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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.