Fifty cities provided information on vacancy rates (in %) for local apartments in the following relative frequency distribution.
Vacancy Rate (in percent) Frequency
0 < x ≤ 3 5
3 < x ≤ 6 5
6 < x ≤ 9 10
9 < x ≤ 12 20
12 < x ≤ 15 10
Calculate the average vacancy rate. (Round your answer to 2 decimal places.)
To calculate the average vacancy rate, we need to find the weighted average of the vacancy rates.
First, we need to find the midpoint of each class interval. The midpoint is calculated by taking the average of the upper and lower limits of the class interval.
For the first class interval 0 < x ≤ 3, the midpoint is (0 + 3) / 2 = 1.5
For the second class interval 3 < x ≤ 6, the midpoint is (3 + 6) / 2 = 4.5
For the third class interval 6 < x ≤ 9, the midpoint is (6 + 9) / 2 = 7.5
For the fourth class interval 9 < x ≤ 12, the midpoint is (9 + 12) / 2 = 10.5
For the fifth class interval 12 < x ≤ 15, the midpoint is (12 + 15) / 2 = 13.5
Next, we need to calculate the weighted sum of the vacancy rates. This is done by multiplying each vacancy rate by the frequency and summing these values.
For the first class interval, the weighted sum is 1.5 * 5 = 7.5
For the second class interval, the weighted sum is 4.5 * 5 = 22.5
For the third class interval, the weighted sum is 7.5 * 10 = 75
For the fourth class interval, the weighted sum is 10.5 * 20 = 210
For the fifth class interval, the weighted sum is 13.5 * 10 = 135
Finally, we calculate the average vacancy rate by dividing the weighted sum by the total number of cities.
The total number of cities is 5 + 5 + 10 + 20 + 10 = 50
The weighted sum of the vacancy rates is 7.5 + 22.5 + 75 + 210 + 135 = 450
The average vacancy rate is 450 / 50 = 9.
Therefore, the average vacancy rate is 9.00%.
To calculate the average vacancy rate, we need to find the midpoint of each class interval and multiply it by its corresponding frequency. Then, we sum up these products and divide by the total frequency.
First, we calculate the midpoint for each class interval by taking the average of the lower and upper limits:
0 < x ≤ 3: (0 + 3) / 2 = 1.5
3 < x ≤ 6: (3 + 6) / 2 = 4.5
6 < x ≤ 9: (6 + 9) / 2 = 7.5
9 < x ≤ 12: (9 + 12) / 2 = 10.5
12 < x ≤ 15: (12 + 15) / 2 = 13.5
Next, we calculate the sum of the products of the midpoints and frequencies:
(1.5 * 5) + (4.5 * 5) + (7.5 * 10) + (10.5 * 20) + (13.5 * 10) = 30 + 22.5 + 75 + 210 + 135 = 472.5
Finally, we divide this sum by the total frequency, which is the sum of all frequencies:
5 + 5 + 10 + 20 + 10 = 50
Therefore, the average vacancy rate is:
472.5 / 50 = 9.45
Rounded to two decimal places, the average vacancy rate is 9.45%.