Final answer:
To find the probability that a ski resort has an average of 1500 to 3000 customers per weekday over four days, we calculate the standard deviation of the sample mean, find the z-scores for both 1500 and 3000 customers, and then use the standard normal distribution to find the associated probabilities.
Step-by-step explanation:
To calculate the probability that a ski resort averages between 1500 and 3000 customers per weekday over the course of four weekdays, we need to first understand the distribution of the mean number of customers for four days. The central limit theorem tells us that the mean of a sample drawn from a population with any shape of distribution approaches a normal distribution if the sample size is large enough. In this case, the ski resort gets an average of 2000 customers per weekday with a standard deviation of 800 customers, and we assume the underlying distribution is normal.
When considering the average over four days, the mean remains the same, but the standard deviation of the sample mean (σx-bar) is equal to the population standard deviation (σ) divided by the square root of the sample size (n), so σx-bar = 800 / √4 = 800 / 2 = 400 customers.
Next, we use the z-score formula which is z = (X - μ) / σx-bar, where X is the value for which we are finding the probability, μ is the mean, and σx-bar is the standard deviation of the sample mean. To find the probability of being between 1500 and 3000, we calculate the z-scores for each and then use the standard normal distribution to find the probabilities for these z-scores.
For X = 1500, z = (1500 - 2000) / 400 = -1.25, and for X = 3000, z = (3000 - 2000) / 400 = 2.5. Consulting standard normal distribution tables or using a calculator with a normal distribution function, we can find the probabilities corresponding to these z-scores and then calculate the probability of being between them. The final answer will be the probability corresponding to z = 2.5 minus the probability corresponding to z = -1.25.