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# System prepended metadata

title: PID-729

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Given the frequency distribution of heights (in cms) of $390$ cows, with class marks $CM1=112$ and $CM2=117$, and frequencies $fi: 6\ 25\ 48\ 72\ 116\ 60\ 38\ 22\ 3$, we can determine the class interval size, class limits, class boundaries, and class marks as follows:

a. The class interval size is equal to the difference between the two class marks, which is:

$$\text{Class interval size} = CM2 - CM1 = 117 - 112 = 5 cm$$

b. To determine the class limits, we need to find the upper and lower limits of each class. We can start by finding the lower limit of the first class, which is given by:

$$\text{Lower limit of first class} = CM1 - (\text{class interval size}/ 2) = 112 - (5 / 2) = 109.5 \text{ cm}.$$

The upper limit of the first class is found by adding the class interval size to the lower limit:

\begin{aligned}
\text{Upper limit of first class} &\text{= Lower limit of first class + class interval size}\\
& = 109.5 + 5 = 114.5 \text{ cm}.
\end{aligned}

We can repeat this process to find the lower and upper limits of each class. The class limits are:

Class 1: 109.5 - 114.5 cm
Class 2: 114.5 - 119.5 cm
Class 3: 119.5 - 124.5 cm
Class 4: 124.5 - 129.5 cm
Class 5: 129.5 - 134.5 cm
Class 6: 134.5 - 139.5 cm
Class 7: 139.5 - 144.5 cm
Class 8: 144.5 - 149.5 cm
Class 9: 149.5 - 154.5 cm

c. The class boundaries are found by adding and subtracting half of the class interval size from the class limits. The class boundaries are:

Class 1: 109 - 115 cm
Class 2: 114 - 120 cm
Class 3: 119 - 125 cm
Class 4: 124 - 130 cm
Class 5: 129 - 135 cm
Class 6: 134 - 140 cm
Class 7: 139 - 145 cm
Class 8: 144 - 150 cm
Class 9: 149 - 155 cm

d. The class marks are simply the midpoints of each class interval. The class marks are:

Class 1: (109.5 + 114.5) / 2 = 112 cm
Class 2: (114.5 + 119.5) / 2 = 117 cm
Class 3: (119.5 + 124.5) / 2 = 122 cm
Class 4: (124.5 + 129.5) / 2 = 127 cm
Class 5: (129.5 + 134.5) / 2 = 132 cm
Class 6: (134.5 + 139.5) / 2 = 137 cm
Class 7: (139.5 + 144.5) / 2 = 142 cm
Class 8: (144.5 + 149.5) / 2 = 147 cm
Class 9: (149.5 + 154.5) / 2 = 152 cm

Therefore, the class interval size is 5 cm, the class limits are given above, the class boundaries are given above, and the class marks are given above.