Question

Question 1

    Snowfalls occur randomly and independently over the course of winter in a Nebraska city. The average is one snowfall every 3 days.

    a) What is the probability of 5 snowfalls in 2 weeks? Carry answer to the nearest ten-thousandths

    b) What is the probability of a snowfall today? Carry answer to the nearest ten-thousandths

Question 2

    After observing the number of children checking out books, a librarian estimated the following probability distribution of x, the number of books checked out per visit.

        x                   f(x)

      1                   .05

        2                  .15

        3                  .15

        4                  .25

        5                 .20

        6                  .10

        7                  .10

    a) what is the probability that a child will check out more than four books?

    b) What is the probability that a child will check out at least two books?

Question 3

    Hits on a personal website occur quite infrequently. They occur randomly and independently with a average of five per week.

    a) What is the probability that the site gets 10 or more hits a week? Carry answer to the nearest ten-thousandths

    b) What is the probability that the site gets 20 or more hits in two weeks? Carry answer to the nearest ten-thousandths

Question 4

    A sign on the gas pumps of a chain of gasoline stations encourages customers to have their oil checked with the claim that one of four cars needs to have oil added. If this is true, What is the probability of the following events. Binomial

    a) One out of the next four cars needs oil? Carry answer to the nearest ten-thousandths

    b) Two out of the next eight cars need oil? Carry answer to the nearest ten-thousandths

    c) Three out of the next 12 cars need oil? Carry answer to the nearest ten-thousandths

Question 6

    A random variable is uniformly distributed between 5 and 25

    a) Find P(x > 25)

    b) Find P(10 < x < 15) (Carry answer to the nearest Hundredths)

    c) Find P (5.0 < x < 5.1) (Carry answer to the nearest thousandths)

Question 7

    Find the probabilities

    a) P(z < 1.50) (Carry answer to the nearest ten-thousandths)

    b) P(z < -1.60) (Carry answer to the nearest ten-thousandths)

    c) P(-1.40 < z < .60) (Carry answer to the nearest ten-thousandths)

Question 8

          A new gas-electric hybrid car has recently hit the market. The distance traveled on 1 gallon of fuel is normally distributed with a meran of 65 miles and a standard deviation of 4 miles. Find the probability that

    a) The car travels more than 70 miles per gallon? (Carry answer to the nearest ten-thousandths)

    b) The car travels less than 60 miles per gallon? (Carry answer to the nearest ten-thousandths)

    c) The car travels between 55 and 70 miles per gallon? (Carry answer to the nearest ten-thousandths)

Question 9

    The amount of time it takes for a student to complete a statistics quiz is uniformly distributed between 30 and 60 minutes. One student is selected at random.

    a) What is the probability that the student requires more than 55 minutes to complete the quiz? Carry answer to the nearest ten-thousandths

    b) What is the probability that the student completes the quiz in a time between 30 and 40 minutes? Carry answer to the nearest ten-thousandths

    e) What is the probability that the student completes the quiz in exactly 37.23 minutes? Carry answer to the nearest ten-thousandths

Question 11

    The lifetime of an alkaline battery is exponentially distributed with a mean of 20

    a) What is the probability that the battery will last between 10 and 15 hours? Carry answer to the nearest ten-thousandths

    b) What is the probability that a battery will last for more than 30 hours? Carry answer to the nearest ten-thousandths

Question 12

    Assume that x is a binomial random variable with n = 100 and p =.45. Use a normal approximation to find

    a) P(x ≤ 45)

    b) P(40 ≤ x ≤ 50)

    c) P(x ≥ 38)

    Note: Carry standard deviation to the nearest hundredths

Question 13

    A recent study involving attrition rates at a major university has shown that 43% of all incoming freshmen do not graduate within 4 years of entrance. If 200 freshman are randomly sampled this year and their progress through college is followed.

    a) What is the approximate probability that no less than half will graduate within the next 4 years? Carry answer to the nearest ten-thousandths

    b) What is the approximate probability that the number of sampled freshmen graduating within 4 years will be between 80 and 105? Carry answer to the nearest ten-thousandths

Question 14

    The number of pizzas consumes per month by university students is normally distributed with a mean of 10 and a standard deviation of 3.

    a) What proportion of students consume more than 12 pizzas per month. Carry answer to the nearest ten-thousandths

    b) What is the probability that in a random sample of 25 students more than 275 pizzas are consumed? Carry answer to the nearest ten-thousandths

Question 15

    Doctors in the field of general practice earned an average of $100,240 per year. Assume the standard deviation of earnings was $10,750.

    What is the probability that a random sample of 100 general practitioners have a mean income of at least $100,000? Carry answer to the nearest ten-thousandths

Question 16

    The mean weight of oranges in an orange grove is 6.4 ounces, and the standard deviations 0.6 ounce. The owner of the grove has a roadside stand at which boxes of 36 oranges are sold. What proportion of boxed contain at least 223.2 ounces of oranges? Carry answer to the nearest ten-thousandths

Question 5

    You're dressing in the dark because of a power failure. You have four black socks and eight red socks on a drawer. You pull out three socks. (Hypergeometric Probability Distribution)

    a) What is the probability that two are black? Carry answer to the nearest ten-thousandths

Question 10

    x is normally distributed with a mean of 250 and standard deviation of 40. What value of x does the top 15% exceed? Carry answer to the nearest tenth?

Question 17

    Lean trimmed , 3-ounce tenderloin steaks contain an average of 174 calories. Suppose the standard deviation of such steaks is 10 calories. If a person eats one of these steaks each week for a year.

    a) What is the probability that the average number of calories consumed per steak will be less than 175? Carry answer to the nearest ten-thousandths

Answer 1

Solution:

1) Let X be a random variable with represents the snowfall over the course of winter in a Nebraska city.

Given that average is one snowfall every 3 days.

Hence, rate of snowfall = 1/3 snowfall every day.

To obtain the probability we shall consider that X follows poisson distribution with rate λ.

According to poisson process probability of occurrence of x events in time t is given by,

e-t (xt) P(X = 1) =

a) We have to obtain probability of exactly 5 snowfalls in 2 weeks. It means we have to obtain P(X=5)

We have, t = 2 weeks = 14 days,  λ = (1/3) snowfall per day

::P(X = 5) = (e =) (145 5!

::P(X = 5) = 0.17344

The probability of exactly 5 snowfalls in 2 weeks 0.17344.

b) Since, it is given that Snowfalls occur randomly and independently over the course of winter in a Nebraska city and the average is one snowfall every 3 days.

Hence, probability of a snowfall today is 1/3 = 0.33333.

2) X is a random variable which represents the number of books checked out per visit by children.

We are given probability distribution of X.

a) We have to find probability that a child will check out more than four books. It means we have to obtain P(X >4).

::P(X > 4) = P(X = 5) + P(X = 6) + P(X = 7)

::P(X > 4) = 0.20 + 0.10+ 0.10 = 0.40

Probability that a child will check out more than four books is 0.40.

b) We have to find probability that a child will check out at least two books. It means we have to find P(X ≥ 2).

::P(X > 2) = 1- P(X<2) = 1 - P(X = 1)

::P(X > 2) =1-0.05 = 0.95

probability that a child will check out at least two books is 0.95.