Negative Binomial experiment is based on sequences of Bernoulli trials with probability of success p. Let x+m be the number of trials to achieve m successes, and then x has a negative binomial distribution. In summary, negative binomial distribution has the following properties

• Each trial can result in just two possible outcomes. One is called a success and the other is called a failure.
• The trials are independent
• The probability of success, denoted by p, is the same on every trial.
• The experiment consists of m successes, and x+m repeated trials, and the m^th success occurs at the (x+m)^th trial.

1. Write down the probability distribution P(x=k), consistent with the notation here
2. If you are tossing a regular coin repeatedly, what is the probability that the 3^rd head occurs at the 6^th time you toss it?
3. Anne is selling girl scot cookies in her neighborhood with 20 houses. She has a target to sell 10 boxes. Suppose each house has a probability 0.6 to buy one box of her cookies. What is the probability that she sells the last box at the 15^th house? What is the probability that she exhausts all 20 houses?

We would like to know: "What percentage of college students drink alcohol every day?" In a random sample of 500 students, 75 said they drink alcohol every day. Use the data to construct a 95% confidence interval to answer to the question. You MUST show your work to receive full credit.

1. A certain type of pump’s time until failure has an exponential distribution with a mean of 5000 hours. State answers rounded to two decimal places.
   1. What is the probability it will still be operating after 3000 hours of service?
   2. What is the probability that it will fail prior to 6000 hours of service?
   3. If it has been running for 3000 hours, what is the probability it will still be running 2000 hours later?

11.) According to an avid aquarist, the average number of fish in a 20-gallon tank is 10, with a standard deviation of two. His friend, also an aquarist, does not believe that the standard deviation is two. She counts the number of fish in 15 other 20-gallon tanks. Based on the results that follow, do you think that the standard deviation is different from two? Data: 11; 10; 9; 10; 10; 11; 11; 10; 12; 9; 7; 9; 11; 10; 11

Degrees of freedom

test statistic

pvalue

accept or reject

An urn contains 5 blue and 3 yellow beads. Find the probability of drawing (b, b, y, b, b) in that order.

1) with replacement

2) without replacement

Repeat for the states (b, b, b, b, b,) and (y, y, y, b, b, y).

Jane wants to estimate the proportion of students on her campus who eat cauliflower. After surveying 29 ​students, she finds 4 who eat cauliflower. Obtain and interpret a 95​% confidence interval for the proportion of students who eat cauliflower on​ Jane's campus using Agresti and​ Coull's method. LOADING... Click the icon to view Agresti and​ Coull's method.

Please explain how to do 1a and b throughly, thank you!

1A) A psychologist specializing in marriage counseling claims that, among all married couples, the proportion p for whom her communication program can prevent divorce is at least 76%. In a random sample of 225 married couples who completed her program, 164 of them stayed together. Based on this sample, can we reject the psychologist's claim at the 0.01 level of significance?Perform a one-tailed test. Then fill in the table below.

Carry your intermediate computations to at least three decimal places and round your answers as specified in the table.

1B) Citrus Rental is a popular car rental agency that has a history of having too few cars available, so that its available cars are overdriven. The mean monthly mileage over the years for Citrus cars has been about 1600 miles per month. Recently, though, Citrus purchased thousands of new cars, and the company claims that the average mileage of its cars is now less than in the past. To test this, a random sample of 12 recent mileages of Citrus cars was taken. The mean of these 12 mileages was 1565 miles per month, and the standard deviation was 210 miles per month. Assume that the population of recent monthly mileages of Citrus cars is normally distributed. At the 0.1 level of significance, can it be concluded that the mean recent monthly mileage, μ, of Citrus cars is less than 1600 miles per month?

Perform a one-tailed test. Then fill in the table below.

Carry your intermediate computations to at least three decimal places and round your answers as specified in the table.

Section I
You toss a coin and roll a die simultaneously. If the coin shows heads, the experiment outcome is equal to the value shown on the die. If the coin shows tails, the experiment outcome is equal to twice the value shown on the die. Assume that the coin and the die are fair. Let ? be 1 if the coin shows heads and 2 if the coin shows tails, ?be the outcome of rolling the die, and ? the outcome of the experiment. Notice that ?, ?, and ? are random variables.

