Simon has a six-sided die that he suspects lands on the number 6 more frequently than would be predicted by chance alone. He amuses himself one day by repeatedly rolling the die and recording whether the outcome is a 6 or not. Out of 150 rolls, a 6 occurs 28 times. Simon decides to test whether the die is unfair, i.e., whether 6 does indeed occur more frequently than would be predicted by chance alone. He chooses a significance level of 5%.

a. What are the null and alternative hypotheses?

b. What is the value of the test statistic?

c. What is the p-value? Give an expression involving a probability, not just a final answer.

d. State your conclusions in the language of the problem.

e. Give a 95% confidence interval for the probability of rolling a 6 using Simon’s die.

Baseball’s World Series is a maximum of seven games, with the winner being the first team to win four games. Assume that the Atlanta Braves and the Minnesota Twins are playing in the World Series and that the first two games are to be played in Atlanta, the next three games at the Twins’ ballpark, and the last two games, if necessary, back in Atlanta. Taking into account the projected starting pitchers for each game and the home field advantage, the probabilities of Atlanta winning each game are as follows: Game 1 2 3 4 5 6 7 Probability of win 0.60 0.55 0.48 0.45 0.48 0.55 0.50 Conduct a simulation study with 1,000 trial. Using the summary statistics gathered, answer the following questions:

1. What is the probability that the Atlanta Braves win the World Series?

2. What is the average number of games played regardless of winner?

Please Double Check answers I've recived 3 wrong answers on three diffrent questions today thank you

A leading magazine (like Barron's) reported at one time that the average number of weeks an individual is unemployed is 17 weeks. Assume that for the population of all unemployed individuals the population mean length of unemployment is 17 weeks and that the population standard deviation is 2 weeks. Suppose you would like to select a random sample of 98 unemployed individuals for a follow-up study.

Find the probability that a single randomly selected value is greater than 16.8. P(X > 16.8) = (Enter your answers as numbers accurate to 4 decimal places.)

Find the probability that a sample of size n = 98 is randomly selected with a mean greater than 16.8. P(M > 16.8) = (Enter your answers as numbers accurate to 4 decimal places.)

a.  When a fair die is thrown, numbers 1 through 6 would appear with equal probability of 1/6. The mean is 3.5 and variance is 2.92.

If 57 dice are thrown, what is the variance of their sum?

b .

When a fair die is thrown, numbers 1 through 6 would appear with equal probability of 1/6. The mean is 3.5 and variance is 2.92.

If 40 dice are thrown, what is the mean of their sum?

c.

Let X be a Normal random variable with mean 30 and standard deviation 8.  Find its 60th percentile (i.e. find a number y such that P(X < y) = 0.6)

(provide one digit to the right of decimal point)

d.

Let X be a Normal random variable with mean 30 and standard deviation 8.  Find P(32 < X < 40).   (provide four digits to the right of the decimal point)

Suppose you are used to passing by, on average, 7 turtles on your one-mile walking route. You wish to come up with a pmf for the number of turtles Y that you will pass by on future walks on this route. Suppose you assume that with each 5-foot stride you will pass by a turtle with probability 7/1056 and that the event of your passing by a turtle in one stride is independent of the event of your passing by a turtle in any other stride. Recall: 1 mile = 5280 ft .

(a) Under these assumptions, what is the probability distribution of Y ?

(b) Under these assumptions, give EY

. (c) Under these assumptions, give P(Y ≤ 3).

(d) Now compute P(Y ≤ 3) treating Y as though it were a Poisson(λ) rv with λ = 7

(e) Explain what you think is the point of this question

Question 6: Winston, a dog, loves to play fetch. He catches each ball mid-air independently with probability 0.4. Write a simulation in R in which you throw Winston five balls and compute how many of the five balls he catches mid-air. Repeat this simulation N=5,000 times.

a) Plot a histogram of the number of catches Winston has made in each series of five catches.

b) What proportion of the time does Winston catch exactly two balls mid-air?

c) What proportion of the times does Winston catch exactly three balls mid-air?

d) Can you derive a formula to describe the probability of catching X=0,1,2,3,4,5 balls mid-air? How well does this formula match the proportions in your histogram? Hint: You will need to use the Combination Rule for (d).

According to a brief report by the Australian Institute of Health and Welfare, a child with a head circumstance less than the third percentile is considered to have a condition called microcephaly, which may indicate reduced brain volume.2 (squared) In other words, in a large group of children, roughly 3% of them have such a condition. In this question, consider a group of 50 children randomly selected nationally, and let X represent the number of children among this group who have microcephaly.

(a) Write down the possible values of X.

(b) What is a suitable distribution for X? Explain your answer by showing that the assumption(s) of this distribution is (are) met. For the distribution chosen, you must state the parameter(s) as well.

(c) Manually calculate the probability that there are two children who have microcephaly in the sample.

(d) Manually calculate the probability that there is more than one child with microcephaly in the sample.

The Higher Education Research Institute at UCLA collected data from 203,967 incoming first-time, full-time freshmen from 270 four-year colleges and universities in the U.S. 71.3% of those students replied that, yes, they believe that same-sex couples should have the right to legal marital status. Suppose that you randomly select freshman from the study until you find one who replies "yes." You are interested in the number of freshmen you must ask.

Construct the probability distribution function (PDF). Stop at

x = 6.

(Round your answers to four decimal places.)

What is the probability that you will need to ask fewer than three freshmen? (Round your answer to four decimal places.)

_________

A statistical program is recommended.

ONE-SAMPLE T TEST :

The following observations are on stopping distance (ft) of a particular truck at 20 mph under specified experimental conditions.

The report states that under these conditions, the maximum allowable stopping distance is 30. A normal probability plot validates the assumption that stopping distance is normally distributed.

a) Determine the probability of a type II error when α = 0.01,σ = 0.65, and the actual value of μ is 31 (use statistical software or Table A.17). (Round your answer to three decimal places.)

b) Repeat this for μ = 32. (Round your answer to three decimal places.)

c) What is the smallest sample size necessary to have α = 0.01 and β = 0.10 when μ = 31 and σ = 0.65? (Round your answer to the nearest whole number.)