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The binomial distribution is a discrete probability distribution that models the number of successes in a sequence of independent experiments. In our restaurant example, each customer represents a single experiment. The success is when a customer orders apple pie, defined by a probability, denoted by \( p \). Here, \( p = 0.20 \) or 20%.
This distribution is applicable because:

• Each customer decision is independent.
• There are only two outcomes: ordering or not ordering apple pie.
• The probability of success (ordering pie) is the same for each trial/customer.

The binomial distribution helps us determine the expected number of successes—here, pie orders—and the spread or variability around this expectation.

Standard deviation is a measure used to indicate the extent of deviation for a group as a whole. In the context of a binomial distribution, it measures the spread or 'spreadoutness' of the distribution.
The formula used here is \( \sigma = \sqrt{np(1-p)} \), where \( n \) is the total number of trials (400 customers) and \( p \) is the probability (0.20). This calculates to \( \sigma = 8 \) in our example.
This value tells us that most daily apple pie orders will vary within a few units from the average of 80, forming part of our confidence interval calculations. Understanding this variability is key to predicting outcomes and preparing for daily operations.

The margin of error quantifies the amount of random sampling error in a survey's results. In this example, it reflects the uncertainty around the proportion of customers ordering apple pie.
For a 95% confidence interval, the margin of error is calculated as \( ME = Z \times \sqrt{\frac{p(1-p)}{n}} \) where \( Z = 1.96 \) for a 95% confidence level. The smaller the margin, the more precise our estimate.
In our case, we aim for a margin of error of \( 0.01 \) or 1 percentage point around 20%. Adjusting the sample size allows us to shrink the margin, providing a more accurate approximation of the true order percentage within the desired range.

Determining the appropriate sample size is crucial for achieving reliable survey results. By increasing the sample size, we generally reduce the margin of error, allowing for greater confidence in the results.
To find the adequate sample size needed for a precision level, we rearrange the margin of error formula: \( n = \frac{Z^2 \times p(1-p)}{ME^2} \). In our apple pie scenario, we set the margin of error to 0.01 (1%) for a more defined estimate. With \( p = 0.20 \), we calculated the necessary sample size to be approximately 6146.
This shows that to confidently narrow our estimates of pie orders between the 19% and 21% mark, a larger customer base is needed. Balancing the cost and practicality of surveying more customers against the desired accuracy is an essential consideration.