Let's break this down step by step.

1. Mean number of correct answers (μ)

This is a binomial distribution problem. In this case, the number of trials $n = 12$ (because there are 12 questions), and the probability of success on each trial (getting the right answer) is $p = \frac{1}{6}$, since there are 6 possible answers and only one is correct.

The formula for the mean (μ) of a binomial distribution is: $μ = n \times p$ Substitute the values: $μ = 12 \times \frac{1}{6} = 2$ So, the mean number of correct answers is 2.

2. Standard deviation (σ)

The formula for the standard deviation (σ) of a binomial distribution is: $σ = \sqrt{n \times p \times (1 - p)}$ Substitute the values: $σ = \sqrt{12 \times \frac{1}{6} \times \left(1 - \frac{1}{6}\right)} = \sqrt{12 \times \frac{1}{6} \times \frac{5}{6}}$ $σ = \sqrt{\frac{60}{36}} = \sqrt{\frac{5}{3}} \approx 1.291$ So, the standard deviation is approximately 1.29 (rounded to two decimal places).

3. Range for usual number of correct answers

For the usual values, we use the rule that the "usual" values are within 2 standard deviations of the mean. This means the interval is: $\mu - 2\sigma \quad \text{to} \quad \mu + 2\sigma$ Substitute the values: $2 - 2(1.29) \quad \text{to} \quad 2 + 2(1.29)$ $2 - 2.58 \quad \text{to} \quad 2 + 2.58$ $-0.58 \quad \text{to} \quad 4.58$ Since the number of correct answers must be a whole number, the range of usual values is: $[0, 4]$ Thus, the usual values for the number of correct answers are [0, 4].

4. Would it be unusual to get at least 8 correct answers?

To determine whether it's unusual to get at least 8 correct answers, we need to look at how likely this is. The usual number of correct answers falls within the range [0, 4], and 8 is far above this range.

The probability of getting 8 or more correct answers in a binomial distribution with $n = 12$ and $p = \frac{1}{6}$ is very small. Therefore, getting 8 correct answers is unusual.

Summary:

• Mean number of correct answers: 2
• Standard deviation: 1.29
• Usual values: [0, 4]
• It would be unusual to get at least 8 correct answers by guessing.

Do you need more details on any part of the explanation or have further questions?

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Here are some related questions to expand your understanding:

1. How do you calculate the probability of getting exactly 3 correct answers using a binomial distribution?
2. What would the mean and standard deviation be if the number of questions were doubled to 24?
3. How does the binomial distribution change if the probability of guessing correctly changes from 1/6 to 1/4?
4. What are the implications of changing the number of answer choices (e.g., to 4 or 5 choices) on the mean and standard deviation?
5. How can you use the normal approximation to the binomial distribution in this case?

Tip: Always remember that binomial distributions have a fixed number of trials and two outcomes (success or failure), making them useful for modeling situations like this.