11-23. Assume that a given State has regulations specifying that the fluoride lev-
els in water may not exceed 1.5 milligrams per liter. Your are given the assignment
to analyze the following sample of fluoride levels, in milligrams per liter, taken over
a 45 day period.
.753 .945 .883 .721 .812 .731 .833 .891 .792
.860 .890 .782 .923 .858 1.01 .842 .890 .825
.843 .849 .772 1.05 .972 .910 .732 .799 .897
.855 .830 .761 .934 .942 .890 .782 .835 .899
.977 .891 .824 .837 .792 .843 .844 .803 .943
(a) Use class intervals about the class marks M={. 73 ,. 78 ,. 83 ,. 88 ,. 93 ,. 98 , 1. 03 }where
±. 025 is added to each class mark to form the class interval. Find and plot the
frequency and cumulative frequency distribution for this data.
(b) Find the mean and variance associated with the given data.
(c) If Xis a random variable representing the fluoride level from the above sample,
then approximate the following probabilities.
(i)P(X≤.88 ) (ii)P(. 78 < X ≤.93 ) (iii)P(X > .83 )
11-24. (Binomial distribution)
Let pdenote the probability of an event happening (success) and q= 1 −pdenote
the probability of an event not happening (failure) in a single trial. To study success
or failure of an event in n-trials one usually first calculates (p+q)n.
(a) Show that
(p+q)n=
(
n
0
)
qn+
(
n
1
)
pqn−^1 +···+
(
n
x
)
pxqn−x+···+
(
n
n
)
pn
where the term f(x) =
(
n
x
)
pxqn−xdenotes the probability density function rep-
resenting the probability that the event will happen exactly xtimes in ntrials
and there are n−xfailures, with x= 0, 1 , 2 ,.. ., n an integer.
(b) Find the probability of getting exactly 2 heads in 5 tosses of a fair coin.
(c) Find the probability of getting at least 2 heads in 5 tosses of a fair coin.
(d) Find the probability of getting at least 4 heads in 6 tosses of a fair coin.
