Let's go through the steps for solving the remaining parts of this binomial distribution problem.

Given:

• Number of trials, $n = 867$
• Probability of success, $p = 28\% = 0.28$
• The mean, $\mu = 242.8$ (which is correct)

1. Standard Deviation ($\sigma$):

The formula for the standard deviation of a binomial distribution is:

$\sigma = \sqrt{n \cdot p \cdot (1 - p)}$

Substituting the given values:

$\sigma = \sqrt{867 \cdot 0.28 \cdot (1 - 0.28)}$ $\sigma = \sqrt{867 \cdot 0.28 \cdot 0.72}$ $\sigma = \sqrt{867 \cdot 0.2016}$ $\sigma = \sqrt{174.7872}$ $\sigma \approx 13.22$

So, the standard deviation $\sigma$ is approximately 13.22.

2. Range Rule of Thumb (Minimum and Maximum Usual Values):

Using the range rule of thumb, the minimum and maximum usual values are calculated as:

• Minimum usual value: $\mu - 2\sigma$
• Maximum usual value: $\mu + 2\sigma$

Substitute $\mu = 242.8$ and $\sigma = 13.22$:

• Minimum usual value: $\mu - 2\sigma = 242.8 - 2 \cdot 13.22 = 242.8 - 26.44 = 216.36 \approx 216$
• Maximum usual value: $\mu + 2\sigma = 242.8 + 2 \cdot 13.22 = 242.8 + 26.44 = 269.24 \approx 269$

Thus, the interval for usual values is $[216, 269]$.

Final Answers:

1. Standard deviation ($\sigma$): $13.22$
2. Usual values interval: $[216, 269]$

Would you like any further details or clarifications?

Here are 5 related questions to deepen your understanding:

1. How does the probability of success ($p$) affect the mean and standard deviation of a binomial distribution?
2. What happens to the range of usual values if the standard deviation increases?
3. Can you explain why the formula for standard deviation includes both $p$ and $(1 - p)$?
4. How would the binomial distribution change if the number of trials $n$ were halved?
5. How would you use this binomial distribution to calculate the probability of exactly 250 successes?

Tip: Always verify that the probability and number of trials are within reasonable bounds for a binomial distribution—extreme values might require alternative statistical approaches.