Fifty cities provided information on vacancy rates (in %) in local apartments in the following frequency distribution.

Question

Vacancy Rate (%) Relative Frequency
0 < x ≤ 3 0.14
3 < x ≤ 6 0.20
6 < x ≤ 9 0.36
9 < x ≤ 12 0.20
12 < x ≤ 15 0.10

a-1. Construct the frequency distribution, the cumulative frequency distribution, and the cumulative relative frequency distribution. (Round "Cumulative Relative Frequency" answers to 2 decimal places.)

a-2. How many of the cities had a vacancy rate of more than 6% but no more than 9%?

a-3. How many of the cities had a vacancy rate of 9% or less?

a-4. What proportion of the cities had a vacancy rate of more than 9%? (Round your answer to 2 decimal places.)

b. Which statement is correct regarding the shape of the distribution using a histogram?

multiple choice
The distribution is relatively symmetric.
The distribution is positively skewed.
The distribution is negatively skewed.
Symmetry or skewness cannot be determined with the available information..

Answer 1

a-1. To construct the frequency distribution, we can use the given relative frequencies to calculate the frequencies for each category:

Vacancy Rate (%) Relative Frequency Frequency
0 < x ≤ 3 0.14 (0.14)(50) = 7
3 < x ≤ 6 0.20 (0.20)(50) = 10
6 < x ≤ 9 0.36 (0.36)(50) = 18
9 < x ≤ 12 0.20 (0.20)(50) = 10
12 < x ≤ 15 0.10 (0.10)(50) = 5

The frequency distribution is as follows:

Vacancy Rate (%) Frequency
0 < x ≤ 3 7
3 < x ≤ 6 10
6 < x ≤ 9 18
9 < x ≤ 12 10
12 < x ≤ 15 5

To construct the cumulative frequency distribution, we simply add up the frequencies from the frequency distribution:

Vacancy Rate (%) Frequency Cumulative Frequency
0 < x ≤ 3 7 7
3 < x ≤ 6 10 17
6 < x ≤ 9 18 35
9 < x ≤ 12 10 45
12 < x ≤ 15 5 50

To construct the cumulative relative frequency distribution, we add up the relative frequencies:

Vacancy Rate (%) Relative Frequency Cumulative Relative Freq.
0 < x ≤ 3 0.14 0.14
3 < x ≤ 6 0.20 0.34
6 < x ≤ 9 0.36 0.70
9 < x ≤ 12 0.20 0.90
12 < x ≤ 15 0.10 1.00

a-2. To find the number of cities with a vacancy rate of more than 6% but no more than 9%, we subtract the cumulative frequency for 6 < x ≤ 9 from the cumulative frequency for 3 < x ≤ 6. In this case, it is 18 - 10 = 8 cities.

a-3. To find the number of cities with a vacancy rate of 9% or less, we can use the cumulative frequency for 9 < x ≤ 12, which is 45 cities.

a-4. To find the proportion of cities with a vacancy rate of more than 9%, we can subtract the cumulative relative frequency for 9 < x ≤ 12 from 1: 1 - 0.9 = 0.1. So, 10% of the cities had a vacancy rate of more than 9%.

b. Symmetry or skewness cannot be determined with the available information.