Supplementary Problems
(a) Arrange the numbers 12, 56, 42, 21, 5, 18, 10, 3, 61, 34, 65, and 24 in an array and (b) determine the range.
Table 2.14 shows the frequency distribution for the number of minutes per week spent watching TV by 400 junior high students. With reference to this table determine:
(a) The upper limit of the fifth class
(b) The lower limit of the eighth class
(c) The class mark of the seventh class
(d) The class boundaries of the last class
(e) The class-interval size
(f) The frequency of the fourth class
(g) The relative frequency of the sixth class
(h) The percentage of students whose weekly viewing time does not exceed 600 minutes
(i) The percentage of students with viewing times greater than or equal to 900 minutes
(j) The percentage of students whose viewing times are at least 500 minutes but less than 1000 minutes
Viewing Time
Number of
(minutes)
Students
300–399
14
400–499
46
500–599
58
600–699
76
700–799
68
800–899
62
900–999
48
1000–1099
22
1100–1199
The Correct Answer and Explanation is :
Part 1: Array and Range of Numbers
(a) Arrange the numbers 12, 56, 42, 21, 5, 18, 10, 3, 61, 34, 65, and 24 in an array:
To arrange the numbers in ascending order:
Array (ascending order):
[ 3, 5, 10, 12, 18, 21, 24, 34, 42, 56, 61, 65 ]
(b) Determine the range:
The range is the difference between the largest and smallest numbers in the dataset.
Range = Largest value − Smallest value
Range = 65 − 3 = 62
Part 2: Frequency Distribution Analysis
Here is the frequency distribution table based on the information provided:
| Viewing Time (minutes) | Number of Students |
|---|---|
| 300–399 | 14 |
| 400–499 | 46 |
| 500–599 | 58 |
| 600–699 | 76 |
| 700–799 | 68 |
| 800–899 | 62 |
| 900–999 | 48 |
| 1000–1099 | 22 |
| 1100–1199 | Unknown |
(a) Upper limit of the fifth class (600–699 minutes):
The upper limit of the fifth class is 699.
(b) Lower limit of the eighth class (1000–1099 minutes):
The lower limit of the eighth class is 1000.
(c) Class mark of the seventh class (900–999 minutes):
The class mark is the midpoint of the class interval. For the seventh class (900–999):
Class mark = (\frac{900 + 999}{2} = \frac{1899}{2} = 949.5)
(d) Class boundaries of the last class (1100–1199 minutes):
To find the class boundaries, subtract 0.5 from the lower limit and add 0.5 to the upper limit.
Class boundaries:
Lower boundary = 1100 − 0.5 = 1099.5
Upper boundary = 1199 + 0.5 = 1199.5
(e) Class-interval size:
The class interval size is the difference between the lower limit of any class and the lower limit of the next class.
Class interval size = 400 − 300 = 100 minutes
(f) Frequency of the fourth class (600–699 minutes):
The frequency of the fourth class is 76 students.
(g) Relative frequency of the sixth class (800–899 minutes):
Relative frequency is calculated as the frequency of the class divided by the total number of students. The total number of students is 400.
Relative frequency of the sixth class = (\frac{62}{400} = 0.155)
(h) Percentage of students whose weekly viewing time does not exceed 600 minutes:
To find this, we sum the frequencies for all classes that have a viewing time of 600 minutes or less (i.e., classes 300–399, 400–499, 500–599).
Total students with ≤ 600 minutes = (14 + 46 + 58 = 118)
Percentage = (\frac{118}{400} \times 100 = 29.5\%)
(i) Percentage of students with viewing times greater than or equal to 900 minutes:
To find this, we sum the frequencies of the classes 900–999, 1000–1099, and 1100–1199.
Total students with ≥ 900 minutes = (48 + 22 + 0 = 70) (assuming 1100–1199 has a frequency of 0 for this question)
Percentage = (\frac{70}{400} \times 100 = 17.5\%)
(j) Percentage of students whose viewing times are at least 500 minutes but less than 1000 minutes:
To find this, we sum the frequencies for the classes 500–599, 600–699, 700–799, and 800–899.
Total students with ≥ 500 and < 1000 minutes = (58 + 76 + 68 + 62 = 264)
Percentage = (\frac{264}{400} \times 100 = 66\%)
Summary
In this analysis, we have found the following values for the given frequency distribution:
- Range of numbers: 62
- Class details (upper limit, lower limit, class marks, etc.)
- Relative and percentage frequencies for different time intervals.
This provides a thorough analysis of the distribution of students’ viewing times and highlights key statistical properties like frequencies and relative percentages.