What is the probability that 16 people enter the security

Question

Martial Rutherphore Sop

Question 1. In the spring, a biologist and her team placed radio collars on 35 grizzly bears living in Southwestern Alberta in an attempt to study their migration habits. In the fall, she and her team return to the area and observe 13 grizzly bears from a helicopter, and count how many of these bears had a radio- collar.

Identify the probability model that best models this count.

- A. Binomial distribution
- B. Uniform distribution
- C. Poisson distribution
- D. Normal distribution
- E. Bernoulli distribution
- F. Hypergeometric distribution
- G. none of the above

Question 2. A pollster working for a certain political candidate is to randomly call 400 persons, and ask if they will vote for this political candidate. The pollster then counts how many of the 400 persons indicate they will vote for said candidate.

Identify the probability model that best models this count.

- A. Poisson distribution
- B. Uniform distribution
- C. Binomial distribution
- D. Bernoulli distribution
- E. Hypergeometric distribution
- F. Normal distribution
- G. none of the above

Question 3. A student waited at a bus stop. To pass the time, he randomly picked vehicles passing by and counted the number of drivers who are using a cell phone while driving.

Identify the probability model that best models this count.

- A. Poisson distribution
- B. Hypergeometric distribution
- C. Bernoulli distribution
- D. Uniform distribution
- E. Binomial distribution
- F. Normal distribution
- G. none of the above

Question 4. The manager of the emergency room (ER) at a large hospital is to count the number of patients arriving at the ER to receive medical attention, per hour. He randomly picks 12 one-hour time intervals, and counts the number of people arriving to receive medical attention in each of these 12 one-hour time intervals.

He notices that this count fluctuates from one-hour to the next.

Identify the probability model that best models such fluctuations.

- A. Random distribution
- B. Poisson distribution
- C. Uniform distribution
- D. Bernoulli distribution
- E. Hypergeometric distribution
- F. Binomial distribution
- G. none of the above

Question 5. A lottery consists of picking 5 numbers from 1 to 45, the winner of this lottery has to match all numbers chosen. That is, his/her 5 numbers have to match the 5 numbers taken from 1 to 45.

A player of this lottery is to count how many of his/her numbers match the winning 5-number combination.

Identify the probability model that best models this count.

- A. Hypergeometric distribution
- B. Uniform distribution
- C. Bernoulli distribution
- D. Poisson distribution
- E. Binomial distribution
- F. Normal distribution
- G. none of the above

Question 6. A person is to walk into a casino and play a certain game. The chance the person will win the game is 0.4. Once they play the first game, win or lose, they are to play the game 3 more times for a total of 4 games.

A random variable X is to count how many of the 4 games the gambler wins.

(a) Finish the probability distribution of X below. Use four decimals in each of your entries.

X	0	1	2	3	4
P(X = x)

(b) From the distribution you found in part (a), what can you say about the distribution of X ?

The distribution of X is
- ?
- symmetrical
- skewed to the right
- skewed to the left
, with an mean of ____ games won and a standard deviation of ___ games won.

Question 7. A recent survey of post-secondary education stu- dents in Canada revealed that 74% are confident in their ability to manage their finances.

You are to randomly pick 35 post-secondary education stu- dents across the country, and ask each the following question:

Are you confident in your ability to manage your finances?

You have defined the random variable X to represent the number, out of 35 post-secondary students chosen, who respond YES.

(a) What is the probability that 23 of these post-secondary students respond YES?

P(X = 23) = _____ (use four decimals in your answer)

(b) What is the probability that between 14 and 20 of these students will respond YES?

P(14 ≤ X ≤ 20) = (use four decimals in your answer)

(c) How many of the 35 post-secondary students would you expect to respond YES? Find the standard deviation as well.

E(X ) = µ_X = (use two decimals in your answer)

SD(X ) = σ_X = (use two decimals in your answer)

(d) What can you say about the distribution of the X ? Select the most appropriate answer.

A. The distribution of X -values is skewed to the right.
B. The distribution of X -values is approximately sym- metrical.
C. The distribution of X -values is unknown.
D. The distribution of X -values is skewed to the left.

