.
The prob refers to the probability of the specific event occurring.
The lower.tail is set to true by default, which means by default it will show the probability below the
specified value.
The log.p is set to false by default, which means probabilities are shown as log of the given input.
2) You can use the dbinom() command (function) in R to determine the probability of getting 0 heads
when you flip a fair coin 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 heads when you flip the coin four times. TIP: after
you run the first dbinom() command, press the up arrow and make a small change and run it again.
probability of getting exactly 1 head:
> dbinom(1, size=4, prob=0.5)
[1] 0.25
probability of getting exactly 2 heads:
> dbinom(2, size=4, prob=0.5)
[1] 0.375
probability of getting exactly 3 heads:
> dbinom(3, size=4, prob=0.5)
[1] 0.25
probability of getting exactly 4 heads:
> dbinom(4, size=4, prob=0.5)
[1] 0.0625
,3) Use the pbinom() function in R to show the cumulative probability of getting 0, 1, 2, 3, or 4 heads
when you flip the coin 4 times (this is the same as finding the probability than the value is less than or
equal to 0, 1, 2, 3, or 4.)
probability of getting no more than 0 heads:
> pbinom(0, size=4, prob=0.5)
[1] 0.0625
probability of getting no more than 1 head:
> pbinom(1, size=4, prob=0.5)
[1] 0.3125
probability of getting no more than 2 heads:
> pbinom(2, size=4, prob=0.5)
[1] 0.6875
probability of getting no more than 3 heads:
> pbinom(3, size=4, prob=0.5)
[1] 0.9375
probability of getting no more than 4 heads:
>pbinom(4, size=4, prob=0.5)
[1] 1
4) The following R command will show the probability of exactly 6 successes in an experiment that has
10 trials in which the probability of success for each trial is 0.5:
dbinom(6, size=10, prob=0.5)
True
, 5) Read Yakir (2011, pp. 68-69) carefully to review the meaning of the pbinom function (related to
tests that a value will be “equal to” versus “less than or equal to” a criterion value). What is the
probability of getting fewer than 2 heads when you flip a fair coin 3 times (round to 2 decimal places) ?
> pbinom(1, size=3, prob=0.5)
[1] 0.5
6) What is the probability of getting no more than 3 heads when you flip a fair coin 5 times (be sure to
understand the wording differences between this question and the previous one—round to 2 decimal
places)?
> pbinom(3, size=5, prob=0.5)
[1] 0.8125 (or 0.81 when rounded to 2 decimal places)
7) What is the expectation for the time interval between customers entering the store? Be sure to
specify the units of measurement in your answer (see Yakir, 2011, pp. 79-80). Round to 3 decimal
places:
The equation that calculates the expectation for an exponential distribution is 1/lambda, which means
the expectation in this case is 0.25 minutes or 15 seconds.
8) What is the variance of the time interval? Be sure to specify the units of measurement in your
answer. Round to 3 decimal places:
The variance is calculated by 1/(lambda^2). Hence, 1/(4^2) is 0.063 minutes or 3.75 seconds.
9) The pexp() function is introduced at the bottom of Yakir, 2011, p. 79, and there are some tips above.
What is the probability that the time interval between customers entering the store will be less than
a) Write a brief definition of the word "quartile" as we have used it in previous weeks. Be sure to
provide a citation:
A quartile, which contains the root word “quarter,” does exactly as its root word suggests. It is the
numbers in a set of an organized data (from lowest to highest) that sorts them in four sections (The
Difference between Quantile and Quartile, n.d.).
b) Write a brief definition of the word "quantile" as it might be used in statistics. Be sure to provide a
citation (do not cut and paste... use your own words to summarize what you discovered):
A quantile is a less specified version of quartile. A quantile can a set of numbers in accordance to any
metric that the person specifies, which doesn’t necessarily have to be in quarters. (The Difference
between Quantile and Quartile, n.d.).
