The following sample data represent the per kg cost (in kwacha) of raw materials used in processing batches of a chemical feedstock.
12.01 14.90 11.24 14.40 12.87
11.30 16.98 15.23 12.38 12.90
12.29 11.51 12.06 15.06 11.16
10.12 14.82 17.02 12.56 13.95
11.13 12.14 13.41 15.67 12.17
12.15 12.57 14.21 13.21 16.50
13.81 15.58 11.10 12.03 11.20
12.27 17.05 12.57 13.11 13.31
13.84 11.62 12.33 16.31 12.74
14.25 13.63 13.34 13.43 14.78
A) construct a frequency distribution with class width=K2. 00 starting with K10. 00, and give both the frequency and the cumulative relative frequency for each class.
B) compute the mean, the variance, the mode and the median.
A) To construct a frequency distribution with a class width of K2.00 starting with K10.00, we need to determine the class intervals.
First, we determine the minimum and maximum values in the data:
Minimum value = 10.12
Maximum value = 17.05
Next, we determine the range of the data:
Range = Maximum value - Minimum value
Range = 17.05 - 10.12
Range = 6.93
To determine the number of classes, we use the formula:
Number of classes = (Range / Class width) + 1
Class width = K2.00
Number of classes = (6.93 / 2.00) + 1
Number of classes = 3.465 + 1
Number of classes = 4.465 (round up to 5 classes)
Now we determine the class intervals:
Class interval = Range / Number of classes
Class interval = 6.93 / 5
Class interval = 1.386 (round up to 1.40)
Starting with K10.00, the class intervals will be:
10.00 - 11.40
11.40 - 12.80
12.80 - 14.20
14.20 - 15.60
15.60 - 17.00
Now we count the frequency of each class:
10.00 - 11.40: 11.16, 11.30, 11.10, 10.12, 11.13, 11.20, 11.51, 11.14 (Frequency: 8)
11.40 - 12.80: 12.01, 12.29, 12.06, 11.82, 12.57, 12.14, 12.57, 12.33, 12.57, 12.74 (Frequency: 10)
12.80 - 14.20: 14.90, 12.87, 13.95, 13.81, 13.21, 13.34, 13.43 (Frequency: 7)
14.20 - 15.60: 14.40, 15.06, 15.67, 15.58, 13.84 (Frequency: 5)
15.60 - 17.00: 16.98, 16.50, 17.02, 16.31 (Frequency: 4)
Finally, we calculate the cumulative relative frequency for each class:
10.00 - 11.40: Frequency = 8, Cumulative Relative Frequency = 8 / 50 = 0.16
11.40 - 12.80: Frequency = 10, Cumulative Relative Frequency = (8 + 10) / 50 = 0.36
12.80 - 14.20: Frequency = 7, Cumulative Relative Frequency = (8 + 10 + 7) / 50 = 0.5
14.20 - 15.60: Frequency = 5, Cumulative Relative Frequency = (8 + 10 + 7 + 5) / 50 = 0.6
15.60 - 17.00: Frequency = 4, Cumulative Relative Frequency = (8 + 10 + 7 + 5 + 4) / 50 = 0.68
B) To compute the mean, variance, mode, and median:
Mean = sum of all values / number of values
Mean = (12.01 + 14.90 + ... + 13.34 + 14.78) / 50
Variance = sum of (each value - mean)^2 / (number of values - 1)
Variance = ((12.01 - mean)^2 + (14.90 - mean)^2 + ... + (13.34 - mean)^2 + (14.78 - mean)^2) / (50 - 1)
Mode = the value that appears most frequently in the dataset
Median = the middle value of the ordered dataset (if the number of values is odd) or the average of the two middle values (if the number of values is even)