Allen's hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther.† Suppose a small group of 12 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.)

(b) What conditions are necessary for your calculations? (Select all that apply.)

σ is knownσ is unknownn is largenormal distribution of weightsuniform distribution of weights

(c) Interpret your results in the context of this problem.

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.The probability that this interval contains the true average weight of Allen's hummingbirds is 0.20.     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 to the true average weight of Allen's hummingbirds is equal to the sample mean.The probability that this interval contains the true average weight of Allen's hummingbirds is 0.80.

(d) Find the sample size necessary for an 80% confidence level with a maximal margin of error  E = 0.14 for the mean weights of the hummingbirds. (Round up to the nearest whole number.)
hummingbirds

Allen's hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther.† Suppose a small group of 20 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.) normal distribution of weights σ is unknown uniform distribution of weights n is large σ is known (c) Interpret your results in the context of this problem. 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 0.20. 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. The probability to the true average weight of Allen's hummingbirds is equal to the sample mean. The probability that this interval contains the true average weight of Allen's hummingbirds is 0.80. (d) Find the sample size necessary for an 80% confidence level with a maximal margin of error E = 0.06 for the mean weights of the hummingbirds. (Round up to the nearest whole number.) hummingbirds

1. Consider the following events for a college student being selected at random. A = student is a hockey player B = student is majoring in kinesiology. Translate the following phrase into symbols: The probability that the student is not a hockey player and is majoring in kinesiology.

Select one:

a. P(AcandB)

b. P(AandBc)

c. P(Bc|A)

d. P(A|Bc)

2. Suppose you roll two fair dice, one that is purple, and one other is violet. Each die can have numbers from 1 to 6. Determine the probability of getting a sum of 5, which means you add the number from each die together obtain the sum of 5.

Select one:

a. 4/36

b. 2/36

c. 5/36

d. 3/36

3. Suppose you have a jar that includes six balls, two gold balls, three purple balls, and one orange ball. You will draw the balls without replacement. Determine the probability of getting a gold ball on the first draw and a purple ball on the second draw. Note that the following probabilities are expressed in terms of products of non-reduced fractions.

Select one:

a. (2/6)(2/6)(2/6)(2/6)

b. (3/6)(3/6)(3/6)(3/6)

c. (2/6)(3/5)(2/6)(3/5)

d. (3/6)(2/5)

(1 point) A biologist captures 24 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 ?X.  

Suppose there are 113 grizzly bears in the population.

(a) What is the probability that of the 15 grizzly bears observed, 3 had radio collars? Use four decimals in your answer.

?(?=3)=P(X=3)=

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(b) Find the probability that between 3 and 8 (inclusive) of the 15 grizzly bears observed were radio collared?

?(3≤?≤8)=P(3≤X≤8)=

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(use four decimals)

(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.

?(?)=E(X)=

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(use two decimals)

??(?)=SD(X)=

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(use two decimals)  

(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 4 and 8 (inclusive) ".

Given this information, what is the probability that the biologist saw 7 radio-collared grizzly bears?

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(use four decimals in your answer)

Allen's hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther.† Suppose a small group of 12 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 = 0.28 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.)

(b) What conditions are necessary for your calculations? (Select all that apply.)

normal distribution of weightsn is largeuniform distribution of weightsσ is knownσ is unknown

(c) Interpret your results in the context of this problem.

The probability that this interval contains the true average weight of Allen's hummingbirds is 0.80.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 0.20.The probability to the true average weight of Allen's hummingbirds is equal to the sample mean.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 = 0.08 for the mean weights of the hummingbirds. (Round up to the nearest whole number.)
hummingbirds

The new Fore and Aft Marina is to be located on the Ohio River near Madison, Indiana. Assume that Fore and Aft decides to build a docking facility where one boat at a time can stop for gas and servicing. Assume that arrivals follow a Poisson probability distribution, with an arrival rate of 4 boats per hour, and that service times follow an exponential probability distribution, with a service rate of 8 boats per hour. The manager of the Fore and Aft Marina wants to investigate the possibility of enlarging the docking facility so that two boats can stop for gas and servicing simultaneously.

Note: Use P₀ values from Table 11.4 to answer the questions below.

