Final answer:
Using the normal distribution and z-scores, we find that 99.74% of rents are between $470 and $1,970, 0.13% are less than $470, and 15.87% are greater than $1,470.
Step-by-step explanation:
To answer the question about the percentage of rents within certain ranges, we need to use z-scores and the normal distribution table. Z-scores are calculated by subtracting the mean from the observation and then dividing by the standard deviation. In this case, the mean is $1,220 and the standard deviation is $250.
Calculating Z-Scores
For $470:
z = (470 - 1220) / 250 = -3
For $1,970:
z = (1970 - 1220) / 250 = 3
Using the z-table, we find that the area to the left of z = 3 is nearly 0.9987, and the area to the left of z = -3 is nearly 0.0013.
To find the percentage of rents between $470 and $1,970, we subtract the smaller area from the larger area:
0.9987 - 0.0013 = 0.9974
Multiplying by 100, we get 99.74% of rents fall between $470 and $1,970.
Percentage of Rents Less Than $470
The area to the left of z = -3 is 0.0013, or 0.13%.
Percentage of Rents Greater Than $1,470
For $1,470:
z = (1470 - 1220) / 250 = 1
Using the z-table, the area to the left of z = 1 is approximately 0.8413.
Since we want the area greater than $1,470, we subtract this from 1:
1 - 0.8413 = 0.1587
Multiplying by 100, we get 15.87% of rents are greater than $1,470.