Explain the difference between a discrete and a continuous
random variable.
1) Explain the difference between a discrete and a
continuous random variable. Give two examples of each type of random variable.
2) Determine whether each of the distributions given below
represents a probability distribution. Justify your answer.
a) x 1 2 3 4
P (x) 1/8 1/8 3/8 1/8
b) x 3 6 8
P (x) 0.2 0 1
c) x 20 30 40 50
P (x) 0.3 0.2 0.1 0.4
3) Four cards are selected, one at a time, from a standard
deck of 52 cards. Let x represent the number of aces drawn in a set of 4 cards.
a. If this experiment is completed without replacement,
explain why x is not a binomial random variable.
b. If this experiment is completed with replacement, explain
why x is a binomial random variable.
4) How does the bell-shaped curve for the sampling distribution
of sample means for samples of size n = 100 compare to the bell-shaped curve
for the sampling distribution of sample means for samples of size n = 60?
5) What are the characteristics of the normal distribution?
Why is the normal distribution important in statistical analysis? Provide an
example of an application of the normal distribution.
6) In your own words describe the standard normal
distribution. Explain why it can be used to find probabilities for all normal
distributions.
7) Explain why the normal distribution can be used as an
approximation to the binomial distribution. What conditions must be met to use
the normal distribution to approximate the binomial distribution? Why is a
correction for continuity necessary?
8) Consider a binomial distribution with 15 identical
trials, and a probability of success of 0.5
a. Find the probability that x = 2 using the binomial tables
b. Use the normal approximation to find the probability that
x = 2
9) The diameters of oranges in a certain orchard are
normally distributed with a mean of 5.26 inches and a standard deviation of
0.50 inches.
a. What percentage of the oranges in this orchard have
diameters less than 4.5 inches?
b. What percentage of the oranges in this orchard is larger
than 5.12 inches?
c. A random sample of 100 oranges is gathered and the mean
diameter obtained was 5.12. If another sample of 100 is taken, what is the
probability that its sample mean will be greater than 5.12 inches?
d. Why is the z-score used in answering (a), (b), and (c)?
e. Why is the formula for z used in (c) different from that
used in (a) and (b)?
10) Assume that the population of heights of male college
students is approximately normally distributed with mean m of 68 inches and
standard deviation s of 3.75 inches. A random sample of 16 heights is obtained.
a. Describe the distribution of x, height of the college
student.
b. Find the proportion of male college students whose height
is greater than 70 inches.
c. Describe the distribution of x, the mean of samples of
size 16.
d. Find the mean and standard error of the distribution.
e. Find P ( > 70)
f. Find P ( < 67)
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