Final answer:
The probability that 11 or more customers choose the leading brand can be calculated using the binomial probability formula, summing up the probabilities from 11 to 15 successes. The calculation can get complex and is often done with statistical software.
Step-by-step explanation:
To calculate the probability that 11 or more customers out of a sample of 15 choose the leading dishwasher detergent brand with a 30% market share, we can use the binomial probability formula. Since the market share is 30%, the probability of a customer choosing the leading brand is 0.3, and the probability of not choosing it is 0.7. For binomial probability, the formula is P(X=k) = (n choose k) * (p^k) * ((1-p)^(n-k)), where 'n' is the number of trials, 'k' is the number of successes, and 'p' is the probability of success.
However, since we need the probability of 11 or more customers choosing the leading brand, we need to calculate the sum of probabilities from 11 to 15 successes. This requires either calculating each individually and summing them up or using statistical software or a table to find the cumulative probability. Calculating this by hand can be quite laborious due to the necessity of computing several binomial coefficients and probabilities.
In practical situations, because we are dealing with the tail of the distribution, it may be preferred to utilize the normal approximation to the binomial distribution if the sample size is large enough and the success probability is not too close to 0 or 1. For a sample size of 15, the exact binomial calculation would typically be used directly.