The number 81 is written as a sum of three natural numbers 81 = a + b + c (the triple (a,b,c) is ordered; e.g., the decompositions 81 = 1 + 1 + 79 and 81 = 1 + 79 + 1 are different. Also, assume that all the decompositions have equal probability.) What is the probability that there exists a triangle with sides a, b, and c? (PLEASE EXPLAIN THE STEPS AND WHY YOU KNOW TO DO STEPS, THANKS)

Sampling is a process used in statistical analysis in which a predetermined number of observations are taken from a larger population. The methodology used to sample from a larger population depends on the type of analysis being performed but may include simple random sampling or systematic sampling. In a student survey, illustrate how the four probability (random) sampling techniques and four non-probability (non-random) sampling techniques can be used ?

A tire company finds the lifespan for one brand of its tires is normally distributed with a mean of 48,400 miles and a standard deviation of 5000 miles.

If the manufacturer is willing to replace no more than 10% of the tires, what should be the approximate number of miles for a warranty?

What is the probability that a tire will last more than 52,000 miles?  

What is the probability that a mean of 25 tires will last less than 47,000 miles?

At a certain airport, 75% of the flights arrive on time. A sample of 10 flights is studied. (Assume flights either arrive on time or delayed.)

Type your answer with 3 significant digits if necessary. (2.333, 0.0332, 0.000322).

a) What is the probability that at least 8 flights arrive on time?

b) What is the probability that 7 flights on time and 3 flights are delayed?

c) What is the standard deviation of the number of flights on time?

Can you explain with R Code:

The probability that an electronic component fails in the first day of operation is 0.005. 400 items are tested independently and whether they fail or not after a day will be recorded.

(a)What is the distribution of the number of items that fail?

(b) What is the probability that at least two items fail?

(c) Give the Poisson approximation for a, and compute the approximate answer to part b based on the Poisson approximation?

Comment on the accuracy of the approximation.

10) The overbooking problem: A plane has a capacity of 150 passengers. The airline, which knows that the industry standard is that one person out of 12 is a no-show at the airport, sells 160 tickets. a) What is the probability that all the passengers that show up at departure will be accommodated? b) What is the maximum number of tickets that the airline should sell so that they should be able to accommodate all the passengers with probability at least equal to 90%?

I have a coin of unknown fairness. Devise a test procedure to test its fairness; that is, to decide whether or not the coin is fair. Take the null hypothesis to be "the coin is fair". Pick a level of significance, and number of tosses to perform.

(a) What is the probability that I will conclude the coin is unfair, whereas in fact it is fair?

(b) Assuming in reality, the probability that the coin will produce heads is 0.55. What is theprobability that I will conclude that the coin is fair?

the visit of a customer at a restaurant follows a Poisson process with a rate of 3 arrivals per week. A day with no visits to the restaurant is called a risky day.

a. Find the expected number of risky days in a week

b.  Given that a risky day was observed on a Sunday, what is the probability that the next risky day will appear on the following Wednesday?

c. Find the probability that the 4th day, which is Thursday, of the week is the second risky day.

Four fair dices were rolled.

Part(a) How many possible outcomes there will be, if the order of dices are considered and their faces (number of points) are recorded?

Part(b) How many possible outcomes there will be, if the sum of the points of the four dices are recorded?

Part (c) What is the probability of getting a result with the sum exactly equals to 6?

Part(d) What is the probability of getting a result with the sum no less than 6?

1. Suppose that a random sample of size 64 is to be selected from a population with mean 40 and standard deviation 5.

a. What is the mean of the ¯xx¯ sampling distribution?

b. What is the standard deviation of the ¯xx¯ sampling distribution?

c. What is the approximate probability that ¯xx¯ will be within 0.5 of the population mean μμ?

d. What is the approximate probability that ¯xx¯ will differ from μμ by more than 0.7?

2. A Food Marketing Institute found that 45% of households spend more than $125 a week on groceries. Assume the population proportion is 0.45 and a simple random sample of 190 households is selected from the population. What is the probability that the sample proportion of households spending more than $125 a week is between 0.23 and 0.49?

Answer =  (Enter your answer as a number accurate to 4 decimal places.)

3. Based on historical data, your manager believes that 29% of the company's orders come from first-time customers. A random sample of 193 orders will be used to estimate the proportion of first-time-customers. What is the probability that the sample proportion is less than 0.31?

Answer =  (Enter your answer as a number accurate to 4 decimal places.)