To calculate the mean ($\mu$) and standard deviation ($\sigma$) for a binomial distribution, we use the following formulas:

• Mean ($\mu$): $\mu = n \times \pi$

• Standard deviation ($\sigma$): $\sigma = \sqrt{n \times \pi \times (1 - \pi)}$

Given:

• $n = 40$
• $\pi = 0.70$

Let's compute the mean and standard deviation.

Step 1: Calculate the mean

$\mu = 40 \times 0.70 = 28.00$

Step 2: Calculate the standard deviation

$\sigma = \sqrt{40 \times 0.70 \times (1 - 0.70)} = \sqrt{40 \times 0.70 \times 0.30}$

Let's compute that next.### Results:

• Mean ($\mu$): $28.00$
• Standard deviation ($\sigma$): $2.8983$

Would you like more details or have any other questions?

Here are 5 related questions you could explore:

1. How does the binomial distribution change when $\pi$ is closer to 0.5?
2. What happens to the standard deviation when $n$ increases?
3. How can you use the binomial distribution to calculate probabilities?
4. How is the binomial distribution related to the normal distribution?
5. Can we use this binomial setup to model real-world scenarios like success rates in a process?

Tip: The binomial distribution assumes independent trials, where the probability of success remains constant for each trial.