20% of a restaurants customers order the chef's special. 230 customers are anticipated to dine at the restaurant tonight. The standard deviation for this binomial distribution is 
(round your answer to 3 decimal places).

To find the standard deviation of a binomial distribution, we use the formula:

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

Where:
- \( n \) is the number of trials (customers in this case)
- \( p \) is the probability of success (ordering the chef's special)

Given:
- \( n = 230 \)
- \( p = 0.20 \)

First, let's calculate \( 1 - p \):

\[ 1 - p = 1 - 0.20 = 0.80 \]

Next, we substitute these values into the standard deviation formula:

\[ \sigma = \sqrt{230 \cdot 0.20 \cdot 0.80} \]

Now, we perform the calculations inside the square root:

\[ 230 \cdot 0.20 = 46 \]
\[ 46 \cdot 0.80 = 36.8 \]

So, the standard deviation is:

\[ \sigma = \sqrt{36.8} \approx 6.065 \]

Therefore, the standard deviation for this binomial distribution is approximately \( 6.065 \) (rounded to three decimal places).

Would you like more details or have any questions? Here are some related questions to expand your understanding:

1. What is the mean number of customers ordering the chef's special?
2. How do you interpret the standard deviation in the context of this problem?
3. What is the probability that exactly 50 customers will order the chef's special?
4. How would the standard deviation change if the probability of ordering the chef's special increased to 30%?
5. How would you calculate the variance for this distribution?
6. What are the characteristics of a binomial distribution?
7. How does the standard deviation help in understanding the spread of the data?
8. What assumptions must be true for this distribution to be considered binomial?

**Tip:** In a binomial distribution, the variance is the square of the standard deviation. Understanding both variance and standard deviation can give deeper insights into data variability.

Learn how to calculate the standard deviation for a binomial distribution problem where 20% of restaurant customers are expected to order the chef's special. Understand the formula and steps involved in finding the standard deviation with detailed calculations and explanations. Suitable for students or individuals learning about binomial distributions and probability.

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