The problem involves a binomial distribution. Now, the binomial distribution models the number of successes in a series of trials, where each trial has a binary outcome (success or failure). In this case, the success (given by the percentage) is identifying a person of a particular ethnic group, and we have 220 trials.
At first, we convert the percentage (20%) given to us into a decimal, which gives us 0.20. This gives us our probability of success (p), hence, p = 0.20.
To find the mean (µ) of a binomial distribution, we use the formula:
µ = n * p
Where ‘n’ is the number of trials (sample_size), which is 220 in this case, and ‘p’ is the probability of success calculated earlier.
So, µ = 220 * 0.20 = 44.00
Next, we will calculate the standard deviation (σ). The standard deviation of a binomial distribution is given by the formula:
σ = sqrt[n * p * (1 - p)]
Here, ‘sqrt’ denotes the square root, ‘n’ is the number of trials, ‘p’ is the probability of success and ‘1 - p’ gives us the probability of failure.
So, σ= sqrt[220 * 0.20 * (1 - 0.20)] = 5.93
So, the mean of this size sample of the binomial distribution is 44.00 and the standard deviation is 5.93.