Reference no: EM132229077
Continuous Probability Function Questions -
For each probability and percentile problem, draw the picture.
Q1. Consider the following experiment. You are one of 100 people enlisted to take part in a study to determine the percent of nurses in America with an R.N. (registered nurse) degree. You ask nurses if they have an R.N. degree. The nurses answer "yes" or "no." You then calculate the percentage of nurses with an R.N. degree. You give that percentage to your supervisor.
a. What part of the experiment will yield discrete data?
b. What part of the experiment will yield continuous data?
Q2. When age is rounded to the nearest year, do the data stay continuous, or do they become discrete? Why?
The Uniform Distribution Questions -
For each probability and percentile problem, draw the picture.
Q3. A random number generator picks a number from one to nine in a uniform manner.
a. X ∼________
b. Graph the probability distribution.
c. f(x) =_________
d. µ =_______
e. σ = ______
f. P(3.5 < x < 7.25) =_______
g. P(x > 5.67) = ________
h. P(x > 5|x > 3) =_________
i. Find the 90th percentile.
Q4. A subway train on the Red Line arrives every eight minutes during rush hour. We are interested in the length of time a commuter must wait for a train to arrive. The time follows a uniform distribution.
a. Define the random variable. X =______
b. X ∼______
c. Graph the probability distribution.
d. f(x) = ______
e. µ =______
f. σ =_______
g. Find the probability that the commuter waits less than one minute.
h. Find the probability that the commuter waits between three and four minutes.
i. Sixty percent of commuters wait more than how long for the train? State this in a probability question, similarly to parts g and h, draw the picture, and find the probability.
Q5. The Sky Train from the terminal to the rental-car and long-term parking center is supposed to arrive every eight minutes. The waiting times for the train are known to follow a uniform distribution. What is the average waiting time (in minutes)?
a. zero
b. two
c. three
d. four
Q6. The Sky Train from the terminal to the rental-car and long-term parking center is supposed to arrive every eight minutes. The waiting times for the train are known to follow a uniform distribution. The probability of waiting more than seven minutes given that a person has waited more than four minutes is?
a. 0.125
b. 0.250
c. 0.5
d. 0.75
Q7. Suppose that the value of a stock varies each day from $16 to $25 with a uniform distribution.
a. Find the probability that the value of the stock is more than $19.
b. Find the probability that the value of the stock is between $19 and $22.
c. Find the upper quartile - 25% of all days the stock is above what value? Draw the graph.
d. Given that the stock is greater than $18, find the probability that the stock is more than $21.
Q8. The number of miles driven by a truck driver falls between 300 and 700, and follows a uniform distribution.
a. Find the probability that the truck driver goes more than 650 miles in a day.
b. Find the probability that the truck drivers goes between 400 and 650 miles in a day.
c. At least how many miles does the truck driver travel on the furthest 10% of days?
The Exponential Distribution Questions -
Q9. Suppose that the useful life of a particular car battery, measured in months, decays with parameter 0.025. We are interested in the life of the battery.
a. Define the random variable. X =________________.
b. Is X continuous or discrete?
c. X ∼ ______
d. On average, how long would you expect one car battery to last?
e. On average, how long would you expect nine car batteries to last, if they are used one after another?
f. Find the probability that a car battery lasts more than 36 months.
g. Seventy percent of the batteries last at least how long?
Q10. The time (in years) after reaching age 60 that it takes an individual to retire is approximately exponentially distributed with a mean of about five years. Suppose we randomly pick one retired individual. We are interested in the time after age 60 to retirement.
a. Define the random variable. X =_________________.
b. Is X continuous or discrete?
c. X ∼ _______
d. µ = ______
e. σ = _______
f. Draw a graph of the probability distribution. Label the axes.
g. Find the probability that the person retired after age 70.
h. Do more people retire before age 65 or after age 65?
i. In a room of 1,000 people over age 80, how many do you expect will NOT have retired yet?
Q11. The average lifetime of a certain new cell phone is three years. The manufacturer will replace any cell phone failing within two years of the date of purchase. The lifetime of these cell phones is known to follow an exponential distribution. The decay rate is:
a. 0.3333
b. 0.5000
c. 2
d. 3
Q12. Suppose that the longevity of a light bulb is exponential with a mean lifetime of eight years.
a. Find the probability that a light bulb lasts less than one year.
b. Find the probability that a light bulb lasts between six and ten years.
c. Seventy percent of all light bulbs last at least how long?
d. A company decides to offer a warranty to give refunds to light bulbs whose lifetime is among the lowest two percent of all bulbs. To the nearest month, what should be the cutoff lifetime for the warranty to take place?
e. If a light bulb has lasted seven years, what is the probability that it fails within the 8th year.
Q13. At an urgent care facility, patients arrive at an average rate of one patient every seven minutes. Assume that the duration between arrivals is exponentially distributed.
a. Find the probability that the time between two successive visits to the urgent care facility is less than 2 minutes.
b. Find the probability that the time between two successive visits to the urgent care facility is more than 15 minutes.
c. If 10 minutes have passed since the last arrival, what is the probability that the next person will arrive within the next five minutes?
d. Find the probability that more than eight patients arrive during a half-hour period.
References -
1. The Uniform Distribution - McDougall, John A. The McDougall Program for Maximum Weight Loss. Plume, 1995.
2. The Exponential Distribution - Data from the United States Census Bureau. Data from World Earthquakes, 2013.