Some Stastistics Questions, step by step explanations if possible! Thanks! 1. Ri

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Some Stastistics Questions, step by step explanations if possible! Thanks!

1. Richard has just been given a 4-question multiple-choice quiz in his history class. Each question has four answers, of which only one is correct. Since Richard has not attended class recently, he doesn't know any of the answers. Assuming that Richard guesses on all four questions, find the indicated probabilities. (Round your answers to three decimal places.)

(a) What is the probability that he will answer all questions correctly?

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(b) What is the probability that he will answer all questions incorrectly?

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(c) What is the probability that he will answer at least one of the questions correctly? Compute this probability two ways. First, use the rule for mutually exclusive events and the probabilities shown in the binomial probability distribution table.

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(d) Then use the fact that P(r 1) = 1 P(r = 0)?

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(e) Compare the two results. Should they be equal? Are they equal? If not, how do you account for the difference?

They should be equal, but may differ slightly due to rounding error is the correct answer.

(f) What is the probability that Richard will answer at least half the questions correctly?

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2. Let x be a random variable that represents white blood cell count per cubic milliliter of whole blood. Assume that x has a distribution that is approximately normal, with mean = 6400 and estimated standard deviation = 2850. A test result of x < 3500 is an indication of leukopenia. This indicates bone marrow depression that may be the result of a viral infection.

(a) What is the probability that, on a single test, x is less than 3500? (Round your answer to four decimal places.)

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(b) Suppose a doctor uses the average x for two tests taken about a week apart. What can we say about the probability distribution of x?

The probability distribution of x is approximately normal with x = 6400 and x = 2015.25. is the correct answer.

(c) What is the probability of x < 3500? (Round your answer to four decimal places.)

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(d) Repeat part (b) for n = 3 tests taken a week apart. (Round your answer to four decimal places.)

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(e) Compare your answers to parts (a), (b), and (c). How did the probabilities change as n increased?

The probabilities decreased as n increased. is the correct answer.

(f) If a person had x < 3500 based on three tests, what conclusion would you draw as a doctor or a nurse?

It would be an extremely rare event for a person to have two or three tests below 3,500 purely by chance. The person probably has leukopenia. is the correct answer.

3. The taxi and takeoff time for commercial jets is a random variable x with a mean of 8.1 minutes and a standard deviation of 2.9 minutes. Assume that the distribution of taxi and takeoff times is approximately normal. You may assume that the jets are lined up on a runway so that one taxies and takes off immediately after the other, and that they take off one at a time on a given runway.

(a) What is the probability that for 35 jets on a given runway, total taxi and takeoff time will be less than 320 minutes? (Round your answer to four decimal places.)

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(b) What is the probability that for 35 jets on a given runway, total taxi and takeoff time will be more than 275 minutes? (Round your answer to four decimal places.)

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(c) What is the probability that for 35 jets on a given runway, total taxi and takeoff time will be between 275 and 320 minutes? (Round your answer to four decimal places.)

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4. In the following problem, check that it is appropriate to use the normal approximation to the binomial. Then use the normal distribution to estimate the requested probabilities.

It is estimated that 3.4% of the general population will live past their 90th birthday. In a graduating class of 751 high school seniors, find the following probabilities. (Round your answers to four decimal places.)

(a) 15 or more will live beyond their 90th birthday

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(b) 30 or more will live beyond their 90th birthday

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(c) between 25 and 35 will live beyond their 90th birthday

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(d) more than 40 will live beyond their 90th birthday

Explanation / Answer

Q.1 (a) Out of 4 answers, 1 is correct so P(correct) = 0.25

(a) What is the probability that he will answer all questions correctly? so it is a binomial probability question where out of 4, 4 are correct.

P(all answer correctly) = 4C4 (0.25)4 = 0.0039025

(b) What is the probability that he will answer all questions incorrectly?

P (all answers incorrectly) = 4C4 (0.75)4 = 0.3164

(c) What is the probability that he will answer at least one of the questions correctly?

(i) The rule for mutually exclusive events

P ( X >=1) = 4C1 (0.25)1 (0.75)3 + 4C2 * (0.25)2 (0.75)2 + 4C3 * (0.25)3 (0.75)  + 4C4 * (0.25)4   

= 0.6836

(ii) binomial calculator P(x>=1) = P(X =1; 4; 0.25) = 0.6836

(d) P ( X >=1) = 1 - P(none of the answers correctly) = 1 - P( X=0) = 1 - 4C0 (0.75)4 = 1 - 0.3164 = 0.6836

(e) Compare the two results. Should they be equal? Are they equal? If not, how do you account for the difference?

All three tests show equal results and there is no difference.

(They should be equal, but may differ slightly due to rounding error is the correct answer.)

(f) What is the probability that Richard will answer at least half the questions correctly?

P(X>=2) = P(2) + P(3) + P(4) = 4C2 * (0.25)2 (0.75)2 + 4C3 * (0.25)3 (0.75)  + 4C4 * (0.25)4   

= 0.2617

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