Win/Loss and With/Without Joe: Joe plays basketball for the Wildcats and missed some of the season due to an injury. The win/loss record with and without Joe is summarized in the contingency table below.

Observed Frequencies: O_i's

The Test: Test for a significant dependent relationship between the outcome of the game (win/lose) and whether or not Joe played. Conduct this test at the 0.01 significance level.

(a) What is the null hypothesis for this test?

H₀: The outcome of the game and whether or not Joe plays are dependent variables. H₀: The outcome of the game and whether or not Joe plays are independent variables.      H₀: The probability of winning is dependent upon whether or not Joe plays.

(b) What is the value of the test statistic? Round to 3 decimal places unless your software automatically rounds to 2 decimal places.

χ²

=

(c) Use software to get the P-value of the test statistic. Round to 4 decimal places unless your software automatically rounds to 3 decimal places.
P-value =

(d) What is the conclusion regarding the null hypothesis?

reject H₀ fail to reject H₀    

(e) Choose the appropriate concluding statement.

We have proven that Joe causes the team to do better. The evidence suggests that the outcome of the game is dependent upon whether or not Joe played.     There is not enough evidence to conclude that the outcome of the game is dependent upon whether or not Joe played. We have proven that the outcome of the game is independent of whether or not Joe played.

Losses follow a Poisson frequency distribution with a mean of 2 per year. The amount of a loss is 1,2, or 3 with each having a probability of 1/3. Loss amounts are independent of the number of losses and from each other. An insurance policy covers all losses in a year subject to an annual aggregate deductible of 2. Calculate the expected claim payments for this insurance policy.

An insurance company insures large number of independent individual houses. The expected
average loss for each house for 1 year period is $600, and the standard deviation of the average
loss is $100. Let us assume that the average loss follows the normal distribution. Using
Normdist() function in Excel, calculate the probability that the average loss will exceed $850 (show the detailed steps)

An urn contains 4 green balls, five blue balls, and seven red balls. You remove five balls at random without replacement. Let X be the random variable that counts the number of green balls in your sample. a) Find the probability mass function p(x) describing the distribution of X. b) Find the mean and variance of X

Physic chemistry questions.

True or false. Justify the answer.

1. The cyclic boundary condition is the one which allows to quantize the "l" quantic number in rigid rotor.

2. When the potencial in the monodimensional box's walls isn't infinite, the particle inside has not null probability to be outside of the box.

3. For the hydrogen atom, 5f orbitals doesn't exist.

6. [11 marks] A special committee of five professionals is formed from a group of four lawyers, three doctors and three teachers. At least three lawyers are needed in the special committee. Define X as the number of teachers in the special committee.

1. Construct the probability distribution of the random variable X.
2. Calculate the expected of X and provide an interpretation of it.
3. Calculate the standard deviation of X.

When studying radioactive material, a nuclear engineer found that over 365 days, 1,000,000 radioactive atoms decayed to 973,133 radioactive atoms, so 26,867 atoms decayed during 365 days.

a. Find the mean number of radioactive atoms that decayed in a day.

b. Find the probability that on a given day, 50 radioactive atoms decayed.

Thanks in advance!

Suppose that in a batch of 500 components, 20 are defective and the rest are good. A sample of 10 components is selected at random with replacement, and tested. Let X denote the number of defectives in the sample.

a. What is the PMF of X? State the distribution, its parameters, and give the equation for its PMF with the correct parameters.

b. What is the probability that the sample contains at least one defective component?

Pennsylvania has a lottery entitled “Pick 4.” To win, a player must correctly match four digits from a daily lottery in which four digits are selected.

Question #1 – What is the probability of winning if you play just one number?

Question #2 - In the Pick 4 lottery, for a bet of $100, the payoff is $5000. What is the value of winning? Is it worth it?

A survey has a success rate of 75%. Suppose that the surgery is performed on three patients.

1. What is the probability that the survey is successeful on at least 2 patients?

2. On average, how many surgeries can we expect of successful if performed on a group of three patients?

3. What is the standard deviation of the number of succeful surgeries performed on a group of three patients?