Final answer:
You are asked to find the probability that the sample proportion of first-time customers is between 0.21 and 0.34, given that 34% of orders typically come from first-time customers. You would use the binomial probability formula for exact calculations if the sample size is available or apply the normal approximation for large samples.
Step-by-step explanation:
The probability question based on the manager's belief that 34% of orders come from first-time customers is: What is the probability that in a sample, the proportion of first-time customers would be between 0.21 and 0.34?
To find the probability that the sample proportion is between 0.21 and 0.34, we can use the binomial probability distribution. Assuming we have a large enough sample size, we can apply the normal approximation to the binomial distribution to calculate this probability. However, some more information such as the sample size would be needed to provide a precise answer using the binomial formula:
Binomial probability formula:
P(X = k) = (n choose k) * p^k * (1-p)^(n-k)
But to answer this question, we would use normal approximation or use a calculator with the binomial distribution function to find P(0.21 < p < 0.34), assuming 'n' is the sample size and 'p' is the probability of a first-time customer order.