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A binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent yes/no experiments, each with the same probability of success. It is commonly used in situations where there are only two possible outcomes, often labeled "success" and "failure."

In our exercise, the context revolves around likely voters and their opinions on Congress's performance.

• The number of trials, denoted as \( n \), represents the 45 voters surveyed.
• The probability of success, \( p \), is 0.11, since 11% of voters think Congress is doing a good or excellent job.

With these parameters, we can use the binomial distribution to examine how voter opinions might vary given the constant probability of success with each voter.

The mean and standard deviation are key components in understanding the behavior of a binomial distribution. Let's look at each:

- **The Mean (µ):** This indicates the expected number of successes in your trials. You can calculate it using the formula: \ \[ \mu = n \cdot p \] For this scenario, \( \mu = 45 \times 0.11 = 4.95 \).

- **The Standard Deviation (σ):** This gives us insight into the variability or spread from the mean. It's calculated as follows: \ \[ \sigma = \sqrt{n \cdot p \cdot (1-p)} \] Substituting in the current values, we find the standard deviation, but note that it is usually computed only when normal approximation conditions are satisfied.

The probability of success in a binomial distribution refers to the likelihood of a single trial resulting in a success. It is symbolized by \( p \). In our context, a success is defined as a voter thinking Congress is doing a good or excellent job, which is reported to be 11%.

It’s important to realize that:

• Each trial (or decision to survey a voter) is independent of the others.
• The probability of success remains constant for every trial, at 0.11 in this case.

These conditions are crucial for any distribution to be classified as binomial, and they frame the way we set up our calculations for mean and standard deviation.

Normal approximation is a method used to approximate the binomial distribution with a normal one when the sample size is large enough. This conversion is helpful for complex calculations and easier interpretations.

The two main conditions to check whether normal approximation is applicable are:

• \( n \cdot p \geq 5 \)
• \( n \cdot (1-p) \geq 5 \)

Let's compute these for our problem:

• \( n \cdot p = 45 \times 0.11 = 4.95 \), which is less than 5.
• \( n \cdot (1-p) = 45 \times 0.89 = 40.05 \), which is more than 5.

Since \( n \cdot p \) does not meet the necessary threshold of 5, the normal approximation cannot be used for this binomial distribution. This demonstrates why understanding and checking these conditions is vital in probabilistic models.