The number of calls that come into a small mail-order company follows a Poisson distribution. Currently, these calls are serviced by a single operator. The manager knows from past experience that an additional operator will be needed if the rate of calls exceeds 15 per hour. The manager observes that 6 calls came into the mail-order company during a randomly selected 15-minute period. a. If the rate of calls is actually 20 per hour, what is the probability that 9 or more calls will come in during a given 15-minute period? b. If the rate of calls is really 30 per hour, what is the probability that 9 or more calls will come in during a given 15-minute period? c. Based on the calculations in parts a and b, do you think that the rate of incoming calls is more likely to be 20 or 30 per hour? d. Would you advise the manager to hire a second operator ? Explain.

The longest "run" of S's in the sequence SSFSSSSFFS has length 4, corresponding to the S's on the fourth, fifth, sixth, and seventh positions. Consider a binomial experiment with n = 4, and let ybe the length (number of trials) in the longest run of S's. (Round your answers to four decimal places.)

(a) When p = 0.5, the 16 possible outcomes are equally likely. Determine the probability distribution of y in this case (first list all outcomes and the y value for each one).

Calculate μ_y.
μ_y =  

(b) Repeat Part (a) for the case p = 0.7.

Calculate μ_y.
μ_y =  

(c) Let z denote the longest run of either S's or F's. Determine the probability distribution of z when p = 0.5.

An ordinance requiring that a smoke detector be installed in all previously constructed houses has been in effect in a particular city for 1 year. The fire department is concerned that many houses remain without detectors. Let p= the true proportion of such houses having detectors and suppose that a portion of random sample of 25 homes is inspected. If the sample strongly indicates that fewer than 80% of all houses have a detector, the fire department will campaign for a mandatory inspection program. Because of the costliness of the program, the department prefers not to call for such inspections unless denote the number of homes with detectors among the 25 sampled. Consider rejecting the claim that p ≥ 0.8 if x ≤ 15.

1. What is the probability that the claim is rejected when the actual value of p is 0.8?
2. What is the probability of not rejecting the claim when p = 0.7? When p = 0.6?
3. How do the “error probabilities” of parts (a) and (b) change if the value 15 in the decision rubble is replaced by 14?

A lady claimed that she is able to tell whether the tea was added first or the milk was added first to a cup. To test this idea, a statistician proposed to give her ten cups of tea, each made in random order (tea first or milk first) without telling her which is which. Assume each cup is independent. Let X equal to the number of cups that the lady identified correctly.

(a) Suppose that she is just randomly guessing, with a 50-50 percent chance. Find P(X = 7) .

(b) Suppose again that she is randomly guessing, with a 50-50 percent chance. This experiment will be stopped early if she cannot correctly identify at least one cup among the first three cups. What is the probability that this experiment continues beyond three cups?

(c) Suppose she is indeed able to tell 9% of the time. Find the probability she correctly identifies at least 7 cups.

Through research, we know that 65% of teens list Mountain Dew as their favorite soft drink. Let’s assume that the soft drink preference of one teen is independent of another teen’s preferred soft drink.

(a) Suppose I’m going to pick teens repeatedly until I find one that favors Mountain Dew. Let X count the number of teens I have to pick in order to find that Mountain Dew preferring teen. Calculate P(X = 4).

(b) (Bonus:) I want to find 3 teens that favor Mountain Dew. I’m going to continue to pick teens until that happens. What is the probability that it take me 4 picks to find my third Mountain Dew favoring teen?

(c) Suppose I pick 3 teens at random. What is the probability that...

i. ... all 3 favor Mountain Dew?

ii. ... at least one favors Mountain Dew?

All airport passengers at the Capital City Airport must pass through a security screening area before proceeding to the boarding area. The airport has three screening stations available, and the facility managers must decide how many to open at any particular time. The average time for processing one passenger at each screening station is 0.5 minutes. On Saturday morning the arrival rate is 3.3 passengers per minute. Assume that processing times at each screening station follow an exponential distribution and those arrivals following a Poisson distribution. Suppose two of the three screening stations are open on Saturday morning. Show steps to find:

    e. What is the probability of five passengers in the screening stations?

Willow Brook National Bank operates a drive-up teller window that allows customers to complete bank transactions without getting out of their cars. On weekday mornings, arrivals to the drive-up teller window occur at random, with an arrival rate of 30 customers per hour or 0.5 customers per minute.

(a)

What is the mean or expected number of customers that will arrive in a four-minute period?

(b)

Assume that the Poisson probability distribution can be used to describe the arrival process. Use the arrival rate in part (a) and compute the probabilities that exactly 0, 1, 2, and 3 customers will arrive during a four-minute period. (Round your answers to four decimal places.)

(c)

Delays are expected if more than three customers arrive during any four-minute period. What is the probability that delays will occur? (Round your answer to four decimal places.)

A first year statistics class took part in a simple experiment. Each person took their pulse rate. They then flipped a coin and, if it came up tails, they ran in place for one minute. Everyone then took their pulse a second time. Of the 92 students in the class, 32 ran in place. In repetitions of this experiment, the number that run in place should have a binomial distribution with n = 92 trials and probability of success p = 1/2. a. Simulate 100 repetitions of this experiment using the commands given below. In how many of the 100 experiments did 32 or fewer people have to run? b. Make a histogram of the 100 observations and describe its important characteristics (shape, location, spread and outliers). c. Using software, calculate the probability of getting 32 or fewer tails in 92 tosses of a fair coin. d. Does it seem likely that only 32 of the 92 students got tails? Give a reasonable explanation for what happened.

The number of calls that come into a small mail-order company follows a Poisson distribution. Currently, these calls are serviced by a single operator. The manager knows from past experience that an additional operator will be needed if the rate of calls exceeds 15 per hour. The manager observes that 6 calls came into the mail-order company during a randomly selected 15-minute period.

a. If the rate of calls is actually 20 per hour, what is the probability that 9 or more calls will come in during a given 15-minute period?

b. If the rate of calls is really 30 per hour, what is the probability that 9 or more calls will come in during a given 15-minute period?

c. Based on the calculations in parts a and b, do you think that the rate of incoming calls is more likely to be 20 or 30 per hour?

d. Would you advise the manager to hire a second operator ? Explain.

A thermocouple is used to measure a known temperature of 25oC. A set of measurements is taken. 29.6% of the measurements are found within ±1 oC of 25oC (the sample mean value is 25oC). We assume the temperature to be a random variable following a Gaussian distribution function. Please answer the below questions and upload your work as a single file (preferably PDF or jpeg). You need to show ALL work.

Question 1. Show that the standard deviation of the thermocouple σ is equal to 2.63oC. Show all steps to obtain the result. Use this value of the standard deviation σ in the subsequent questions.

Question 2. What is the probability that a temperature reading is greater than 27oC? Show all steps to obtain the result.

Question 3. What is the probability that a temperature reading is less than 26.5oC? Show all steps to obtain the result.

Question 4. Determine the 95% confidence interval of the mean temperature μ. Note that the number of pressure readings is 50.