The number of people arriving at an emergency room follows a Poisson distribution with a rate of 7 people per hour.

a. What is the probability that exactly 5 patients will arrive during the next hour? (3pts)

b. What is the probability that at least 5 patients will arrive during the next hour? (5pts)

c. How many people do you expect to arrive in the next two hours? (4pts)

d. One in four patients who come to the emergency room in a hospital. Calculate the probability that during the next 2 hours exactly 20 people will arrive and less than 7 will be hospitalized (8pts)

Suppose that customers arrive at a bank at a rate of 10 per hour. Assume that the number of customer arrivals X follows a Poisson distribution.

A. Find the probability of more than 25 people arriving within the next two hours using the Poisson mass function.

B. Find the probability of more than 25 people arriving within the next two hours using the normal approximation to the Poisson.

C. Compute the percent relative difference between the exact probability computed in part 1 and the normal approximation computed in part 2, i.e., 100 × (Approx−Exact)/Exact %. Comment on the adequacy of the approximation.

Twenty percent of the customers of a grocery store use an express checkout. Consider six randomly selected customers.

a) What is the probability that exactly one of the six customers uses the express checkout? (round to three decimal places)

b) What is the probability that none of the six use express checkout? (round to three decimal places)  

c) What is the probability that at least two of the customers use express checkout? (round to three decimal places)

d) Letting X denote the number among the six who use express checkout, what is the mean value of X? (round to one decimal place)  

The quality assurance engineer of televisions inspects TV's from a large factory. She randomly selects a sample of 18 TV's from the factory to inspect. Assume that 33% of the TV's in the lot are silver. Let Y be the number of silver TV's selected.

A) find the probability that 2 of the TV's selected are silver.

B) find the probability that less than 4 of the TV's selected are silver.

C) find the probability that between 3.5 and 7.5 of the TV's selected are silver.

D) find the mean of Y.

E) find the standard deviation of Y.

All decimals to the ten-thousandths place please.

A batch of 400 sewing machines contains 6 that are defective.

(a) Three sewing machines are selected at random, without replacement. What is the probability that the second sewing machine is defective, but the first and third are acceptable?

(b) Three sewing machines are selected at random, without replacement. What is the probability that the third sewing machine is defective, given that the first two are acceptable?

(c) Sewing machines are selected, this time with replacement. What is the probability that the tenth machine selected is the first defective one?

(d) Sewing machines are selected, with replacement. What is the expected number of machines selected until you find two defective ones?

Consider below dataframe, df2, which provides the number of males and females in a population as well as their socio-economic breakdown. Provide python codes to calculate following probabilities:

• (2.a) For a randomly selected person from this population, probability of being female ("F")
• (2.b) For a randomly selected person from this population, probability of being working class ("working")
• (2.c) For a randomly selected person from this population, joint probability of being male ("M") and middle class ("middle")

df2 = pd.DataFrame({"M": [30,250,20], "F":[10,150,40]}, index=["upperMiddle", "middle", "working"])

df2

b. Read the three problems listed below and determine: (1) what type of probability distribution would be used to solve each problem and why; (2) pick one problem from below and provide a detailed solution with an explanation; (3) indicate which problem you selected to solve in your subject line.

i. The reading speed of sixth-grade students is approximately normal, with a mean speed of 125 words per minute and a standard deviation of 24 words per minute. Find and interpret the probability that a randomly selected sixth-grade student reads less than 100 words per minute.

ii. A Wendy’s manager performed a study to determine a probability distribution for the number of people, X, waiting in a line during lunch. The results were as follows. Find and interpret the probability that 10 or more people are waiting in line for lunch?

iii. Clarinex-D is a medication whose purpose is to reduce the symptoms associated with a variety of allergies. In clinical trials of Clarinex-D, 5% of the patients in the study experienced insomnia as a side effect. As random sample of 20 Clarinex-D users is obtained, and the number of patients who experienced insomnia is recorded. Find and interpret probability that 3 or fewer experienced insomnia as a side effect.

. A bank has recently begun a new credit program. Customers meeting a certain credit requirement can obtain a credit card accepted by participating area merchants that carries a discount. Past data show that about 35% of all applicants for this card are rejected. If we let X represent the number of applicants who are approved for this credit card, the probabilities for the values of X can be calculated using the binomial probability distribution. Please answer the following questions based on the information given in this problem.

a. If 15 individuals apply for the credit card today what is the probability that none of them will be rejected? Show your work.                                           (1 point)                                                                                                                     

b. If 15 individuals apply for the credit card today what is the probability of at most 5 of them being rejected? Show your work.                                      (2 points)     

c. If 15 individuals apply for the credit card today what is the probability of more than 12 of them being approved? Show your work.                         (2 points)           

d. If 15 individuals apply for the credit card today what is the probability of less than half of them being approved? Show your work.                      (2 points)                                                                                                                                                                                                                                                  

e. If 15 individuals apply for the credit card today what is the probability of at least 6 of them being rejected? Show your work.                                          (2 points)           

f. How many rejections are expected out of the 15 applications?       (1 point)     

g. Calculate and interpret the standard deviation for the number of approvals out of the next 15 applications for the credit card. Show your work

If a seed is planted, it has a 45% chance of growing into a healthy plant. If 144 randomly selected seeds are planted, answer the following.

a) Which is the correct wording for the random variable?

  Select an answer rv X = the probability that a randomly selected seed grows into a healthy plant rv X = grows into a healthy plant rv X = a randomly selected seed that grows into a healthy plant rv X = the number of 144 randomly selected seeds that grow into a healthy plant rv X = all seeds that grow into a healthy plant rv X= the number of randomly selected seeds that grow into a healthy plant rv X = a randomly selected seed

b) Pick the correct symbol: ? X̄ σ p s n N X μ  = 144

c) Pick the correct symbol: ? X X̄ p σ μ s n N  = 0.45

d) What is the probability that exactly 61 of them grow into a healthy plant?

Round final answer to 4 decimal places. 0.548

e) What is the probability that less than 61 of them grow into a healthy plant?

Round final answer to 4 decimal places.

f) What is the probability that more than 61 of them grow into a healthy plant?

Round final answer to 4 decimal places.

g) What is the probability that exactly 64 of them grow into a healthy plant?

Round final answer to 4 decimal places.

h) What is the probability that at least 64 of them grow into a healthy plant?

Round final answer to 4 decimal places.

i) What is the probability that at most 64 of them grow into a healthy plant

Round final answer to 4 decimal places.