5.29 Catalog Age lists the top 17 U.S. firms in annual catalog sales. Dell Computer is number one followed by IBM and W.W. Grainger. Of the 17 firms on the list, 8 are in some type of computer-related business. Suppose four firms are randomly selected. a. What is the probability that none of the firms is in some type of computer-related business? b. What is the probability that all four firms are in some type of computer-related business? c. What is the probability that exactly two are in non-computer-related business?

Y is a Binomial random variable where, Y = The number of days in a week someone goes to the gym.

Where a week has 7 days and the probability of someone going to the gym on any given day is .65.  What is the probability that someone goes to the gym at least 3 days out of the week?

Hint: This is a cumulative probability, so you need to add up the probabilities of Y equaling all the possible values up to and including 3.

P(Y <= 3) = ?

(A) 0.2627

(B) 0.4694

(C) 0.3672

(D) 0.4718

Customers arrive at a grocery store at an average of 2.1 per minute. Assume that the number of arrivals in a minute follows the Poisson distribution. Provide answers to the following to 3 decimal places.

Part a)

What is the probability that exactly two customers arrive in a minute?



Part b)

Find the probability that more than three customers arrive in a two-minute period.



Part c)

What is the probability that at least seven customers arrive in three minutes, given that exactly two arrive in the first minute?

question c is not 0.442

In any given year, one in three Americans over the age of 65 will experience a fall. If you have three living grandparents over the age of 65, and assuming that the probability of a fall for each grandparent is independent:

a. What is the probability that none of the three grandparents will experience a fall? Provide your answer as a decimal between 0 and 1. Hint: Your sample size is 3, what is the number of successes.

b. What is the probability that one or more grandparents will experience a fall? Provide your answer as a decimal between 0 and 1.

5.69 Cyberbullying. An online survey, in partnership with Habbo, was conducted to study cyberbullying among 13- to 25-year-olds in the United Kingdom. It was reported that 62% of the young people had received nasty private messages on a smartphone social network app. You randomly samplefour young people from the United Kingdom and ask them if they’ve received nasty messages. Let X be the number who say Yes.

1. a) What are n and p in the binomial distribution of X?

2. b) Find the probability of each possible value of X, and draw a probability histogram for this

   distribution.

3. c) Find the mean number of positive responders and mark the location of this value on your histogram.

   My editions:

4. d) Based on your answer to part b, what is the probability that you select at least 3 young people who say Yes? Is it appropriate to use the normal distribution to find this probability?

5. e) Suppose that you are instead sampling from 200 students. Now, X is Binomially distributed with n=200 and p=0.62. You want to find the probability that at least 140 young people say Yes. Is it appropriate to use the Normal Approximation? Why or why not?

6. f) Find the probability that at least 140 young people say Yes in a sample of 200.

7. g) Rework your answer to part f, using the continuity correction.

Osteosarcoma is a relatively rare type of bone cancer. It occurs most often in young adults, age 10 - 19; it is diagnosed in approximately 8 per 1,000,000 individuals per year in that age group. In New York City (including all five boroughs), the number of young adults in this age range is approximately 1,400,000.

a) What is the expected number of cases of osteosarcoma in NYC in a given year?
b) What is the probability that 15 or more cases will be diagnosed in a given year?
c) The largest concentration of young adults in NYC is in the borough of Brooklyn, where the population in that age range is approximately 450,000. What is the probability of 10 or more cases in Brooklyn in a given year?
d) Suppose that in a given year, 10 cases of osteosarcoma were observed in NYC, with all 10 cases occurring among young adults living in Brooklyn. An official from the NYC Public Health Department claims that the probability of this event (that is, the probability of 10 or more cases being observed, and all of them occurring in Brooklyn) is what was calculated in part c). Is the official correct? Explain your answer. You may assume that your answer to part c) is correct. This question can be answered without doing any calculations.
e) Suppose that over five years, there was one year in which 10 or more cases of osteosarcoma were observed in Brooklyn. Is the probability of this event equal to the probability calculated in part c)? Explain your answer.