c) From within interactive R, enter the command shown below (the command shows a help page for
the pbinom command). Provide a very brief description of the arguments that are passed to the
pbinom() command ("arguments" in computer programming are the options that you give to a
function so that the function can calculate what you want it to). Note that one of the arguments is
lower.tail = TRUE, and because there is a value assigned to it with the equals sign, it means that if you
do not enter a new value for lower.tail, it will be set to TRUE by default. Do not type the ">" into R, it
is the command prompt:
> ?pbinom
Typing ?pbinom into R opens a website with the help page that specifies the following.
pbinom(q, size, prob, lower.tail = TRUE, log.p = FALSE)
The q refers to the quantile.
,The size refers to how many times the experiment is done.
The prob refers to the probability of the specific event occurring.
The lower.tail is set to true by default, which means by default it will show the probability below the
specified value.
The log.p is set to false by default, which means probabilities are shown as log of the given input.
2) You can use the dbinom() command (function) in R to determine the probability of getting 0 heads
when you flip a fair coin 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 heads when you flip the coin four times. TIP: after
you run the first dbinom() command, press the up arrow and make a small change and run it again.
probability of getting exactly 1 head:
> dbinom(1, size=4, prob=0.5)
[1] 0.25
probability of getting exactly 2 heads:
> dbinom(2, size=4, prob=0.5)
[1] 0.375
probability of getting exactly 3 heads:
> dbinom(3, size=4, prob=0.5)
[1] 0.25
probability of getting exactly 4 heads:
> dbinom(4, size=4, prob=0.5)
[1] 0.0625
,3) Use the pbinom() function in R to show the cumulative probability of getting 0, 1, 2, 3, or 4 heads
when you flip the coin 4 times (this is the same as finding the probability than the value is less than or
equal to 0, 1, 2, 3, or 4.)
probability of getting no more than 0 heads:
> pbinom(0, size=4, prob=0.5)
[1] 0.0625
probability of getting no more than 1 head:
> pbinom(1, size=4, prob=0.5)
[1] 0.3125
probability of getting no more than 2 heads:
> pbinom(2, size=4, prob=0.5)
[1] 0.6875
probability of getting no more than 3 heads:
> pbinom(3, size=4, prob=0.5)
[1] 0.9375
probability of getting no more than 4 heads:
>pbinom(4, size=4, prob=0.5)
[1] 1
4) The following R command will show the probability of exactly 6 successes in an experiment that has
10 trials in which the probability of success for each trial is 0.5:
dbinom(6, size=10, prob=0.5)
True
, 5) Read Yakir (2011, pp. 68-69) carefully to review the meaning of the pbinom function (related to
tests that a value will be “equal to” versus “less than or equal to” a criterion value). What is the
probability of getting fewer than 2 heads when you flip a fair coin 3 times (round to 2 decimal places) ?
> pbinom(1, size=3, prob=0.5)
[1] 0.5
6) What is the probability of getting no more than 3 heads when you flip a fair coin 5 times (be sure to
understand the wording differences between this question and the previous one—round to 2 decimal
places)?
> pbinom(3, size=5, prob=0.5)
[1] 0.8125 (or 0.81 when rounded to 2 decimal places)
7) What is the expectation for the time interval between customers entering the store? Be sure to
specify the units of measurement in your answer (see Yakir, 2011, pp. 79-80). Round to 3 decimal
places:
The equation that calculates the expectation for an exponential distribution is 1/lambda, which means
the expectation in this case is 0.25 minutes or 15 seconds.
8) What is the variance of the time interval? Be sure to specify the units of measurement in your
answer. Round to 3 decimal places:
The variance is calculated by 1/(lambda^2). Hence, 1/(4^2) is 0.063 minutes or 3.75 seconds.
9) The pexp() function is introduced at the bottom of Yakir, 2011, p. 79, and there are some tips above.
What is the probability that the time interval between customers entering the store will be less than
