An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (1)
Construct a grouped frequency distribution table with class width of 10.
Calculate the mean age using the formula: Assumed Mean + (Sum of (Frequency * Deviation) / Total Frequency) = 54.5 + 50 − 95 = 52.6 .
Determine the class containing the upper 20% (10 head teachers) using cumulative frequencies.
Interpolate to find the recommended retirement age: 60 + 9 40 − 35 ∗ 10 ≈ 65.56 .
65.56
Explanation
Construct Grouped Frequency Distribution Table First, we need to organize the given data into a grouped frequency distribution table. To do this, we determine the range, number of classes, and class width. The data ranges from 14 to 91, giving a range of 77. Using Sturges' rule, the number of classes is approximately 6.64, so we choose 9 classes for convenience, each with a width of 10.
Grouped Frequency Distribution The classes are: 10-20, 20-30, 30-40, 40-50, 50-60, 60-70, 70-80, 80-90, and 90-100. We then count the number of head teachers (frequency) falling into each class. This gives us the following frequency distribution:
Class
Frequency
10-20
1
20-30
3
30-40
7
40-50
10
50-60
14
60-70
9
70-80
3
80-90
2
90-100
1
Calculate the Mean Age Next, we calculate the mean age using the assumed mean of 54.5. We find the midpoint of each class, subtract the assumed mean from each midpoint to get the deviation, multiply each deviation by its corresponding frequency, and sum these products. Finally, we divide the sum by the total frequency (50) to get the mean deviation, which we add to the assumed mean to find the actual mean.
The class midpoints are: 15, 25, 35, 45, 55, 65, 75, 85, 95. The deviations from the assumed mean (54.5) are: -39.5, -29.5, -19.5, -9.5, 0.5, 10.5, 20.5, 30.5, 40.5. The products of frequency and deviation (fd) are: -39.5, -88.5, -136.5, -95.0, 7.0, 94.5, 61.5, 61.0, 40.5. The sum of fd is -95.0. The mean deviation is -95.0 / 50 = -1.9. The actual mean is 54.5 + (-1.9) = 52.6.
Determine the Recommended Retirement Age To find the recommended retirement age, we need to determine the age corresponding to the upper 20% of the head teachers. Since there are 50 head teachers, the upper 20% corresponds to 10 head teachers. We need to find the age such that 40 head teachers are younger than that age.
Looking at the cumulative frequencies, we see:
Class
Frequency
Cumulative Frequency
10-20
1
1
20-30
3
4
30-40
7
11
40-50
10
21
50-60
14
35
60-70
9
44
70-80
3
47
80-90
2
49
90-100
1
50
The cumulative frequency reaches 35 in the 50-60 class, so the 40th head teacher falls within the 60-70 class. We use interpolation to find the age:
L + f n − c f × h
Where: L = Lower boundary of the class (60) n = Position of the value (40) c f = Cumulative frequency of the class before (35) f = Frequency of the class (9) h = Class width (10)
60 + 9 40 − 35 × 10 = 60 + 9 5 × 10 = 60 + 5.555... = 65.555...
Therefore, the recommended retirement age is approximately 65.56.
Final Answer The mean age of the head teachers is 52.6, and the recommended retirement age based on the upper 20% is approximately 65.56.
Examples
Understanding the age distribution and retirement trends of professionals, like head teachers, can help in workforce planning and policy development. For example, knowing the mean age helps in budgeting for training programs, while estimating retirement age helps in planning for replacements and succession. This type of analysis is common in human resources and educational administration to ensure a stable and experienced workforce.