A prototype automotive tire has a design life of 38,500 miles with a standard deviation of 2,500 miles. The manufacturer tests 60 such tires. On the assumption that the actual population mean is 38,500 miles and the actual population standard deviation is 2,500 miles, find the probability that the sample mean will be less than 36,000 miles. Assume that the distribution of lifetimes of such tires is normal.

(a) Let X = number of miles on a single tire. Write the question above in terms of this variable X.

(b) Using the software tool above, find the probability stated on part (a)

(c) Using the software tool above, graph the probability of stated on part (b)

2. An automobile battery manufacturer claims that its midgrade battery has a mean life of 50 months with a standard deviation of 6 months. Suppose the distribution of battery lives of this particular brand is approximately normal. On the assumption that the claims are true, find the probability that a randomly selected battery of this type will last less than 48 months. (Use the software link for every question)

(a) Let X = number of months a battery will last. Write the question above in terms of this variable X

(b) Find the probability that a single battery of this type will last less than 48 months.

(c) Find the probability that the mean of a random sample of 36 batteries will be less than 48 months.

(d) Why do you think the values from part (b) and part (c) are different? Explain.

. Consider the 68 words in the following two sentences to be modeled as random variables. The sentences contain words of 1 letter length to 10 letter length. Thus random variable x lies in the range 1 ≤ x ≤ 11

        “A single link flexible arm is a dynamic system with the first eigenvalue equal to zero and giving the primary rigid body motion and the eigenvalues greater than zero giving flexural vibration that may occur during the response. The object is to drive the arm tip to a constant steady state position in as fast a time as possible while keeping the arm tip vibration to a minimum.”

(A) Develop a bar graph for the number of words with a specific number of letters.

For example, in the phrase “This is an example for the type of words related to this problem”:   3 two letter words, 2 three letter words, 3 four letter words, 1 five letter word, 3 seven letter words.

(B) Calculate the probability density distribution and show it bar form. Use 1 for the transition from probability to probability density which makes these two the same.

(C)    Determine the mean μ and standard deviation σ.

(D)   Use part B result and determine the probability that a word falls between

            μ-σ and μ+σ.

   (E) If the system is modeled with a continuous normal probability distribution, determine the probability that a word falls between 6 and 9 letters.

Determine whether the following individual events are independent or dependent. Then find the probability of the combined event.

Randomly selecting a four- person committee consisting entirely of Canadians from a pool of 18 Americans and 8 Canadians.

1.)The event of selecting Canadian and the event of selecting a Canadian are a independent or dependent

- The probability of randomly selecting a ​four-person committee consisting entirely of Canadians from a pool of 18 Americans and 8 Canadians is what? ​(Round to four decimal places as​ needed.)

2.) Use the definitions given in the text to find both the odds for and the odds against the following event.

-Flipping 4 fair coins and getting 0 heads.

The odds for getting 0 heads are what to what.​(Type a whole​ number.)

The odds against getting 0 heads are what to what. (Type a whole number)

3,) Determine whether the following individual events are independent or dependent. Then find the probability of the combined event.

Rolling two 4s followed by one 5 on three tosses of a fair die. Choose the correct answer below.​(Type an integer or a simplified​ fraction.)

- The individual events are independent. The probability of the combined event is what?

- The individual events are dependent. The probability of the combined event is what?

Scores for a common standardized college aptitude test are normally distributed with a mean of 482 and a standard deviation of 106. Randomly selected men are given a Prepartion Course before taking this test. Assume, for sake of argument, that the Preparation Course has no effect on people's test scores.

If 1 of the men is randomly selected, find the probability that his score is at least 544.3.
P(X > 544.3) =
Enter your answer as a number accurate to 4 decimal places.

If 14 of the men are randomly selected, find the probability that their mean score is at least 544.3.
P(x-bar > 544.3) =
Enter your answer as a number accurate to 4 decimal places.

If the random sample of 14 men does result in a mean score of 544.3, is there strong evidence to support a claim that the Preapartion Course is actually effective? (Use the criteria that "unusual" events have a probability of less than 5%.)

• Yes. The probability indicates that is is highly unlikely that by chance, a randomly selected group of students would get a mean as high as 544.3 if the Preparation Course has no effect.
• No. The probability indicates that is is possible by chance alone to randomly select a group of students with a mean as high as 544.3 if the Preparation Course has no effect.