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In probability theory, the binomial distribution is a discrete probability distribution. It describes the number of successful outcomes in a fixed number of trials. Each trial is independent and has the same probability of success. This is crucial when conducting experiments or surveys. In the given problem, the success refers to a worker commuting by bicycle. Here, the probability of success, or commuters using a bicycle, is 0.0544. The sample size, which is the number of workers surveyed, is 400. The binomial distribution helps us understand the likelihood of different numbers of people commuting by bicycle and is an essential concept when dealing with similar quantities in large populations.

The normal distribution, also known as the Gaussian distribution, is a continuous probability distribution that is symmetric around the mean. It is characterized by its bell-shaped curve and is used extensively in statistics due to its natural occurrence in many contexts.When working with the binomial distribution for large sample sizes, like in this scenario, the Central Limit Theorem allows us to approximate the binomial distribution with a normal distribution. This is valid when both \(np\) and \(n(1-p)\) are greater than 5.In the exercise, we calculated the mean \(\mu = np = 21.76\) and the standard deviation \(\sigma = \sqrt{np(1-p)} = 4.65\). By transforming the binomial to a normal distribution, we can efficiently compute probabilities.

A z-score tells us how many standard deviations an element is from the mean. It is a way of standardizing results from different data sets for comparison.To calculate a z-score, subtract the mean from the value you're interested in and divide by the standard deviation: \[ Z = \frac{X - \mu}{\sigma} \]In the given problem, we use z-scores to standardize values for better probability calculations. We use continuity correction when transforming discrete variables (like the number of bicycle commuters) into continuous ones. For instance, 22.5 and 27.5 are used instead of whole numbers such as 23 and 27.These z-scores help to find the probability of the number of commuters falling in the specified range by referencing the standard normal distribution.

Probability calculation, particularly using the normal distribution, involves finding the area under the curve. The area corresponds to the probability of obtaining a result within a certain range.To determine the probability that a number or range falls between two z-scores, we look up the probability of each z-score in a standard normal distribution table or use statistical software.For this problem, we first find the probability of \(Z_2\) (which corresponds to 27.5) then subtract the probability of \(Z_1\) (which corresponds to 22.5). The difference gives us the probability of having a number of commuters between 23 and 27. By understanding these steps, you can apply similar techniques to solve other problems of statistical probability.