For a binomial distribution, the mean, variance, and standard deviation are calculated using the following formulas:

1. Mean (μ) = $n \times p$
2. Variance (σ²) = $n \times p \times (1 - p)$
3. Standard Deviation (σ) = $\sqrt{n \times p \times (1 - p)}$

Given:

• $n = 70$
• $p = 0.9$

Let's calculate each one:

1. Mean:

$\mu = n \times p = 70 \times 0.9 = 63$ So, the mean is 63.

2. Variance:

$\sigma^2 = n \times p \times (1 - p) = 70 \times 0.9 \times (1 - 0.9) = 70 \times 0.9 \times 0.1 = 6.3$ So, the variance is 6.3.

3. Standard Deviation:

$\sigma = \sqrt{n \times p \times (1 - p)} = \sqrt{70 \times 0.9 \times 0.1} = \sqrt{6.3} \approx 2.5$ So, the standard deviation is approximately 2.5.

Final Answers:

• Mean (μ) = 63
• Variance (σ²) = 6.3
• Standard Deviation (σ) ≈ 2.5

Would you like more details or clarification on any part? Here are some related questions:

1. What happens to the mean and variance if the probability $p$ increases or decreases?
2. How does the binomial distribution behave when $p$ is very close to 0 or 1?
3. Can you derive the binomial distribution formula from the properties of mean and variance?
4. How does the standard deviation help in understanding the spread of a binomial distribution?
5. What are some real-life examples of binomial distributions?

Tip: Always make sure to check that $n$ is the number of trials and $p$ is the probability of success when applying these formulas!