Question

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.

Answer #1

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)

In a recent poll, 134 registered voters who planned to
vote in the next election were asked if they would vote for a
particular candidate and 80 of these people responded that they
would. We wish to predict the proportion of people who will vote
for this candidate in the election.
a) Find a point estimator of the proportion who would vote for
this candidate.
b) Construct a 90% confidence interval for the true proportion
who would vote for this...

In a recent poll, 134 registered voters who planned to vote in
the next election were asked if they would vote for a particular
candidate and 80 of these people responded that they would. 1. 1.
We wish to predict the proportion of people who will vote for this
candidate in the election.
a) Find a point estimator of the proportion who would vote for
this candidate.
b) Construct a 90% confidence interval for the true proportion
who would vote...

A political candidate has asked you to conduct a poll to
determine what percentage of people support him. If the candidate
only wants a 4% margin of error at a 97.5% confidence level, what
size of sample is needed? When finding the z-value, round it to
four decimal places.
You want to obtain a sample to estimate a population proportion.
At this point in time, you have no reasonable preliminary
estimation for the population proportion. You would like to be...

Suppose that you are offered the following "deal." You roll a
six sided die. If you roll a 6, you win $12. If you roll a 2, 3, 4
or 5, you win $1. Otherwise, you pay $10.
a. Complete the PDF Table. List the X values, where X is the
profit, from smallest to largest. Round to 4 decimal places where
appropriate.
Probability Distribution Table
X
P(X)
b. Find the expected profit. $ (Round to the nearest cent)
c....

Suppose that you are offered the following "deal." You roll a
six sided die. If you roll a 6, you win $9. If you roll a 2, 3, 4
or 5, you win $1. Otherwise, you pay $6
a. Complete the PDF Table. List the X values, where X is the
profit, from smallest to largest. Round to 4 decimal places where
appropriate.
Probability Distribution Table
X
P(X)
b. Find the expected profit. $ (Round to the nearest...

Suppose that you are offered the following "deal." You roll a
six sided die. If you roll a 6, you win $13. If you roll a 4 or 5,
you win $5. Otherwise, you pay $6.
a. Complete the PDF Table. List the X values, where X is the
profit, from smallest to largest. Round to 4 decimal places where
appropriate.
Probability Distribution Table
X
P(X)
b. Find the expected profit. $ ____ (Round to the nearest
cent)
c. Interpret...

Suppose that you are offered the following "deal." You roll a
six sided die. If you roll a 6, you win $20. If you roll a 4 or 5,
you win $1. Otherwise, you pay $8.
a. Complete the PDF Table. List the X values, where X is the
profit, from smallest to largest. Round to 4 decimal places where
appropriate.
Probability Distribution Table
X
P(X)
b. Find the expected profit. $ (Round to the nearest cent)
c. Interpret the...

Suppose that you are offered the following "deal." You roll a
six sided die. If you roll a 6, you win $7. If you roll a 4 or 5,
you win $1. Otherwise, you pay $8.
a. Complete the PDF Table. List the X values, where X is the
profit, from smallest to largest. Round to 4 decimal places where
appropriate.
Probability Distribution Table
X
P(X)
b. Find the expected profit. $ (Round to the nearest cent)
c. Interpret the...

Suppose that you are offered the following "deal." You roll a
six sided die. If you roll a 6, you win $17. If you roll a 4 or 5,
you win $2. Otherwise, you pay $10.
a. Complete the PDF Table. List the X values, where X is the
profit, from smallest to largest. Round to 4 decimal places where
appropriate.
Probability Distribution Table
X
P(X)
b. Find the expected profit. $ (Round to the nearest cent)
c. Interpret the...

PROBLEM #2
Suppose you play a game in which a fair 6 sided die is rolled
once. If the outcome of the roll (the number of dots on the side
facing upward) is less than or equal to 4, you are paid as many
dollars as the number you have rolled. Otherwise, you lose as many
dollars as the number you have rolled.
Let X be the profit from the game (or the amount of money won or
lost per...

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