Step 1 :Given that the mean number of rooms in the hotel chain is 98 and the standard deviation is 4.
Step 2 :The empirical rule, also known as the 68-95-99.7 rule, states that for a normal distribution, almost all data will fall within three standard deviations of the mean. Specifically, 68% of data falls within the first standard deviation, 95% falls within two standard deviations, and 99.7% falls within three standard deviations.
Step 3 :A hotel with more than 102 rooms is more than one standard deviation away from the mean. According to the empirical rule, the probability that a randomly chosen hotel will have more than 102 rooms is the probability that it falls more than one standard deviation above the mean.
Step 4 :To find this, we can subtract the probability that a hotel falls within one standard deviation of the mean (68%) from 100%. This will give us the probability that a hotel falls more than one standard deviation away from the mean, either above or below.
Step 5 :Since we're only interested in hotels with more than 102 rooms (i.e., more than one standard deviation above the mean), we divide this result by 2.
Step 6 :\(mean = 98\)
Step 7 :\(std\_dev = 4\)
Step 8 :\(one\_std\_dev\_above = 102\)
Step 9 :\(prob\_within\_one\_std\_dev = 68\)
Step 10 :\(prob\_more\_than\_one\_std\_dev = 100 - prob\_within\_one\_std\_dev = 32\)
Step 11 :\(prob\_more\_than\_102\_rooms = prob\_more\_than\_one\_std\_dev / 2 = 16.0\)
Step 12 :Final Answer: The probability that a randomly chosen hotel will have more than 102 rooms is \(\boxed{16\%}\).