The table below shows the number of male and female students enrolled in nursing at a university for a certain semester. A student is selected at random. Complete parts? (a) through? (d).

?(a) Find the probability that the student is male or a nursing major.P(being male or being nursing major)=

?(Round to the nearest thousandth as? needed.)

?(b) Find the probability that the student is female or not a nursing major.P(being female or not being a nursing major)=

?(Round to the nearest thousandth as? needed.)

?(c) Find the probability that the student is not female or a nursing major. P(not being female or being a nursing major)=

?(Round to the nearest thousandth as? needed.)

?(d) Are the events? "being male" and? "being a nursing? major" mutually? exclusive? Explain.

A. Yes, because there are 98males majoring in nursing.

B. No, because there are 98 males majoring in nursing.

C. No, because one? can't be male and a nursing major at the same time.

D. Yes, because one? can't be male and a nursing major at the same time.

USING EXCEL

Now your task is to plot the cumulative probability density function with the information given in number 2. First, open Worksheet #3. Then enter “=” in cell A17 of your new worksheet and then click on cell A17 of your second worksheet and hit return. Then copy cell A17 down the A column as before.

              Next, you’re to compute the cumulative probabilities for x = 0, 1, 2, …. 20, beginning in cell B17 and ending in cell B37, using BINOM.DIST, of course.

              Once you’ve done that, prepare a histogram-like graph of the cumulative probability distribution, as you did in question 2. Label the vertical axis “P(X ≤ x); the horizontal axis, “x = 0 1 2 3 4 5 … 20”; and the graph, “Exercise 4, #3 Discrete CDF (p = ,35)”. (You’ll want to add in some “major” horizontal gridlines as well.)

              Finally, below your graph, please answer the following question: why do cumulative probability functions of discrete variables take on this stepwise form?

A biologist captures 22 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 ?.

Suppose there are 116 grizzly bears in the population.

(a) What is the probability that of the 15 grizzly bears observed, 5 had radio collars? Use four decimals in your answer. ?(?=5)=

(b) Find the probability that between 5 and 8 (inclusive) of the 15 grizzly bears observed were radio collared?

?(5≤?≤8)

(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.

?(?)=

??(?)=

(d) The biologist gets back from the helicopter observation expedition, and was asked the question: How many radio collared grizzly bears did you see? The biologist cannot remember exactly, so responds " somewhere between 5 and 9 (inclusive) ".

Given this information, what is the probability that the biologist saw 6 radio-collared grizzly bears?

A beginning unicyclist is attempting to unicycle from his garage to the end of his driveway—a distance of 200 feet. His skill is such that, once mounted on the unicycle, he will fall within a distance of x feet from his starting point with probability given by x/250 − x2/5002, for 0 < x < 500. Let X be the distance from the unicyclist’s starting point at which he falls.
(a) GivethecdfFX(x)ofX forallx∈R.
(b) GivethepdffX(x)ofX forallx∈R.
(c) With what probability does the unicyclist fall before reaching the end of his driveway?
(d) With what probability does the unicyclist reach the end of his driveway before falling?
(e) What is the maximum distance the unicyclist is capable of reaching?
(f) Find the median of X.
(g) Find EX.
(h) Find the standard deviation of X.
(i) Let Y = (200 − X)/3 represent the remaining distance, in yards, to the end of the driveway from the garage (if he passes the end of the driveway, Y will be a negative number).
i. Find EY .
ii. Find Var Y .

If you know that CAPMAS (Central Agency for Public Mobilization and Statistics), which is the National Statistical Office of Egypt, regularly announce the Households Income Expenditure and Consumption (HIECS) survey. HIECS I the survey that is used to estimate the households’ income and their expenditure, as well as other parameters. The following table provides the distribution of the randomly selected 25 thousand Households across different annual expenditure categories in 2018.

(A) What is the probability that households in Egypt have an annual expenditure of less than L.E. 32,000? (Round answer to 2 decimal places) ...

(B) What is the probability that households in Egypt have an annual expenditure between L.E. 17,000 and L.E. 67,000? (Round answer to 2 decimal places) ...