1. What is the probability that the boat dock will be idle? Round your answer to four decimal places.
   
   P₀ =  
   
2. What is the average number of boats that will be waiting for service? Round your answer to four decimal places.
   
   L_q =  
   
3. What is the average time a boat will spend waiting for service? Round your answer to four decimal places.
   
   W_q =  hours
   
4. What is the average time a boat will spend at the dock? Round your answer to four decimal places.
   
   W =  hours
   
5. If you were the manager of Fore and Aft Marina, would you be satisfied with the service level your system will be providing?
   
   Yes  
   
   Why or why not? Round your answers to whole numbers.
   
   Because the average wait time is  seconds. Each channel is idle  % of the time.

Ontario has decided to allow outdoor activities amid COVID 19 with some restrictions. Dr. David Williams, Chief Medical Officer of health, is recommending that people wear face masks and maintain physical distancing in public places. You have estimated that 80% of people outdoors would be wearing a face mask. To decide for yourself, you have conducted a survey on June 28^th in one of the “Active” outdoor zones in Toronto.

(a) On June 28^th you have sampled 25 people that were outdoor to see if they are wearing face masks.

i) Calculate the probability that more than 15 outdoor people were wearing face masks.

ii) Calculate the probability that at least 10 but no more than 14 outdoor people were wearing face masks.

(b) On another day you have sampled 80 people that were outdoor who are wearing face masks.

i) Find the expected number of people that are outdoor who are wearing face masks.

ii) Find the variance of the people that are outdoor who are wearing face masks.

(c) Using the solution you found in part b), determine the following:

i) E(4X+2)

ii) V(2X+3)

(d) On another day, you decided to sample 6 people that were outdoor, and you think that 77% will be wearing a facemask. What is the probability that 4 outdoor people will be wearing a facemask?

Allen's hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther.† Suppose a small group of 12 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 = 0.40 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 limitupper limitmargin of error

(b)

What conditions are necessary for your calculations? (Select all that apply.)

n is largeuniform distribution of weightsσ is unknownnormal distribution of weightsσ is known

(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 an 80% 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 a 20% 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 = 0.09 for the mean weights of the hummingbirds. (Round up to the nearest whole number.)

hummingbirds

According to the National Institute of Allergy and Infectious Diseases, approximately 6% of U.S. children 4 years of age or younger have a food allergy. A day care program has capacity for 10 children in that age range. Assume that the children attending the day care program are independent. Let the random variable X be the number of children in this day care who have a food allergy.

1. Which distribution does X follow?

a. Normal distribution with mean 8 and variance 0.07

b. Binomial distribution with n = 10 and p = 0.94

c. Binomial distribution with n = 10 and p = 0.06

d. Poisson distribution with mean and variance 0.07

2. What are the approximate values of the mean and standard deviation of the distribution of X?

a. Mean = 0.08 and standard deviation = 0.0049

b. Mean = 0.60 and standard deviation = 0.7509

c. Mean = 0.60and standard deviation = 0.564

d. Mean = 0.56 and standard deviation = 0.7217

3. What is the probability that at most/best 4 children have a food allergy?

a. 0.9998

b. 0.9988

c. 0.1250

d. 0.0700

4. What is the probability that at least one of the ten children has a food allergy?

a. 0.8965

b. 0.5596

c. 0.4425

d. 0.4614

5. What is the probability P(X < 2)?

a. 0.9853

b. 0.8965

c. 0.8824

d. 0.8884

Suppose that you are taking a multiple choice test (consisting of only 4-answer-choice questions) for which you have mastered 60% of the material. When you actually take the test, if you know the answer of a question, then you will answer it correctly for sure; if you do not know the answer, however, you can still guess the answer (meaning that you will have 25% chance of getting the correct answer). Finally, assume that your answer to each question is independent of the others.

(a) For a given question on the test, what is the probability that you answer it correctly?

(b) Suppose that each multiple-choice question is worth of 10 points (assuming no partial credits). What is the probability that you will get at least 90 points (out of 100 points) over the 10 questions?

(c) Suppose that you enter a slightly different test. It is still of the same subject and the same format (i.e., 4-answer-choice questions). You will be given one question at a time. If you answered the question correctly, the test continues; if you answered a question wrong, the test stops. (Then you will be graded based on how many questions you can answer correctly.) What is the probability that you will have the chance to answer at least 3 questions?

(d) What is the average number of questions you will encounter in the test in (c)?