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Suppose that a technology task force is being formed to study technology awareness among instructors. Assume that ten people will be randomly chosen to be on the committee from a group of 28 volunteers, 20 who are technically proficient and eight who are not. We are interested in the number on the committee who are not technically proficient. a. In words, define the random variable X. b. List the values that X may take on. C. Give the distribution of \(X . X \sim\) ____ (___,____) d. How many instructors do you expect on the committee who are not technically proficient? e. Find the probability that at least five on the committee are not technically proficient. f. Find the probability that at most three on the committee are not technically proficient.

Step by step solution

01

Define the Random Variable

The random variable \( X \) is defined as the number of instructors on the committee who are not technically proficient.

02

List Possible Values of X

Since the committee consists of 10 members and a maximum of 8 volunteers are not proficient, \( X \) can take values from 0 to 8, i.e., \( X = 0, 1, 2, ..., 8 \).

03

Identify the Distribution and Parameters

The distribution of \( X \) can be modeled as hypergeometric since we are selecting without replacement. So, \( X \sim \text{Hypergeometric}(N=28, K=8, n=10) \) where \( N \) is the total number of volunteers, \( K \) is the number of non-proficient volunteers, and \( n \) is the number of selections made.

04

Calculate the Expected Value of X

The expected value \( E(X) \) for a hypergeometric distribution is calculated as \( E(X) = n \frac{K}{N} = 10 \frac{8}{28} = \frac{80}{28} \approx 2.857 \).

05

Probability At Least Five Are Not Proficient

To find the probability that at least five committee members are not proficient, calculate \( P(X \geq 5) = P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8) \). Use the hypergeometric probability mass function to find these probabilities and sum them to get the total probability.

06

Probability At Most Three Are Not Proficient

To find the probability that at most three committee members are not proficient, calculate \( P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) \). Again, use the hypergeometric formula to find each probability and sum them to get the probability.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Random Variable

In statistics, a random variable is a numerical description of the outcome of a random phenomenon. For our specific exercise involving the technology task force, the random variable \( X \) is defined as the number of instructors on the committee who are not technically proficient. This means that each possible outcome of the committee selection process, in terms of the non-proficient instructors chosen, is represented by a possible value of \( X \).
It's crucial to understand that random variables provide a way to quantify the outcomes of random processes, offering a bridge between real-world phenomena and mathematical analysis.

Expected Value

The expected value of a random variable gives a measure of the center of the distribution of the random variable, often referred to as its average or mean. For a hypergeometric distribution, like the one in our exercise, the expected value \( E(X) \) can be calculated using the formula:
\[ E(X) = n \frac{K}{N} \]
where \( n \) is the number of selections made, \( K \) is the number of non-proficient volunteers, and \( N \) is the total number of volunteers. In our case, this becomes \( E(X) = 10 \frac{8}{28} \approx 2.857 \).

• This means that, on average, we can expect about 2.857 instructors on the committee to be not technically proficient.
• The expected value helps to give us an overall sense of what might happen, even though the actual number may vary due to the randomness involved in the selection.

Probability

Probability measures the likelihood of an event occurring. In our task force example, the hypergeometric distribution is used to determine various probabilities because we are selecting without replacement.
To calculate specific probabilities, such as the probability that at least five committee members are not proficient, we need to use the hypergeometric probability mass function. This function takes into account the lack of replacement in our selections.

• For instance, to find \( P(X \geq 5) \), we calculate the sum of probabilities \( P(X = 5) \), \( P(X = 6) \), \( P(X = 7) \), and \( P(X = 8) \).
• Similarly, for probabilities like \( P(X \leq 3) \), we sum \( P(X = 0) \), \( P(X = 1) \), \( P(X = 2) \), and \( P(X = 3) \).

This rigorous approach ensures that our probability calculations accurately reflect the parameters and constraints set by our specific situation.

Selection Without Replacement

The concept of selection without replacement is key to understanding why a hypergeometric distribution applies to this exercise. In selection without replacement, once an item (or in this case, a volunteer) is chosen, it cannot be chosen again. This affects the probabilities of subsequent selections since the total number of items decreases with each choice.

• This method of selection impacts the probabilities and expected values because it means each choice is dependent on previous choices, creating a chain of related events.
• In the context of the task force selection, this approach means that selecting one person who is not proficient decreases the odds of selecting another non-proficient person on subsequent draws, thereby altering the probability distribution.

Understanding selection without replacement is crucial for correctly applying the hypergeometric distribution to solve the problem.