(C) What is the probability that households in Egypt have an annual expenditure greater than L.E. 32,000? (Round answer to 2 decimal places) ...

Suppose the length of textbooks in a library follows a bimodal distribution with a little right skewness (very mild). The mean of this distribution is 512 pages with a standard deviation of 390 pages.

For each of the following i) draw a picture. ii) label the picture with 2 axes (underneath). iii) label the shorthand for the new distribution. iv) Find the z-score. v) Find the answer.  

a1) What is the probability that a random sample of 36 textbooks has an average of 445.2 pages or less?

a2) What is the probability that a random sample of 49 textbooks has an average higher than 613.3 pages?

a3) What is the probability that a random sample of 30 textbooks has an average number of pages between 400 and 500?

a4) Do you think you could answer a1 - a4 with a sample of just 2 textbooks? Why or why not?

a5) Do you think you could answer a1-a4 with a sample of 25 textbooks? Why or why not?

a6) Describe the central limit theorem in a paragraph.  

a7) What is the formula for standard error? And what is it in relation to the central limit theorem.  

Problem 3. A quality-control engineer wants to check whether (in accordance with specifications) 95% of the concrete beams shipped by his company pass the strength test (i.e., the strength is greater or equal to 32). To this end, he randomly selects a sample of 20 beams from each large lot ready to be shipped and passes the lot if at most one of the 20 selected beams fails the test; otherwise, each of the beams in the lot is checked. Let the random variable X be the number of selected beams that pass the test.
1. Find the probability that all 20 selected beams pass the test.
2. Find the probability that 2 beams in the sample fail the test.
3. Find the probability that between 17 to 19 beams in the sample pass the test.
4. Find the probabilities that the quality-control engineer will commit the error of holding a lot for further inspection even though 95% of the beams strength is greater or equal to 32 (in accordance with specifications).

Hint: The quality-control engineer hold a lot if 2 or more beams in the sample fail the test.

QUESTION 3

A recent study shows that on average 20% of employees in the work population prefer their vacation time during March break. Vincent Company employs 56 people. Use the normal approximation to the binomial distribution to answer the questions below:

Required:

1. Determine the expected value, and the standard deviation of that value, of the number of Vincent Company employees who would prefer their vacation time during March break. ( 3 marks)

2. What is the probability that more than ten Vincent Company employees will prefer their vacation during March break? Sketch a normal curve and shade the desired area of your diagram. ( 4 marks)

3. What is the probability that exactly ten Vincent Company employees will prefer their vacation during March break? Sketch a normal curve and shade the desired area of your diagram.   ( 4 marks)
4. What is the probability that more than five, but less than nine, Vincent Company employees will prefer their vacation during March break? Sketch a normal curve and shade the desired area of your diagram.   ( 4 marks)

A roulette wheel has 38 slots, numbered 0, 00, and 1 to 36. The slots 0 and 00 are colored green, 18 of the others are red, and 18 are black.The dealer spins the wheel and at the same time rolls a small ball along the wheel in the opposite direction. The wheel is carefully balanced so that the ball is equally likely to land in any slot when the wheel slows. Gamblers can bet on various combinations of numbers and colors. (a)If you bet on “red,” you win if the ball lands in a red slot. What is the probability of winning with a bet on red in a single play of roulette? (b)You decide to play roulette four times, each time betting on red. What is the distribution of X, the number of times you win? (c)If you bet the same amount on each play and win on exactly four of the eight plays, then you will “break even.” What is the probability that you will break even? (d)If you win on fewer than four of the eight plays, then you will lose money. What is the probability that you will lose money?

You have developed a self-service kiosk capable of serving about 15 clients per hour. You have been told that the average rate of customers using this kiosk is about 10 customers per hour. You also know that the number of customers who approach the kiosk per hour follows the Poisson distribution.

1. Write out the pmf of the Poisson RV in this case and solve for 20 customers approaching the kiosk.

2. Use an R function to find a probability for the above.

3. Have R generate random numbers following the above distribution for 100,000 intervals. What is the maximum number of customers approaching the kiosk in your simulation?