Question 8. A biologist captures 21 grizzly bears during the spring, and fits each with a radio collar. At the end of summer, the biologist is to observe 15 grizzly bears from a helicopter, and count the number that are radio collared. This count is represented by the random variable X .

Suppose there are 119 grizzly bears in the population.

(a) What is the probability that of the 15 grizzly bears observed, 4 had radio collars? Use four decimals in your answer.

P(X = 4) = ____

(b) Find the probability that between 3 and 7 (inclusive) of the 15 grizzly bears observed were radio collared?

P(3 ≤ X ≤ 7) = ____(use four decimals)

(c) How many of the 15 grizzly bears observe from the helicopter does the biologist expect to be radio-collared? Provide the standard deviation as well.

E(X ) = ___ (use two decimals)

SD(X ) = ____ (use two decimals)

(d) The biologist gets back from the helicopter observation expedition, and was asked the question: How many radio col- lared grizzly bears did you see? The biologist cannot remember exactly, so responds " somewhere between 5 and 8 (inclusive) ".

Given this information, what is the probability that the biol- ogist saw 6 radio-collared grizzly bears?
(use four decimals in your answer)

Question 9. Your statistics professor hands you a fair die, with each side having the same chance of being the uppermost face. You are asked to toss the die 7 times and count the number of times the die shows a topside of six.

This count is represented by the random variable X .

(a) Complete the probability distribution of X below. Use four decimals in each of your entries.

X	0	1	2	3	4	5	6
P(X = x)								0.00

(b) From the distribution you found in part (a), what can you say about the distribution of X ?

The distribution of X is
- ?
- symmetrical
- skewed to the right
- skewed to the left
, with an mean of ____ sixes and a standard deviation of ____ sixes.

(Enter your answers to two decimals.)

(c) As requested, you have tossed the die 7-times and observed X = 6 sixes. If you were to repeat the 7-tosses of this die, what is the probability you will observe at least the same number of sixes as this? Enter your answer to four decimal places.

(d) Looking at your initial outcome of X = 6 and the chance of observing at least this outcome again, what does this say about the fairness of the die? Select the most appropriate answer.

A. The die appears not to be fair. X = 6 is an unusual event.
B. The distribution of the random variable X appears to be skewed to the right.
C. The distribution of the random variable X appears to be roughly symmetrical.
D. The die appears to be fair. X = 6 is not an unusual event.
E. The die appears to favor an outcome of a six. X = 6 is an unusual event.

Question 10. On a typical day at a local popular mall, the number of shoplifters caught by mall security fluctuates from day to day, with an average of 8.5 caught per day.

Suppose you are to count the number of shoplifters apprehended at this mall today. (Assume today is a typical day at the mall.)

Part (a) What is the probability that mall security apprehended 7 shoplifters?

(use four decimals in your answer)

Part (b) What is the probability that mall security will apprehend at least 7 shoplifters. Enter your answer to four decimals.

Part (c) Find the probability that between 6 and 12 shoplifters will be apprehended. Enter your answer to four decimals.

Part (d) Think about the distribution of the number of shoplifters mall security apprehends on a typical day. What can you say about this distribution? Select the most appropriate reason below.

A. The distribution of values is skewed to the right, with a mean of 8.5 shoplifters and a standard deviation of 2.91547594742265 shoplifters.

B. The distribution of values is skewed to the right, with a mean of 8.5 shoplifters and a variance of 2.91547594742265 shoplifters.

C. The distribution of values is roughly symmetri- cal, with a mean of 8.5 shoplifters and a variance of 2.91547594742265 shoplifters2.

D. The number of shoplifters apprehended each day is 8.5.

E. The distribution of values is roughly symmetrical, with a mean of 8.5 shoplifters and a standard deviation of 2.91547594742265 shoplifters.

Part (e) On a typical day, the mall is open from 10:00am to 9:00pm, a total of 11 hours.You are interested in the number of shoplifters apprehended in any given hour.

How many shoplifters would you expect mall security to catch in any given hour? Provide the variance of this count as well.

