Three couples and two single individuals have been invited to an investment seminar and have agreed to attend. Suppose the probability that any particular couple or individual arrives late is 0.39 (a couple will travel together in the same vehicle, so either both people will be on time or else both will arrive late). Assume that different couples and individuals are on time or late independently of one another. Let X = the number of people who arrive late for the seminar.

(a) Determine the probability mass function of X. [Hint: label the three couples #1, #2, and #3 and the two individuals #4 and #5.] (Round your answers to four decimal places.)

(b) Obtain the cumulative distribution function of X. (Round your answers to four decimal places.)

Use the cumulative distribution function of X to calculate

P(3 ≤ X ≤ 6).

This is a question answered by R.

**In this problem, we'll use simulation to think about survey sampling. Suppose I want to plan a survey to learn what percentage of students prefer coffee to tea.**

a. **Let *X* be the number of students in my sample that prefer coffee. If I survey *n* students, and the true proportion of students that prefer coffee is *p*, then we can model *X* as a Binomial(*n*, *p*) random variable. If I survey ten students, what is the theoretical probability that more than 60% of my sample prefer coffee? Your answer should be a formula or an R command you could use to answer this question for any p**

c. **We can also use simulation to explore this situation. Generate 1000 samples from a Binomial(10, 0.5) random variable X. What proportion of generated samples have sample proportions greater than 60%? Using your formula from a. with p = 0.5, calculate the theoretical probability of observing a sample proportion greater than 60% and compare. **

Mattera runs a cafeteria lunch service for their employees. Every Tuesday they offer individual portions of lasagna for $6.50 each. They buy the lasagna from Caitlin’s Kitchen for $4.00 apiece. They have been doing this for years, and history tells them that demand for lasagna follows a normal distribution with a mean of 600 portions and a standard deviation of 150 portions. Any leftovers at the end of the lunch period are sold to a local shelter for $2.00 per portion.

a. What is the probability that the demand for lasagna will be within 20% of expected demand?

b. How many portions of lasagna should Mattera order from Caitlin’s in order to maximize their expected profit?

c. If they order the quantity that you chose in part b, what is the probability that they will run out of lasagna at lunch?

d. Suppose that Caitlin’s production cost is $2.50 per portion. What order quantity will maximize the expected profit for the supply chain (i.e., Caitlin’s and Mattera working together)?

e. What is the minimum buyback price that Caitlin’s could offer Mattera in order to get them to order the optimal number of portions from part c?

To generate leads for new business, Gustin Investment Services offers free financial planning seminars at major hotels in Southwest Florida. Gustin conducts seminars for groups of 25 individuals. Each seminar costs Gustin $3,500, and the commission for each new account opened is $5,000. Gustin estimates that for each individual attending the seminar, there is a 0.01 probability that he/she will open a new account. (a) Determine the equation for computing Gustin's profit per seminar, given values of the relevant parameters. Profit = (New Accounts Opened × ) – (b) What type of random variable is the number of new accounts opened? (Hint: Review Appendix 11.1 for descriptions of various types of probability distributions.) (c) Choose the appropriate spreadsheet simulation model to analyze the profitability of Gustin's seminars. (I) (II) (III) (IV) Would you recommend that Gustin continue running the seminars? (d) How many attendees (in a multiple of five, i.e., 25, 30, 35, . . .) does Gustin need before a seminar's average expected profit is greater than zero?

Consider a binary channel transmitting bits independently. Each bit is demodulated with a 0 corresponding to 0 volts and a 1 corresponding to 5 volts. Thus, the received random variable v is normally distributed with variance σ² = 1 and mean μ = 0 or μ = 5. The demodulated voltage v is compared to a threshold τ to decide whether a bit is a 0 or 1, i.e., decide that a 1 was sent if v > τ and that a 0 was sent if v < τ. Bits are equally probable so that P ( μ = 0 ) = P ( μ = 5 ) = 1 / 2. Note that an error occurs either if a 0 was sent so that μ = 0 but v > τ and a 1 is decided or if a 1 was sent so that μ = 5 but v < τ and a 0 is decided. What are the conditional probabilities P ( v > τ | μ = 0 ) and P ( v < τ | μ = 5 )? What is the resulting probability of error? What value should one choose for τ to minimize the probability of error? If 1250 bytes (8 bits each) are sent, what is the expected number of errors?

