3. Consider the following probability table for a discrete random variable X.
X P(X=x)
1
0.24
2
0.57
3
0.19
What is the sampling distribution of X based
on n = 2 observed values of X?
4. Craps is a game playing by rolling two fair dice. If the total number of
dots on the first roll of a round is 7 or 11, an event known as a “natural”,
then the player automatically wins that round and a new round begins. The
probability of rolling a natural is 2/9. We will consider 200 rounds of craps.
(a) Let X represent the total
number of rounds in which a player rolls a natural. Name the probability
distribution of X. Give the relevant parameter values associated with
this probability distribution.
(b) What probability distribution can you
use to approximate the probability distribution of X? Explain. Give the
relevant parameter values associated with this probability distribution.
(c) Use your answer in part (b) to
approximate the probability that a player rolls a natural in at most 40 of the
rounds.
(d) Let Pˆ represent the proportion of rounds in which a player
rolls a natural. What is the sampling distribution of Pˆ ? Explain. Give the relevant parameter
values associated with this sampling distribution.
(e) Use your answer from part (d) to
approximate the probability that at most 20% of the rounds end because a player
rolls a natural. (i.e. at most 40/200 rounds)
(f) The actual probability in parts (b)
and (e), to five significant digits, is 0.253977. Which of your two
approximations is better? Explain the discrepancy.
5. Suppose that the body temperatures of healthy male adults, taken orally,
vary according to a normal distribution with a mean of 36.8 °C and a standard
deviation of 0.4 °C. Define random variables as appropriate in answering the
following questions.
(a) What probability a randomly selected
healthy male adult has a body temperature above 36.2 °C?
(b) What proportion of healthy male adults
have a body temperature between 35.7 °C and 37.3 °C?
(c) Twenty percent of healthy male adults
have a body temperature above what value?
(d) Suppose you take a random sample of
four healthy male adults. What is the probability the average body temperature
of these four people is at most 37.13 °C?
6. A certain breed of hen produces eggs with a mean weight of 58 grams and
a standard deviation of 14 grams. Let X represent the mean weight of a
random sample of 72 eggs.
(a) What is the sampling distribution of X
? Explain. Give the relevant parameter values associated with this sampling
distribution.
(b) Approximate the probability that the
mean weight of the 72 eggs is between 55 and 62 grams. (c) Approximate the
probability that the total weight of the 72 eggs is at least 4400 grams.
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