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 19
1,250 < x ≤ 1,500 16
1,500 < x ≤ 1,750 8
1,750 < x ≤ 2,000 14
2,000 < x ≤ 2,250 8
2,250 < x ≤ 2,500 11
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 2,000 but no more than 2,250 people? (Round your answer to 2 decimal places.)
a-3. What proportion of the time was attendance 1,500 or less? (Round your answer to 2 decimal places.)
a-4. What proportion of the time was attendance more than 1,500 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.
To construct the relative frequency distribution, we need to divide the frequency of each attendance range by the total frequency. To construct the cumulative relative frequency distribution, we need to add up the relative frequencies as we go.
First, let's calculate the total frequency:
Total Frequency = 19 + 16 + 8 + 14 + 8 + 11 = 76
a-1. Relative Frequency Distribution:
Attendance Range Frequency Relative Frequency
1,000 < x ≤ 1,250 19 19/76 ≈ 0.25
1,250 < x ≤ 1,500 16 16/76 ≈ 0.21
1,500 < x ≤ 1,750 8 8/76 ≈ 0.11
1,750 < x ≤ 2,000 14 14/76 ≈ 0.18
2,000 < x ≤ 2,250 8 8/76 ≈ 0.11
2,250 < x ≤ 2,500 11 11/76 ≈ 0.14
a-2. The proportion of the time attendance was more than 2,000 but no more than 2,250 people can be calculated by summing up the relative frequencies of the corresponding categories:
Proportion = 0.11
a-3. The proportion of the time attendance was 1,500 or less can be calculated by summing up the relative frequencies of the categories that include 1,500 or less:
Proportion = 0.25 + 0.21 + 0.11 = 0.57
a-4. The proportion of the time attendance was more than 1,500 people can be calculated by summing up the relative frequencies of the categories that are above 1,500:
Proportion = 0.11 + 0.18 + 0.11 + 0.14 = 0.54
b. The correct statement regarding the shape of the distribution using a histogram is: The distribution is not symmetric.
a-1. To construct the relative frequency distribution, divide each frequency by the total number of observations. The cumulative relative frequency is calculated by summing up all the relative frequencies up to each class interval.
Attendance Frequency Relative Frequency Cumulative Relative Frequency
1,000 < x ≤ 1,250 19 19/76 ≈ 0.25 0.25
1,250 < x ≤ 1,500 16 16/76 ≈ 0.21 0.25 + 0.21 ≈ 0.46
1,500 < x ≤ 1,750 8 8/76 ≈ 0.11 0.46 + 0.11 ≈ 0.57
1,750 < x ≤ 2,000 14 14/76 ≈ 0.18 0.57 + 0.18 ≈ 0.75
2,000 < x ≤ 2,250 8 8/76 ≈ 0.11 0.75 + 0.11 ≈ 0.86
2,250 < x ≤ 2,500 11 11/76 ≈ 0.14 0.86 + 0.14 = 1.00
a-2. To find the proportion of the time attendance was more than 2,000 but no more than 2,250 people, we calculate the cumulative relative frequency for the interval 2,000 < x ≤ 2,250. The cumulative relative frequency for this interval is 0.86. So, the proportion of the time attendance was in this range is 0.86.
a-3. To find the proportion of the time attendance was 1,500 or less, we sum up the relative frequencies for the intervals 1,000 < x ≤ 1,250 and 1,250 < x ≤ 1,500. The sum of these relative frequencies is 0.25 + 0.21 = 0.46. So, the proportion of the time attendance was 1,500 or less is 0.46.
a-4. To find the proportion of the time attendance was more than 1,500 people, we calculate the cumulative relative frequency for the interval 1,500 < x ≤ 2,500. The cumulative relative frequency for this interval is 1.00. So, the proportion of the time attendance was more than 1,500 people is 1.00.
b. The statement "The distribution is not symmetric" is correct regarding the shape of the distribution using a histogram.