Given contingency table for simultaneous occurance of two categorical random variables X (levels A,B,C) and Y (levels L1,L2,L3)

Determine the marginal probability P ( X = B ) =  (round to the sthird decimal place)

Determine the conditional probability P ( Y = L 2 | X = B ) =  (round to the second decimal place)

Use R to conduct a chi-square test of independence and thus determine

• Number of degrees of freedom DF=
• Ch-square, round to two decimal places=
• p-value, round to two decimal places=
• Make your conclusion at 5% significance. Enter the correct answer using the following options:  (type the corresponding capital letter, do not type the "dot" at the end)
  1. Not independent at 5% significance level.
  2. Independent with 95% confidence.
  3. Do not reject that they are independent at 5% significance level and reserve judgement. We may accept independence, but with some unknown probability.
  4. Reject both, H0 and H1, the test has failed.
  5. Accept both, H0 and H1, the test has failed.

1. Federal law under Title 49 of the United States Code, Chapter 301, Motor Vehicle Safety Standard took effect on January 1, 1968 and required all vehicles (except buses) to be fitted with seat belts in all designated seating positions. While most states have laws requiring seat belt use today, some people still do not “buckle up.” Let’s assume that 90 % of drivers do “buckle up.” If drivers are randomly stopped to check seat belt usage, answer the following questions and show your work. (Use R where necessary)

1. How many drivers do they expect to stop before finding a driver whose seatbelt is not buckled?

2. What is the probability that the second unbelted driver is in the ninth car stopped?

3. What is the probability that of the first 10 drivers, 8 or more are wearing their seatbelts?

4. If they stop 30 cars during the first hour, find the mean and standard deviation of the number of drivers not expected to be wearing seatbelts?

5. If they stop 120 cars during this safety check, what is the probability they find at least 12 drivers not wearing seatbelts?

Let X be the number of goal chances that result in goals for a football team over n goals chances. During many matches, the team gets goals of 20% of the chances.

a) Explain why it might be reasonable (at least as an approach) to assume that X is binomially distributed in this situation. Do this by going through each of the points that must be met in order for us to use binomial distribution, assess whether it is reasonable that they are met here, and specify any additional assumptions we need to make. solved

Not solved:

What is the probability that the team gets two goals in a match with n = 12 goal chances?
What is the probability that they will get more than 50 targets in a season if they get n = 300 target chances?

Before a new season acquires two new spikes in hopes of increasing the likelihood of scoring on the target chances. During the first 116 goals they received 29 goals.
b) Determine with a hypothesis test whether the probability of scoring on the target chances has increased. Use 5% level. Also calculate the p-value of the test.
What would have been the conclusion of the test if we had used a 10% level?

Question 12

1. A student takes an 8-question, true-false exam and guesses on each question. Find the probability of passing if the lowest passing grade is 6 correct out of 8.

   	a.	0.109
   	b.	0.227
   	c.	0.144
   	d.	0.164

Question 13

1. R.H. Bruskin Associates Market Research found that 30% of Americans do not think having a college education is important to succeed in the business world. If a random sample of 5 Americans is selected, find this probability: At most three people will agree with that statement.

   	a.	0.811
   	b.	0.837
   	c.	0.499
   	d.	0.969

Question 14

1. R.H. Bruskin Associates Market Research found that 30% of Americans do not think having a college education is important to succeed in the business world. If a random sample of 5 Americans is selected, find this probability: At least two people will agree with that statement.

   	a.	0.471
   	b.	0.936
   	c.	0.829
   	d.	0.811

Question 15

1. In a restaurant, a study found that 25% of all patrons smoked. If the seating capacity of the restaurant is 80 people, find the variance of the number of smokers.

   	a.	3.87
   	b.	19.8
   	c.	15
   	d.	4.45

The vendor at Citi Field offers a health pack consisting of apples and oranges. The weight, X, of an apple has a normal distribution with a mean of 9 ounces and a standard deviation of 0.6 ounces. Independent of this, the weight, Y, of an orange has a normal distribution with a mean of 7 ounces and a standard deviation of 0.4 ounces. Suppose the health pack has a random selection of 4 apples with weights

X₁, X₂, X₃, X₄

and 3 oranges with weights

Y₁, Y₂, Y₃

. . Let Xsum be the sum of the apple weights in ounces and let Ysum be the sum of the orange weights in ounces. W = Xsum + Ysum is the random variable representing the total weight of the health pack.


a) What is the probability that Xsum > 38?  

b) What is the probability that Ysum > 22?  

c) What is the expected value of Xsum?  

d) What is the standard deviation of Xsum?  

e) What is the variance of the random variable W?  

f) What is the expected value of W?  

g) What is the standard deviation of W?  

h) What is the probability that W > 59 ounces?  

i) i. If 100 health packs are sold what is the expected number sold which weigh more than 59 ounces?

The manager of a computer retails store is concerned that his suppliers have been giving him laptop computers with lower than average quality. His research shows that replacement times for the model laptop of concern are normally distributed with a mean of 4 years and a standard deviation of 0.4 years. He then randomly selects records on 49 laptops sold in the past and finds that the mean replacement time is 3.8 years.

