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 than 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 larger 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 variable Z has a continuous probability distribution that is Normal with
a mean of 0 and a standard deviation of 1. Find the area
a.
b.
c.
d.
e.
f.
g.
To the left of 1.47
To the left of -2.35
To the right of -2.10
To the right of 2.22
Between 1.23 and 2.34
Between -1.23 and 0.45
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 random 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 random 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.