µ = ____ (use three decimals in your answer)

σ² = ____ (use three decimals in your answer)

Question 11. The number of people entering a security check-in lineup in a 15-minute interval at a medium sized airport can be modeled by the following probability model:

P(X = x) = (e^-20.8(20.8)^x)/x!              x = 0, 1, 2, ....

Part (a) What does 20.8 represent in the probability model?

Select the most appropriate explanation below.

A. 20.8 represents the average number of people who enter the a security check-in lineup every 15-minutes.
B. 20.8 represents a weighted-average of the number of people who enter the a security check-in lineup every 15-minutes.
C. 20.8 is the standard deviation of the distribution of people entering the security check-in lineup every 15- minutes.
D. 20.8 represents how much skewed the distribution of values is.
E. 20.8 is the rate at which people enter the security check-in lineup every 15 minutes.

Part (b) What is the probability that 16 people enter the security check-in lineup in a 15-minute interval? Use four decimals in your answer.

P(X = 16) =

Part (c) What is the probability that at least 4 people will enter the security check-in lineup in a 5-minute interval? Enter answer to four decimals.

Part (d) In the past 15-minutes, you have been told that somewhere between 15 and 18 people, inclusive, have entered the security lineup. What is the probability that this uncertain number is 17?

Question 12. A certain section of a Northern Alberta forest is undergoing a pine-beetle infestation. A biologist has deter- mined that the number of pine-beetle infected trees fluctuates from acre to acre, with an average of 10.3 pine-beetle infected trees per acre.

An acre of this forest is chosen at random and the number of pine-beetle infected pine trees is to be counted.

(a) What is the probability that between 6 and 10, inclusive, infected trees are found?

(use four decimals in your answer)

(b) What is the probability that more than 13 pine beetle infected trees will be found?

(use four decimals)

(c) As a way to combat the infestation, the infected trees are to be sprayed with an insecticide at a cost of $3 for every tree infested with pine beetle(s), plus an overhead fixed cost of $60 for equipment rental. Letting Cost represent the total cost for spraying all the pine-beetle infested trees for a randomly chosen acre of forest.

Find the expected cost of spraying an acre. In addition, find the standard deviation in the cost of spraying an acre.

E(Cost) = $ (use two decimals)
SD(Cost) = $ (use two decimals)

Question 13. A recent poll has suggested that 64 % of Canadians will be spending money - decorations, halloween treats, etc. - to celebrate Halloween this year.

24 Canadians are randomly chosen, and the number that will be spending money to celebrate Halloween is to be counted. This count is represented by the random variable X .

Part (a) What is the probability that 14 of these Canadians indicate they will be spending money to celebrate Halloween?

P(X = 14) = ___ (use four decimals in your answer)

Part (b) What is the probability that between 7 and 14 of these Canadians, inclusive, indicate they will be spending money to celebrate Halloween?
P(7 ≤ X ≤ 14) = ____ (use four decimals in your answer)

Part (c) How many of the 24-Canadians randomly chosen would you expect to indicate they will be spending money to celebrate Halloween? Find the standard deviation as well.

E(X ) = µ_X = ____ (use two decimals in your answer)

SD(X ) = σ_X = ____ (use two decimals in your answer)

Part (d) What is the probability that the 10-th Canadian random chosen is the 6-th to say they will be spending money to celebrate Halloween?

(use four decimals in your answer)

Question 14. A certain airline has 169 seats available for a flight from YYC (Calgary International Airport) to LAX (Los Angeles International Airport). Because people with reserva- tions do not show up for their flight 12% of the time, the airline always overbooks this flight. That is, there are more passengers that have tickets on the flight than there are seats.

Suppose the airline has 181 passengers booked for 169 seats. Assume one person showing up for the flight does not effect others who may, or may not, show up for this flight.

(a) How many people (with tickets) does the airline expect to show up for this flight? Provide the standard deviation as well. Enter your answers to two decimals.

Expected number of the number of people showing up for the flight =

Standard deviation of the number of people who show up for the flight =

(b) When the flight takes off from YYC, what is the probability that there will be 6 seats empty? Enter your answer to four decimals.

(c) What is the probability that a passenger with a flight reservation will not make it to LAX due to overbooking? Use four decimals in your answer.