Assuming that the laptop replacement times have a mean of 4 years and a standard deviation of 0.4 years, find the probability that 49 randomly selected laptops will have a mean replacement time of 3.8 years or less.
P(M < 3.8 years) =  
Enter your answer as a number accurate to 4 decimal places. NOTE: Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

Based on the result above, does it appear that the computer store has been given laptops of lower than average quality?

• Yes. The probability of this data is unlikely to have occurred by chance alone.
• No. The probability of obtaining this data is high enough to have been a chance occurrence.

The College Board National Office recently reported that in 2011–2012, the 547,038 high school juniors who took the ACT achieved a mean score of 515 with a standard deviation of 129 on the mathematics portion of the test (http://media.collegeboard.com/digitalServices/pdf/research/2013/TotalGroup-2013.pdf). Assume these test scores are normally distributed.

1. What is the probability that a high school junior who takes the test will score at least 590 on the mathematics portion of the test? If required, round your answer to four decimal places.
   
   P (x ≥ 590) =
   
2. What is the probability that a high school junior who takes the test will score no higher than 510 on the mathematics portion of the test? If required, round your answer to four decimal places.
   
   P (x ≤ 510) =
   
3. What is the probability that a high school junior who takes the test will score between 510 and 590 on the mathematics portion of the test? If required, round your answer to four decimal places.
   
   P (510 ≤ x ≤ 590) =
   
4. How high does a student have to score to be in the top 10% of high school juniors on the mathematics portion of the test? If required, round your answer to the nearest whole number.
   

The local bakery bakes more than a thousand 1-pound loaves of bread daily, and the weights of these loaves varies. The mean weight is 1.7 lb. and 3 oz., or 856 grams. Assume the standard deviation of the weights is 25 grams and a sample of 32 loaves is to be randomly selected.

(A) This sample of 32 has a mean value of x, which belongs to a sampling distribution. Find the shape of this sampling distribution.

a) skewed right

b) approximately normal   

c) skewed left

d) chi-square

(b) Find the mean of this sampling distribution. (Give your answer correct to nearest whole number.)
_____ grams

(c) Find the standard error of this sampling distribution. (Give your answer correct to two decimal places.) ______


(d) What is the probability that this sample mean will be between 846 and 866? (Give your answer correct to four decimal places.) _______


(e) What is the probability that the sample mean will have a value less than 847? (Give your answer correct to four decimal places.) _______


(f) What is the probability that the sample mean will be within 4 grams of the mean? (Give your answer correct to four decimal places.) _______

Allen's hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther.† Suppose a small group of 10 Allen's hummingbirds has been under study in Arizona. The average weight for these birds is x = 3.15 grams. Based on previous studies, we can assume that the weights of Allen's hummingbirds have a normal distribution, with σ = 0.20 gram. (a) Find an 80% confidence interval for the average weights of Allen's hummingbirds in the study region. What is the margin of error? (Round your answers to two decimal places.) lower limit upper limit margin of error (b) What conditions are necessary for your calculations? (Select all that apply.) σ is known uniform distribution of weights σ is unknown n is large normal distribution of weights (c) Interpret your results in the context of this problem. The probability to the true average weight of Allen's hummingbirds is equal to the sample mean. There is a 20% chance that the interval is one of the intervals containing the true average weight of Allen's hummingbirds in this region. The probability that this interval contains the true average weight of Allen's hummingbirds is 0.80. The probability that this interval contains the true average weight of Allen's hummingbirds is 0.20. There is an 80% chance that the interval is one of the intervals containing the true average weight of Allen's hummingbirds in this region. (d) Find the sample size necessary for an 80% confidence level with a maximal margin of error E = 0.10 for the mean weights of the hummingbirds. (Round up to the nearest whole number.) hummingbirds

n this region.

(d) Find the sample size necessary for an 80% confidence level with a maximal margin of error E = 0.10 for the mean weights of the hummingbirds. (Round up to the nearest whole number.)
hummingbirds

South Central Airlines (SCA) operates a commuter flight between Atlanta and Charlotte. The regional jet holds 50 passengers, and currently SCA only books up to 50 reservations. Past data show that SCA always sells all 50 reservations, but on average, two passengers do not show up for the flight. As a result, with 50 reservations the flight is often being flown with empty seats. To capture additional profit, SCA is considering an overbooking strategy in which they would accept 52 reservations even though the airplane holds only 50 passengers. SCA believes that it will be able to always book all 52 reservations. The probability distribution for the number of passengers showing up when 52 reservations are accepted is estimated as follows:

SCA receives a marginal profit of $100 for each passenger who books a reservation (regardless whether they show up or not). The airline will also incur a cost for any passenger denied seating on the flight. This cost covers added expenses of rescheduling the passenger as well as loss of goodwill, estimated to be $150 per passenger. Develop a spreadsheet simulation model for this overbooking system and simulate the number of passengers that show up for a flight.

1. What is the average net profit for each flight with the overbooking strategy?

2. What is the probability that the net profit with the overbooking strategy will be less than the net profit without overbooking ?

3. Explain how your simulation model could be used to evaluate other overbooking levels such as 51, 53, and 54 and for recommending a best overbooking strategy.

PLEASE SHOW STEP BY STEP EXCEL SIMULATION