A hotel manager believes that 28% of the hotel rooms are booked. If the manager is correct, what is the probability that the proportion of rooms booked in a sample of 895 rooms would differ from the population proportion by less than 3% ? Round your answer to four decimal places.

Answer 1

The population proportion of rooms booked is 28%, so the population standard deviation can be calculated as:

σ = sqrt((p * (1 - p)) / n) = sqrt((0.28 * (1 - 0.28)) / 895) ≈ 0.0148

To find the probability that the proportion of rooms booked in a sample of 895 rooms would differ from the population proportion by less than 3%, we need to find the area under the normal curve within ±3% (0.03) of the population proportion (0.28).

First, we calculate the z-score for the lower bound:

z_lower = (0.28 - 0.03 - 0.28) / 0.0148 = -2.03

Next, we calculate the z-score for the upper bound:

z_upper = (0.28 + 0.03 - 0.28) / 0.0148 = 2.03

Using a standard normal distribution table or calculator, we can determine the proportion within these bounds.

P(-2.03 < Z < 2.03) ≈ 0.9772

Rounding to four decimal places, the probability that the proportion of rooms booked in a sample of 895 rooms would differ from the population proportion by less than 3% is approximately 0.9772.