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As a special promotion for its 16-ounce bottles of water, a water company printed a message on the inside of each cap. Some of the caps read "Better luck next time!" whereas others read "You won!" The company advertised the promotion with the slogan "One in eight wins a prize!" Suppose the company is telling the truth and that every 16-ounce bottle of water has a one-in-eight chance of being a winner. Nine friends each buy one 16-ounce bottle of water at a local convenience store. Let X equal the number of friends who win a prize.

**Part A:** Explain why X is a binomial random variable. (3 points)

**Part B:** Find the mean and standard deviation of X. Interpret each value in context. (4 points)

**Part C:** The store clerk is surprised when three of the friends win a prize. Is this group of friends just lucky, or is the company's one-in-eight claim inaccurate? Compute P(X ≥ 3) and use the result to justify your answer.

Answer :

a) X is a binomial random variable because there is a fixed number of trials, for each trial, the probability of winning is the same, and for each trial there are only two possible outcomes, winning and losing.

b) The mean and the standard deviation of X are given as follows:

  • Mean: 1.125.
  • Standard deviation: 0.99.

This means that out of nine friends, the expected number of those who area expected to win a prize is of 1.125, varying around 0.99.

c) At least 3 friends winning a prize is not unusual, as 3 is less than 2.5 standard deviations above the mean.

How to obtain the mean and standard deviation of the binomial distribution, and what are unusual measures?

The binomial distribution represents the probability of exactly x successes on n repeated trials, with p probability of a success on each trial.

The expected value of the binomial distribution is:

E(X) = np

The standard deviation of the binomial distribution is:

[tex]\sqrt{V(X)} = \sqrt{np(1-p)}[/tex]

The change of winning is of one in eight, hence the parameter p is obtained as follows:

p = 1/8 = 0.125.

There are nine friends, hence the parameter n is given as follows:

n = 9.

Then the mean is of:

E(X) = 9 x 0.125 = 1.125.

The standard deviation is of:

[tex]\sqrt{V(X)} = \sqrt{9 \times 0.125 \times 0.875} = 0.99[/tex]

A measure is considered unusual if it is more than 2.5 standard deviations above the mean, hence the upper bound is given as follows:

More than 1.125 + 2.5 x 0.99 = 3.6 friends.

More can be learned about the binomial distribution at https://brainly.com/question/24756209

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