Question

Use the following information to answer problems 1 through 5: Han Solo guesses randomly at

six multiple choice questions on an exam. Each question has four potential answers.

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7. This scenario can be modeled as a

(a) normal experiment with 4 trials and success probability of 1/6 per trial.

(b) normal experiment with 6 trials and success probability of 1/4 per trial.

(c) binomial experiment with 6 trials and success probability of 1/5 per trial.

(d) binomial experiment with 4 trials and success probability of 1/6 per trial.

(e) binomial experiment with 6 trials and success probability of 1/4 per trial.

8. Complete the following table that represents the probability distribution of

X

= the number of

questions Han guesses correctly.

x

0

1

2

3

4

5

6

P(X = x)

(a) 1/7, 2/7, 3/7, 4/7, 5/7, 6/7, 1

(b) .1780, .5339, .8306, .9624, .9954, .9996, 1

(c) 1/7, 1/7, 1/7, 1/7, 1/7, 1/7, 1/7

(d) .1780, .3560, .2966, .1318, .0330, .0044, .0002

(e) 1/6, 1/6, 1/6, 1/6, 1/6, 1/6

9. What is the probability he will answer at least 4 of the questions correctly?

(a) 0.0376 (b) 10/12 (c) .67% (d) 2/12 (e) 0.0046

10. What is the mean number of questions he will answer correctly?

(a) 0 (b) 1 (c) 1.5 (d) 2 (e) 4

11. What is the standard deviation of the number of questions he will answer correctly?

(a) 0.46 (b) 1 (c) 1.06 (d) 1.5 (e) 1.6

Answer 1

Han Solo guesses randomly at 6 multiple choice questions on an exam. Each question has 4 potential answers.

7. A binomial experiment is a statistical experiment that has the following properties:

• The experiment consists of n repeated trials.
• Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure.
• The probability of success, denoted by P, is the same on every trial.
• The trials are independent; that is, the outcome on one trial does not affect the outcome on other trials.

Hence, given scenario can also be modeled as a binomial experiment.

6 questions are to be answered so 6 trials.

4 options and any of the options can be the possible answer so probability of 1/4.

Therefore,this scenario can be modeled as a binomial experiment with 6 trials and success probability of 1/4 per trial.

8.  The following notation is helpful, when we talk about binomial probability.

• x: The number of successes that result from the binomial experiment.
• n: The number of trials in the binomial experiment.
• P: The probability of success on an individual trial.
• Q: The probability of failure on an individual trial. (This is equal to 1 - P.)
• n!: The factorial of n (also known as n factorial).
• b(x; n, P): Binomial probability - the probability that an n-trial binomial experiment results in exactly x successes, when the probability of success on an individual trial is P.
• : The number of combinations of n things, taken r at a time

Suppose a binomial experiment consists of n trials and results in x successes. If the probability of success on an individual trial is P, then the binomial probability is:

b(x; n, P) = * P^x * (1 - P)^{n - x}

Here, n=6,p=0.25 and x varies as below

The probability distribution of X= the number of questions Han guesses correctly.

x	P(X=x)
0		0.1780
1		0.3560
2		0.2966
3		0.1318
4		0.0330
5		0.0044
6		0.0002

9. What is the probability he will answer at least 4 of the questions correctly?

Probability he will answer at least 4 of the questions correctly=

++=0.0376

10.  What is the mean number of questions he will answer correctly?

Binomial distribution has the following properties:

• The mean of the distribution (μ_x) is equal to n * P .

So, here mean = 6*0.25=1.5

11. What is the standard deviation of the number of questions he will answer correctly?

• The standard deviation (σ_x) is sqrt[ n * P * ( 1 - P ) ].

Hence,  standard deviation = = 1.06

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