It has been suggested that the highest priority of retirees is travel. Thus, a study was conducted to investigate the differences in the length of stay of a trip for pre- and post-retirees. A sample of 692 travelers were asked how long they stayed on a typical trip. The observed results of the study are found below.

With this information, construct a table of expected numbers.

The chi-squared value is ?2= 35.8310290188111

Using α=0.05, consider a test of ?0: length of stay and retirement are not related vs. ??: length of stay and retirement are related. Note that this problem involves a 4×2 contingency table and that the magic number for evaluating such a table at theα=0.05 level of significance is magic number = 7.81.

The final conclusion is

A. Accept ?0. There is not sufficient evidence to conclude that the length of stay is related to retirement.
B. Reject ?0.There is sufficient evidence to conclude that the length of stay is related to retirement.

5. A survey on British Social Attitudes asked respondents if they had ever boycotted goods for ethical reasons (Statesman, January 28, 2008). The survey found that 23% of the respondents have boycotted goods for ethical reasons.

a) In a sample of six British citizens, what is the probability that two have ever boycotted goods for ethical reasons? .

b) In a sample of six British citizens, what is the probability that at least two respondents have boycotted goods for ethical reasons?

c) In a sample of ten British citizens, what is the probability that between 3 and 6 have boycotted goods for ethical reasons?

d) In a sample of ten British citizens, what is the expected number of people that have boycotted goods for ethical reasons? Also find the standard deviation.

A company has three machines. On any day, each working machine breaks down with probability 0.4, independent of other machines. At the end of each day, the machines that have broken down are sent to a repairman who can work on only one machine at a time. When the repairman has one or more machines to repair at the beginning of a day, he repairs and returns exactly one at the end of that day. Let Xn be the number of working machines at the beginning of day n

1. Is {Xn : n ≥ 0} a Markov chain? Why?
2. Given that one machine is working today what is the probability that two machines will be working tomorrow?
3. Given that two machines are working today what is the probability that all machines will be working two days later?

(b) The battery in Ali’s phone runs out at random moments. Over a long period, he found that the battery runs out, on average 4 times in a 30-day period.

(i) Find the probability that the battery runs out fewer than 3 times in a 25-day period.

(ii) Find the probability that the battery runs out more than 50 times in a year (365 days).

(iii) Independently of his phone battery, Ali’s computer battery also runs out at random moments. On average, it runs out twice in a 15-day period. Find the probability that the total number of times that his phone battery and his computer battery run out in a 10-day period is at least 4.

The lifetime (in weeks) of printer toner cartridges is normally distributed with mean 4 and variance 1.5. A consumer buys a number of these cartridges with the intention of replacing them successively as soon as they become empty. The cartridges have independent lifetimes. The cartridges with lifetimes exceeding 5 weeks are classified as “satisfactory”.

(a) Find the approximate 95th percentile for the distribution of the life time of printer toner cartridges.

(b) What is the probability that a randomly selected cartridge is classified as “satisfactory”?

(c) If a sample of 20 cartridges is selected at random and the cartridges are inspected one by one, what is the probability that exactly 8 cartridges are classified as “satisfactory”?

(d) If a customer buys 30 cartridges, calculate the probability that at least 5 of the cartridges have not yet been used by the end of the 90th week.

Organisms are present in ballast water discharged from a ship according to a Poisson process with a concentration of 10 organisms/m³ (the article "Counting at Low Concentrations: The Statistical Challenges of Verifying Ballast Water Discharge Standards"† considers using the Poisson process for this purpose).

(a)

What is the probability that one cubic meter of discharge contains at least 5 organisms? (Round your answer to three decimal places.)

(b)

What is the probability that the number of organisms in 1.5 m³ of discharge exceeds its mean value by more than two standard deviations? (Round your answer to three decimal places.)

(c)

For what amount of discharge would the probability of containing at least 1 organism be 0.995? (Round your answer to two decimal places.)

m³

• Suppose all of the Skittles in the class data set are combined into one large bowl and you are going to randomly select ten Skittles with replacement and count how many are yellow.
  • (a) List the requirements of the binomial probability distribution and show that this meets them, including identifying the values for n and p. (6 points)
  • (b) What is the probability that exactly 4 of the 10 Skittles are yellow? (4 points)
  • (c) What is the probability that at most 2 of the 10 Skittles are yellow? (4 points)
  • (d) For samples of size 10, what is the expected value and standard deviation for the number of yellow skittles that will be included? (4 points)

*Show your work

A person tried by a 3-judge panel is declared guilty if at least 2 judges cast votes of guilty. Suppose that when the defendant is in fact guilty, each judge will independently vote guilty with probability 0.7, whereas when the defendant is in fact innocent, this probability drops to 0.2. If 70 percent of defendants are guilty, compute the conditional probability that judge number 3 votes guilty given that

(a) judges 1 and 2 vote guilty;

(b) judges 1 and 2 cast 1 guilty and 1 not guilty vote;

(c) judges 1 and 2 both cast not guilty votes.

Let Ei , i = 1, 2, 3 denote the event that judge i casts a guilty vote. Are these events independent? Are they conditionally independent? Explain.

The following three games are scheduled to be played at the World Curling Championship one morning. The values in parentheses are the probabilities of each team winning their respective game.
Game 1: Finland (0.2) vs. Canada (0.8)

Game 2: USA (0.3) vs. Switzerland (0.7)

Game 3: Germany (0.4) vs. Japan (0.6)
(a) The outcome of interest is the set of winners for each of the three games. List the complete sample space of outcomes and calculate the probability of each outcome.

(b) Let X be the number of European teams that win their respective games. Find the probability distribution of X.

(c) Find the expected value and variance of X.

(d) If two European teams win their games, what is the probability that Finland is one of them?

Answer the following questions:

1. In the following scenarios, identify whether or not it is uniform, binomial, or hypogeometric probability distribution. Briefly explain your choice.
[Do not solve the problem]

a) A spinner is split into 7 equal sections. Each section is coloured differently. One of the sections is coloured purple. What is the probability that the spinner lands on purple

b) If you roll a die 14 times. What is the probability that you will roll an even number 4 times?

2. a) What is the key characteristic that differentiates a binomial distribution from a hypergeometric distribution?

b) Using cards as the scenario, create a question (that doesn’t need to be answered) that illustrate a binomial distribution

c) Using cards as the scenario, create a question (that doesn’t need to be answered) that illustrate a hypergeometric distribution