Fifty cities provided information on vacancy rates (in %) in local apartments in the following frequency distribution.
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..
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.
a-1. To construct the frequency distribution, cumulative frequency distribution, and cumulative relative frequency distribution, we can use the given information:
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
Frequency distribution:
Vacancy Rate (%) Relative Frequency Frequency
0 < x ≤ 3 0.14 14 = (0.14 * 100)
3 < x ≤ 6 0.20 20 = (0.20 * 100)
6 < x ≤ 9 0.36 36 = (0.36 * 100)
9 < x ≤ 12 0.20 20 = (0.20 * 100)
12 < x ≤ 15 0.10 10 = (0.10 * 100)
Cumulative frequency distribution:
Vacancy Rate (%) Relative Frequency Frequency Cumulative Frequency
0 < x ≤ 3 0.14 14 14
3 < x ≤ 6 0.20 20 34 = 14 + 20
6 < x ≤ 9 0.36 36 70 = 34 + 36
9 < x ≤ 12 0.20 20 90 = 70 + 20
12 < x ≤ 15 0.10 10 100 = 90 + 10
Cumulative relative frequency distribution:
Vacancy Rate (%) Relative Frequency Frequency Cumulative Frequency Cumulative Relative Frequency
0 < x ≤ 3 0.14 14 14 0.14 = (14 / 100)
3 < x ≤ 6 0.20 20 34 0.34 = (34 / 100)
6 < x ≤ 9 0.36 36 70 0.70 = (70 / 100)
9 < x ≤ 12 0.20 20 90 0.90 = (90 / 100)
12 < x ≤ 15 0.10 10 100 1.00 = (100 / 100)
a-2. To find the number of cities with a vacancy rate of more than 6% but no more than 9%, we can subtract the cumulative frequency at 6% from the cumulative frequency at 9%.
Cumulative frequency at 6% = 34
Cumulative frequency at 9% = 70
Number of cities = Cumulative frequency at 9% - Cumulative frequency at 6%
Number of cities = 70 - 34
Number of cities = 36
Therefore, 36 cities had a vacancy rate of more than 6% but no more than 9%.
a-3. To find the number of cities with a vacancy rate of 9% or less, we can use the cumulative frequency at 9%.
Cumulative frequency at 9% = 70
Therefore, 70 cities had a vacancy rate of 9% or less.
a-4. To find the proportion of cities with a vacancy rate of more than 9%, we can subtract the cumulative relative frequency at 9% from 1 (as it represents the total proportion of the cities).
Cumulative relative frequency at 9% = 0.70
Proportion of cities with a vacancy rate of more than 9% = 1 - Cumulative relative frequency at 9%
Proportion of cities with a vacancy rate of more than 9% = 1 - 0.70
Proportion of cities with a vacancy rate of more than 9% = 0.30
Therefore, 0.30 (or 30%) of the cities had a vacancy rate of more than 9%.
b. Based on the given frequency distribution, we cannot determine the shape of the distribution using a histogram. The information provided only gives us the relative frequency for each category, not the actual values.