1. What is the minimal sample space in this experiment? (4 points)

2. Identify the event that the outcome of the experiment is, at least, 10. (4 points)

3. What is the value of ??(? ≤ 9)? (4 points)

4. Whatisthevalueof??(4≤?≤10|?=2)?(4points)

5. Whatisthevalueof??(?=2|4≤?≤10)?(4points)

Two dice are rolled.
a. Make a table showing all of the possible outcomes.

b. How many outcomes give a sum of 7?
c. How many outcomes give a sum of 7 or a sum of 11?

d. How many outcomes have either the first die as a 3 or have an even sum

A six-person committee composed of Alice, Ben, Connie, Dolph, Egbert, and Francisco is to select a chairperson, secretary, and treasurer. Nobody can hold more than one of these positions.

e. How many selections are there in which Dolph is either a chairperson or he is not an officer?

f. How many selections are there in which Ben is either chairperson or treasurer?

g. How many selections are there in which either Ben is chairperson or Alice is secretary?

Seasonal affective disorder (SAD) is a type of depression during seasons with less daylight (e.g., winter months). One therapy for SAD is phototherapy, which is increased exposure to light used to improve mood. A researcher tests this therapy by exposing a sample of SAD patients to different intensities of light (low, medium, high) in a light box, either in the morning or at night (these are the times thought to be most effective for light therapy). All participants rated their mood following this therapy on a scale from 1 (poor mood) to 9 (improved mood). The hypothetical results are given in the following table.

(a) Complete the F-table and make a decision to retain or reject the null hypothesis for each hypothesis test. (Round your answers to two decimal places. Assume experimentwise alpha equal to 0.05.)

X_n is a Markov Chain with state-space E = {0, 1, 2}, and transition matrix

0.4 0.2 0.4

P = 0.6 0.3 0.1

0.5 0.3 0.2

And initial probability vector a = [0.2, 0.3, 0.5]

For the Markov Chain with state-space, initial vector, and transition matrix discuss how we would calculate the follow; explain in words how to calculate the question below.

a) P(X₁ = 0, X₂ = 0, X₃ = 1, X₄ = 2|X₀ = 2)

b) P(X₂ = 2, X₄ = 0, X₅ = 1

A probability distribution of all possible sample means for a particular sample size is the _____.

The mean of all possible sample means is __________________ the population mean.

If we decrease the sample size from 20 to 10, the standard error of the mean will ________________.

If a population follows the normal distribution, what is the shape of the distribution of the sample means?

A _____________________ is a single value computed from sample information used to estimate a population parameter.

A ________________________ is a range of values within which the population parameter is likely to occur.

A _____________________ shows the fraction of a sample that has a particular characteristic.

To construct a confidence interval for a mean, the z distribution is used only when the ________________ is known.

What is the appropriate Z score for a 78% confidence interval?

What is the appropriate Z score for a 62% confidence interval?

Word Bank Below:

( population standard deviation) (sampling distribution of the sample mean) (point estimate) (equal to) (0.88) (confidence interval) (increase) (1.23) (normal distribution) (proportion)

For a certain candy, 15% of the pieces are yellow, 20% are red, 20% are blue, 5% are green, and the rest are brown.

a) If you pick a piece at random, what is the probability that it is​ brown? it is yellow or​ blue? it is not​ green? it is​ striped?

b) Assume you have an infinite supply of these candy pieces from which to draw. If you pick three pieces in a​ row, what is the probability that they are all​ brown? the third one is the first one that is​ red? none are​ yellow? at least one is​ green?

a) The probability that it is brown is ___. (Round to three decimal places as needed.)

The probability that it is yellow or blue is ___. (Round to three decimal places as needed.)

The probability that it it is not green is ___. (Round to three decimal places as needed.)

The probability that it is striped is ___. (Round to three decimal places as needed.)

b) The probability of picking three brown candies is __. (Round to three decimal places as needed.)

The probability of the third one being the first red one is __. (Round to three decimal places as needed.)

The probability that none are yellow is __. (Round to three decimal places as needed.)

The probability of at least one green candy is __. (Round to three decimal places as needed.)