Activity #5
Probability and the Standard Normal
1.
Suppose that the random variable X has a Uniform distribution on the interval
10
to
40
.
a.
Draw the density curve.
b.
What is the probability that X is between
10
and
15
?
c.
What is the probability that X is larger t
han
30
?
d.
Find
P(15 < X < 22)
e.
Find
P(X > 30)
2.
According to Y. Zimmels (1983), the sizes of particles used in sedimentation experiments often
have a uniform distribution. In sedimentation experiments involving mixtures of particles of
various sizes, the large
r particles hinder the movements of the smaller ones. Suppose that
spherical particles have
diameters
that are uniformly distributed between
0.01
and
0.05
centimeters.
a.
Find
P(X < 0.03)
b.
Find
P(0.02 < X < 0.04)
c.
Find
P(X > 0.45)
3.
Assume that the random variab
le Z has a continuous probability distribution that is Normal with
a mean of
0
and a standard deviation of
1
. Find the area
a.
To the left of 1.47
b.
To the left of

2.35
c.
To the right of

2.10
d.
To the right of 2.22
e.
Between 1.23 and 2.34
f.
Between

1.23 and 0.45
g.
To
the left of

0.67 or to the right of 2.33
4.
Assume that the random variable Z has a continuous probability distribution that is Normal with
a mean of
0
and a standard deviation of
1
. Find
a.
P(Z <

0.98)
b.
P(Z >

0.45)
c.
P(

0.12 < Z < 0.34)
5.
Assume that the rand
om variable Z has a continuous probability distribution that is Normal with
a mean of
0
and a standard deviation of
1
. Find
a.
The value of A which has area
0.2468
to the left of it.
b.
The value of A which has area
0.8642
to the right of it
6.
Assume that the ran
dom variable Z has a continuous probability distribution that is Normal with
a mean of
0
and a standard deviation of
1
. Find A so that
a.
P(Z < A) = 0.8765
b.
P(Z < A) = 0.1111
c.
P(Z > A) = 0.8889
d.
P(

A < Z < A) = 0.9750
7.
Use StatCrunch to simulate three data sets
based on the standard normal curve. The sets should
have 100, 1000, and 10000 entries respectively. Plot histograms to see if they appear normal.
Do not use the default settings for this
–
instead set the binwidth to something smaller than
default and
compare results. Besides graphically, also test the normality of the distribution by
sorting the data sets and determining the proportion of the numbers between plus and minus 1
and 2.
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