The manager at a water park constructed the following frequency distribution to summarize attendance in July and August.
Attendance Frequency
1,000 < x ≤ 1,250 7
1,250 < x ≤ 1,500 11
1,500 < x ≤ 1,750 17
1,750 < x ≤ 2,000 5
2,000 < x ≤ 2,250 4
2,250 < x ≤ 2,500 6
a-1. Construct the relative frequency distribution and the cumulative relative frequency distribution (Round your answers to 2 decimal places.)
a-2. What proportion of the time was attendance more than 1,500 but no more than 1,750 people? (Round your answer to 2 decimal places.)
a-3. What proportion of the time was attendance 1,250 or less? (Round your answer to 2 decimal places.)
a-4. What proportion of the time was attendance more than 1,250 people? (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 symmetric.
The distribution is not symmetric.
a-1. To construct the relative frequency distribution, divide each frequency by the total number of observations (sum of all frequencies). To construct the cumulative relative frequency distribution, add up the relative frequencies for each category, starting from the lowest category.
Attendance Range Frequency Relative Frequency Cumulative Relative Frequency
1,000 < x ≤ 1,250 7 0.14 0.14
1,250 < x ≤ 1,500 11 0.22 0.36
1,500 < x ≤ 1,750 17 0.34 0.70
1,750 < x ≤ 2,000 5 0.10 0.80
2,000 < x ≤ 2,250 4 0.08 0.88
2,250 < x ≤ 2,500 6 0.12 1.00
a-2. The proportion of the time attendance was more than 1,500 but no more than 1,750 people is the relative frequency for that category, which is 0.34.
a-3. The proportion of the time attendance was 1,250 or less can be calculated by adding up the relative frequencies for the first two categories: 0.14 + 0.22 = 0.36.
a-4. The proportion of the time attendance was more than 1,250 people can be calculated by subtracting the proportion of the time attendance was 1,250 or less (0.36) from 1: 1 - 0.36 = 0.64.
b. The statement "The distribution is not symmetric" is correct regarding the shape of the distribution using a histogram.
a-1. To construct the relative frequency distribution, we need to find the proportion of each frequency relative to the total number of observations.
To calculate the relative frequency, divide each frequency by the sum of all frequencies:
Attendance Frequency Relative Frequency
1,000 < x ≤ 1,250 7 7 / (7+11+17+5+4+6) = 7 / 50 = 0.14
1,250 < x ≤ 1,500 11 11 / 50 = 0.22
1,500 < x ≤ 1,750 17 17 / 50 = 0.34
1,750 < x ≤ 2,000 5 5 / 50 = 0.10
2,000 < x ≤ 2,250 4 4 / 50 = 0.08
2,250 < x ≤ 2,500 6 6 / 50 = 0.12
To construct the cumulative relative frequency distribution, we sum up the relative frequencies as we go down the table:
Attendance Frequency Relative Frequency Cumulative Relative Frequency
1,000 < x ≤ 1,250 7 0.14 0.14
1,250 < x ≤ 1,500 11 0.22 0.36
1,500 < x ≤ 1,750 17 0.34 0.70
1,750 < x ≤ 2,000 5 0.10 0.80
2,000 < x ≤ 2,250 4 0.08 0.88
2,250 < x ≤ 2,500 6 0.12 1.00
a-2. The proportion of the time attendance was more than 1,500 but no more than 1,750 people is given by the difference in cumulative relative frequency between those two attendance ranges:
Cumulative Relative Frequency for 1,500 < x ≤ 1,750 = 0.70
Cumulative Relative Frequency for 1,000 < x ≤ 1,500 = 0.36
Proportion = 0.70 - 0.36 = 0.34
Therefore, the proportion of the time attendance was more than 1,500 but no more than 1,750 people is 0.34 (34%).
a-3. The proportion of the time attendance was 1,250 or less is given by the cumulative relative frequency for 1,250 < x ≤ 1,500:
Cumulative Relative Frequency for 1,250 < x ≤ 1,500 = 0.36
Therefore, the proportion of the time attendance was 1,250 or less is 0.36 (36%).
a-4. The proportion of the time attendance was more than 1,250 people is given by the difference in cumulative relative frequency between 1,250 < x ≤ 1,500, 1,500 < x ≤ 1,750, 1,750 < x ≤ 2,000, 2,000 < x ≤ 2,250, and 2,250 < x ≤ 2,500:
Cumulative Relative Frequency for 1,250 < x ≤ 1,500 = 0.36
Cumulative Relative Frequency for 1,500 < x ≤ 1,750 = 0.70
Cumulative Relative Frequency for 1,750 < x ≤ 2,000 = 0.80
Cumulative Relative Frequency for 2,000 < x ≤ 2,250 = 0.88
Cumulative Relative Frequency for 2,250 < x ≤ 2,500 = 1.00
Proportion = 1.00 - 0.36 = 0.64
Therefore, the proportion of the time attendance was more than 1,250 people is 0.64 (64%).
b. The statement "The distribution is not symmetric" is correct regarding the shape of the distribution using a histogram.