Stat 321 – HW 4

Due Tuesday, Feb. 12

1) Exercise #12 (p. 98)

Use P(X=x) notation for each probability of interest

2) A teacher of statistics tried out the solitaire game on her computer. She won 74 times out of 444 attempts in the past. Let p=probability she wins a single game.

(a) Based on the data, what is your estimate for p?

Suppose she decides to start playing again. Assume p does not changed and let X=number of games she has to play until her next win.

(b) What are the possible outcomes for X? Can you describe a list of them? Is X discrete?

(d) Calculate P(X=2) using the following steps:

1. For X to equal 2, her first win must occur in her second game. In this case, what must be true about the first game she plays? What is the probability of this outcome?

2. What is the probability she loses on the first game and then wins on the second game? [This is P(X=2).] What must you assume about the games to make this calculation? Do you think this assumption is reasonable?

(e) Calculate P(X=3). (What has to happen for her first win to come in her third game?)

(f) Derive a general expression for P(X=x) in terms of x.

(g) What will the line plot of this probability distribution look like? What will the graph of the cdf look like?

3) Exercise #30 (p. 106-7)

4) Exercise #48 (p. 113-4)

You may use either Appendix Table A.1 or Minitab. Either way, make sure it is very clear what you have done. If you use Minitab, include a copy of the Minitab output.

5) At the beginning of the quarter you were asked to indicate a preference between Coke and Pepsi. We will consider these 35 responses a random sample from the population of all current and previous Stat 321 students.

(a) If the two sodas are equally preferred among all Stat 321 students, what is the probability that a randomly selected student lists Coke as their preference?

(b) Let X=number of students who pick Coke in our sample. If the opinions of students in the class are independent of each other, is X a binomial random variable? (Comment on each of the four requirements.) Identify the parameters of the binomial distribution, assuming that the sodas are equally preferred.

(d) If Coke and Pepsi are equally preferred in the population, what is the expected number of Coke responses in our survey? (Expected value of X.)

(e) Of 35 responses, 24 students indicated that they preferred Coke to Pepsi. Is it possible that Coke and Pepsi are equally preferred in the population, but just by chance, we found a 24/11split in our sample?

(f) Use Minitab to calculate the probability of obtaining 24 or more people preferring Coke in 35 independent trials, still assuming the colas are equally preferred.

(g) Is this probability small enough to convince you that the sample results didn’t happen by “chance” but instead reflects a true preference for Coke by Stat 321 students? Explain.

6) Currently, the first round of the playoffs in Major League Baseball is a best-of-five series where the first team to win three games wins the series. The later rounds are a best-of-seven series (first team to 4 victories wins the series).

(a) Suppose team A has a .6 chance of winning a game and the games in a series are independent of each other. Calculate the probability that team A wins the series.

- Let X represent the number of losses before team A wins the third game.

- What values of X correspond to team A winning the series?

- If you use Minitab, be clear what you have done and include your output.

(b) Repeat the above analysis for a best-of-seven series. Which format is more advantageous to the better team?

7) Exercise #86 (p. 125)