Assuming that in some part of Quebec the probability is 0.00007 that a child will develop cancer. Therefore, the number Z among 28, 572 children that will develop cancer follows a Binomial distribution with parameters p = 0.00007 and n = 28, 572. We would like to use the Poisson distribution to approximate these binomial probabilities.

(a) What is the adequate value of the variance of the Poisson distribution to use in order to approximate the precedent binomial distribution?

(b) Find the probabilities of the 11 first possible values of Z (i.e. Z = 0, 1, . . . , 10) using both the formulas for the binomial distribution and then the Poisson approximation. Plot the two histograms and make a comparison. Is this approximation close enough? Justify!

(c) Use the Poisson probabilities to approximate the binomial probabilities that among 28, 572 children

i) None will develop cancer ii) At most two will develop cancer.

(d) Using Poisson approximation, calculate the probability that at least seven in a sample of ten children will not develop cancer. Is it a good approximation? Justify!

The amount of fill (weight of contents) put into a glass jar of spaghetti sauce is normally distributed with mean μ = 840 grams and standard deviation of σ = 9 grams.

You manage a training program that has a failure rate of 34%. A new class of 6 students just checked in at the end of the fiscal year and you need to graduate at least 3 students. Your boss wants to know how likely it is that you will meet your yearly quota.

Calculate the expected number of students that will pass.Show your work. You can do so via a table or calculation.

Calculate the standard deviation. Show your work. You can do so via a table or calculation.

What is the probability of 4 students passing the training program? Write the calculator function and numbers that you used in your work.

What is the probability that at least half the class will pass? Write the calculator function and numbers that you used to show your work. T/F

Please show your work

A population of values has a normal distribution with μ=134.6μ=134.6 and σ=75.5σ=75.5. You intend to draw a random sample of size n=225n=225.
Find the probability that a sample of size n=225n=225 is randomly selected with a mean between 122 and 129.6.
P(122 < M < 129.6) =
Enter your answers as numbers accurate to 4 decimal places. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

A company produces steel rods. The lengths of the steel rods are normally distributed with a mean of 242.6-cm and a standard deviation of 0.8-cm. For shipment, 6 steel rods are bundled together.
Find the probability that the average length of a randomly selected bundle of steel rods is greater than 243.1-cm.
P(M > 243.1-cm) =
Enter your answer as a number accurate to 4 decimal places. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

A survey asked 1000 randomly selected recently retired people how many people, including themselves, are in a family tree that begins with their parents. Let X represent the number of people. The table below shows the probability distribution of the data. Find the mean and the standard deviation of the probability distribution using Excel.

x   P(x)
1   0.001
2   0.001
3   0.005
4   0.016
5   0.012
6   0.018
7   0.039
8   0.046
9   0.056
10   0.059
11   0.057
12   0.068
13   0.044
14   0.056
15   0.061
16   0.047
17   0.049
18   0.049
19   0.034
20   0.039
21   0.021
22   0.028
23   0.034
24   0.014
25   0.035
26   0.014
27   0.014
28   0.02
29   0.014
30   0.007
31   0.009
32   0.01
33   0.002
34   0.007
35   0.002
36   0.008
37   0.002
38   0.002

A manufacturer knows that their items have a normally distributed lifespan, with a mean of 10.3 years, and standard deviation of 2.4 years.
If 21 items are picked at random, 5% of the time their mean life will be less than how many years?
Give your answer to one decimal place.

A particular fruit's weights are normally distributed, with a mean of 645 grams and a standard deviation of 18 grams.If you pick one fruit at random, what is the probability that it will weigh between 631 grams and 666 grams. (Give answer to 4 decimal places.)

About 9% of the population has a particular genetic mutation. 1000 people are randomly selected.
Find the mean for the number of people with the genetic mutation in such groups of 1000. (Remember that means should be rounded to one more decimal place than the raw data.)

If a seed is planted, it has a 60% chance of growing into a healthy plant.
If 9 seeds are planted, what is the probability that exactly 2 don't grow?
(Give your answer as a decimal rounded to 3 places.)

A study conducted by three law school professors found that asylum seekers in the United States face broad disparities in the nation’s immigration courts. The professors discovered that 54% of refugees who ask for asylum in the San Francisco immigration court win asylum, but only 12% are granted asylum in the Atlanta immigration court. [Source: Julia Preston, “Wide Disparities Found in Judging of Asylum Cases,” The New York Times, May 31, 2007.]

You randomly select 30 refugees who are asking for asylum in the Atlanta immigration court. Let X denote the number of asylum seekers who win their cases.

The probability that exactly two asylum seekers are granted asylum is 1) .0735 / .1747 / .0844 / .1593  .

The probability that at least five asylum seekers are granted asylum is 2) .2882 / .0827 / .1734 / .1431 .

The expected value of X is 3) 16.2 / 3.6 / 7.5 / 4.8   , and the standard deviation of X is 4) 2.3717 / 2.7298 / 1.7799 / 2.0080 / 3.168

Debt and financial risk   Tower Interiors has made the forecast of sales shown in the following table. Also given is the probability of each level of sales.

The firm has fixed operating costs of $75,100 and variable operating costs equal to 60% of the sales level. The company pays $12,600 in interest per period. The tax rate is 40%.

a. Compute the earnings before interest and taxes​ (EBIT) for each level of sales.

b. Compute the earnings per share​ (EPS) for each level of​ sales, the expected​ EPS, the standard deviation of the​ EPS, and the coefficient of variation of​ EPS, assuming that there are 11,600 shares of common stock outstanding.

c. Tower has the opportunity to reduce its leverage to zero and pay no interest. This will require that the number of shares outstanding be increased to 17,400. Repeat part ​(b​) under this assumption.

d. Compare your findings in parts ​(b​) and ​(c​)​, and comment on the effect of the reduction of debt to zero on the​ firm's financial risk.

When events occur in a time period (0, a) with regard to a Poisson process, it is well known that, conditioned on the total number of events k, the joint distribution of the times at with the events occur follows a uniform distribution in (0, a) k. That is, if Xi represents the arrival time of one of them, then Xi ∼ U(0, a) and it is independent of the other X’s.

1. If ten patients visit the Emergency Room (ER) of a hospital between 9 and 10, what is the probability that fifth of them reaches the ER before 9:30?

2. If ten patients visit the ER between 9 and 10, what is the probability that the first 3 of them arrive before 9:20, 4 of them between 9:20 and 9:40, and the last 3 of them after 9:40?

3. If two patients visit the ER of a hospital between 9 and 10, what is the conditional distribution of the time at which the second patient arrived at the ER given that the first patient arrived at time t1?