Question

1. Consider the following game. For 3 dollars I will allow you to roll a die...

1. Consider the following game. For 3 dollars I will allow you to roll a die one time. In return, I will pay you the value of the outcome if your roll. (e.g. you roll a 5 and I pay you 5 dollars.) Let X be the net profit (the value left over after subtracting the buy in).

(a) Create a probability distribution table listing the possible values of X and their corresponding probabilities P(X).

(b) Calculate E(X), the expected value of a single roll. (c) What is the probability that you win 1 or more dollars.

(d) What is the probability that you lose money.

(e) If you were to play the game twice, how much money would you expect to win? That is, calculate E(2X).

2 Navel oranges contain an average of 60ml of juice, with a standard deviation of about 30ml. Taylor, an agricultural researcher, will squeeze a sample of 36 of the oranges.

(a) Describe the sampling distribution of the sample average ¯x of the juice contents. (mean and standard deviation).

(b) Within what interval would you expect the sample average to lie, with probability 0.9?

(c) Calculate the probability with which the sample mean ¯x is less than 57.5 ml.

(d) After the experiment, Taylor would like to make mimosas to share with her colleagues. If the recipe calls for a total of 2.25 liters of orange juice, what is the probability that Taylor will have enough juice to follow the recipe? [Hint: Let X be the random variable representing the sum of the juice of the oranges. You may find σ(n · x) = n · σ(x) useful.]

3. Huey, a political advocate and statistical hobbyist in the town of Sacramento is interested in who will be elected the next mayor. A random sample of 40 citizens gave a response that 24 of them were planning to vote for Candidate A. Huey is now interested in constructing a 98% confidence interval for the proportion of citizens planning to vote for Candidate A.

(a) Find the sample proportion.

(b) Find the critical value Zα/2, and the standard error of the sample proportion.

(c) Use b) to calculate the Margin of Error.

(d) Construct the confidence interval.

4. The interval calculated above is too large for Huey’s liking. He would like to get a larger sample size to reduce the margin of error. Huey is still interested in a 98% confidence interval, but would like the margin of error to be at most 5%. Calculate the sample size needed, using the above sample proportion.

Dear student, we can provide you with the solution to one question & 4 sub-question at a time.

1) a) Here if we roll a die then there is a total of 6 outcomes (1,2,3,4,5,6)  each having a probability of

 Roll of die X P(x) 1 -2 2 -1 3 0 4 1 5 2 6 3 Total

b)

c)The probability that you win 1 or more is

d) you lose money when X <0

The probability that you lose money is

e)

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