The following are the weekly wages in rupees of 30 workers in a firm: 140, 139, 126, 114, 100 , 88, 62, 77, 99, 103, 108, 129, 144, 148, 134, 63, 69, 148, 132, 118, 142, 116, 123, 104, 95, 80 , 85, 106, 123, 133 The firm gave bonus of Rs. 10, 15, 20, 25, 30 and 35 for individuals in the respective salary slab; Exceeding 60 but not exceeding 75; Exceeding 75 but not exceeding 90; and so on up to exceeding 135 but not exceeding 150. Generate the classes and draw the histogram of the data.
Answers
Solution : The frequency of a class can be obtained by simply counting the number of observations that are falling in the interval. Class mid points is given by, Class mid point…
Answer:
The relative frequencies for each class interval is calculated by dividing the number of workers with the wages within this interval by the total number of workers (30).
Step-by-step explanation:
From the above question,
They have given :
The following are the weekly wages in rupees of 30 workers of a firm: 140 139 126 114 100 88 62 77 99 103 108 129 144 148 134 63 69 148 132 118 142 116 123 104 95 80 85 106 123 133 (a) Construct a numerical frequency distribution of weekly wages by using the classes: 62-79, 80-97, 98-115, 116-133 and 134-151, as class intervals. (b) Also show the following points on the same table. (i) Class mid points (ii) Class boundaries (iii) Relative frequencies (iv) Cumulative frequencies
To construct a numerical frequency distribution of the weekly wages of 30 workers of a firm, we can use the class intervals 62-79, 80-97, 98-115, 116-133, and 134-151. The class midpoints would be 70.5, 88.5, 106.5, 124.5 and 143.5, respectively.
The class boundaries calculate by subtracting 0.5 from the lower class limit and adding 0.5 to the higher class limit.
The class boundaries for the intervals would thus be 61.5-80.5, 79.5-98.5, 97.5-116.5, 115.5-134.5, and 133.5-152.5.
The relative frequencies for each class interval is calculated by dividing the number of workers with the wages within this interval by the total number of workers (30).
The cumulative frequencies are calculated by adding each of the frequencies with the frequency of the class that comes before it.
The frequency distribution for the weekly wages is displayed in the table below:
Class Class Class Relative Cumulative
Interval Midpoints Boundaries Frequency Frequency Frequency
62-79 70.5 61.5-80.5 7 0.23 0.23
80-97 88.5 79.5-98.5 8 0.27 0.50
98-115 106.5 97.5-116.5 7 0.23 0.73
116-133 124.5 115.5-134.5 7 0.23 0.96
134-151 143.5 133.5-152.5 1 0.03 1.00
(a) Numerical frequency distribution of weekly wages:
Class Interval (CI) Frequency (f)
62-79 6
80-97 8
98-115 7
116-133 7
134-151 2
Class Boundaries (CB):
For a class interval of the form "a-b", the lower class boundary (LCB) is "a" and the upper class boundary (UCB) is "b".
CI LCB UCB
62-79 61 79
80-97 80 97
98-115 98 115
116-133 116 133
134-151 134 151
Class Mid Points (CMP):
To find the class mid point for a class interval of the form "a-b", the formula is (a + b)/2.
CI CMP
62-79 70.5
80-97 88.5
98-115 106.5
116-133 124.5
134-151 142.5
Relative Frequencies (RF):
Relative frequency for a class is the frequency of that class divided by the total number of observations.
CI f RF
62-79 6 6/30 = 0.2
80-97 8 8/30 = 0.2667
98-115 7 7/30 = 0.2333
116-133 7 7/30 = 0.2333
134-151 2 2/30 = 0.0667
Cumulative Frequencies (CF):
The cumulative frequency for a class is the sum of the frequencies of that class and all previous classes.
CI f CF
62-79 6 6
80-97 8 6 + 8 = 14
98-115 7 6 + 8 + 7 = 21
116-133 7 6 + 8 + 7 + 7 = 28
134-151 2 6 + 8 + 7 + 7 + 2 = 30
(b) The final table combining all the points is as follows:
CI LCB UCB CMP f RF CF
62-79 61 79 70.5 6 0.2 6
80-97 80 97 88.5 8 0.2667 14
98-115 98 115 106.5 7 0.2333 21
116-133 116 133 124.5 7 0.2333 28
134-151 134 151 142.5 2 0.0667 30
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