Questions: A survey asks 1200 workers, "Has the economy forced you to reduce the amount of vacation you plan to take this year?" Fifty-one percent of those surveyed say they are reducing the amount of vacation. Ten workers participating in the survey are randomly selected. The random variable represents the number of workers who are reducing the amount of vacation. Decide whether the experiment is a binomial experiment. If it is, identify a success, specify the values of n, p, and q, and list the possible values of the random variable x. Is the experiment a binomial experiment? No Yes What is a success in this experiment? Selecting a worker who is reducing the amount of vacation Selecting a worker who is not reducing the amount of vacation This is not a binomial experiment.

A survey asks 1200 workers, "Has the economy forced you to reduce the amount of vacation you plan to take this year?" Fifty-one percent of those surveyed say they are reducing the amount of vacation. Ten workers participating in the survey are randomly selected. The random variable represents the number of workers who are reducing the amount of vacation. Decide whether the experiment is a binomial experiment. If it is, identify a success, specify the values of n, p, and q, and list the possible values of the random variable x. Is the experiment a binomial experiment? No Yes What is a success in this experiment? Selecting a worker who is reducing the amount of vacation Selecting a worker who is not reducing the amount of vacation This is not a binomial experiment.
Transcript text: A survey asks 1200 workers, "Has the economy forced you to reduce the amount of vacation you plan to take this year?" Fifty-one percent of those surveyed say they are reducing the amount of vacation. Ten workers participating in the survey are randomly selected. The random variable represents the number of workers who are reducing the amount of vacation. Decide whether the experiment is a binomial experiment. If it is, identify a success, specify the values of $n, p$, and $q$, and list the possible values of the random variable $x$. Is the experiment a binomial experiment? No Yes What is a success in this experiment? Selecting a worker who is reducing the amount of vacation Selecting a worker who is not reducing the amount of vacation This is not a binomial experiment.
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Solution

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Solution Steps

Step 1: Determine if the Experiment is a Binomial Experiment

The experiment consists of selecting 10 workers and determining whether each worker is reducing the amount of vacation they plan to take. This satisfies the criteria for a binomial experiment:

  1. A fixed number of trials (\(n = 10\)).
  2. Each trial is independent.
  3. There are two possible outcomes: success (reducing vacation) or failure (not reducing vacation).
  4. The probability of success (\(p = 0.51\)) is constant for each trial.

Thus, the experiment is a binomial experiment.

Step 2: Define Success

In this experiment, a success is defined as:

  • Selecting a worker who is reducing the amount of vacation.
Step 3: Specify Values of \(n\), \(p\), and \(q\)

The values are as follows:

  • Number of trials: \(n = 10\)
  • Probability of success: \(p = 0.51\)
  • Probability of failure: \(q = 1 - p = 0.49\)
Step 4: List Possible Values of the Random Variable \(x\)

The possible values of the random variable \(x\) (the number of workers reducing vacation) are: \[ x \in \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \]

Step 5: Perform Binomial Distribution Analysis

Using the binomial distribution formulas, we calculate:

  • The probability of exactly 0 successes: \[ P(X = 0) = \binom{10}{0} \cdot (0.51)^0 \cdot (0.49)^{10} = 0.0008 \]
  • The mean (expected value): \[ \mu = n \cdot p = 10 \cdot 0.51 = 5.1 \]
  • The variance: \[ \sigma^2 = n \cdot p \cdot q = 10 \cdot 0.51 \cdot 0.49 = 2.499 \]
  • The standard deviation: \[ \sigma = \sqrt{n \cdot p \cdot q} = \sqrt{10 \cdot 0.51 \cdot 0.49} \approx 1.5808 \]

Final Answer

The experiment is a binomial experiment. A success is defined as selecting a worker who is reducing the amount of vacation. The values are:

  • \(n = 10\)
  • \(p = 0.51\)
  • \(q = 0.49\)
  • Possible values of \(x\): \(\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\)

The probability of exactly 0 successes is \(0.0008\), the mean is \(5.1\), the variance is \(2.499\), and the standard deviation is \(1.5808\).

Thus, the final answer is: \[ \boxed{\text{Yes, success is selecting a worker reducing vacation.}} \]

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