A)

The amounts of nicotine in a certain brand of cigarette are normally distributed with a mean of 0.941 g and a standard deviation of 0.319 g. The company that produces these cigarettes claims that it has now reduced the amount of nicotine. The supporting evidence consists of a sample of 30 cigarettes with a mean nicotine amount of 0.848 g.

Assuming that the given mean and standard deviation have NOT changed, find the probability of randomly selecting 30 cigarettes with a mean of 0.848 g or less.
P(x-bar < 0.848 g) =
Enter your answer as a number accurate to 4 decimal places.

B)

A particular fruit's weights are normally distributed, with a mean of 243 grams and a standard deviation of 10 grams.

If you pick 21 fruit at random, what is the probability that their mean weight will be between 248 grams and 249 grams

C)

A manufacturer knows that their items have a normally distributed lifespan, with a mean of 6.3 years, and standard deviation of 1 years.

If 5 items are picked at random, 2% of the time their mean life will be less than how many years?

Give your answer to one decimal place.

Problem 12-09 (Algorithmic)

A project has four activities (A, B, C, and D) that must be performed sequentially. The probability distributions for the time required to complete each of the activities are as follows:

1. Provide the base-case, worst-case, and best-case calculations for the time to complete the project.
   

   Activity	Base case	Worst case	Best case
   A	_____weeks	_____weeks	_____weeks
   B	weeks	weeks	weeks
   C	weeks	weeks	weeks
   D	weeks	weeks	weeks
   Total	weeks	weeks	weeks

2. Use the random numbers 0.7584, 0.7279, 0.4741 and 0.9526 to simulate the completion time of the project in weeks.
   

   Activity	Random Number	Completion Time
   A	0.7584	____weeks
   B	0.7279	weeks
   C	0.4741	weeks
   D	0.9526	weeks
   Total		weeks

4. For this problem, you’ll compare the hypergeometric and binomial distributions. Suppose there is a sock drawer with N socks, each placed loosely in the drawer (not rolled into pairs). The total number of black socks is m. You take out a random sample of n < m socks. Assume all the socks are the same shape, size, etc. and that each sock is equally likely to be chosen.

(a) Suppose the sampling is done without replacement. Calculate the probability of getting at least 2 black socks (your goal in order to wear matching black socks that day...) under the following conditions:

(i) N = 10, n = 4, m = 5.

(ii) N = 20, n = 4, m = 10.

(iii) N = 40, n = 4, m = 20.

(b) Suppose the sampling is done with replacement (this doesn’t make much sense if you are planning to wear the socks!). Calculate the probability of getting at least two black socks when you sample four socks and the proportion of black socks is 0.5. Compare your answer to those in (a).

The 911 number of the city of Turtle Creek receives emergency calls for a life-support vehicle (LSV) at a mean rate of 15 calls per hour. The interarrival time between these calls has an exponential probability distribution. The time that elapses from the dispatch of an LSV in response to a call until the LSV is available to respond to another call has an exponential probability distribution with a mean of 48 minutes. Turtle Creek defines the average response time as the average time between the receipt of a call and the dispatch of an LSV to attend to this call. Calls are processed on a first-come first-served basis. Turtle Creek wants an LSV fleet of sufficient size to keep average response time to less than 2 minutes.

1. Turtle Creek is considering two different sizes for its fleet of LSVs: 15 LSVs or 20 LSVs. For each of these fleet sizes, compute the average response time (how long, on average, a call has to wait in the queue until an LSV is dispatched to it).
2. Does either of the fleet sizes meet Turtle Creek’s goal of an average response time of 2 minutes.

The 911 number of the city of Turtle Creek receives emergency calls for a life-support vehicle (LSV) at a mean rate of 15 calls per hour. The interarrival time between these calls has an exponential probability distribution. The time that elapses from the dispatch of an LSV in response to a call until the LSV is available to respond to another call has an exponential probability distribution with a mean of 48 minutes. Turtle Creek defines the average response time as the average time between the receipt of a call and the dispatch of an LSV to attend to this call. Calls are processed on a first-come first-served basis. Turtle Creek wants an LSV fleet of sufficient size to keep average response time to less than 2 minutes.

1. Turtle Creek is considering two different sizes for its fleet of LSVs: 15 LSVs or 20 LSVs. For each of these fleet sizes, compute the average response time (how long, on average, a call has to wait in the queue until an LSV is dispatched to it).
2. Does either of the fleet sizes meet Turtle Creek’s goal of an average response time of 2 minutes.