第10类NCERT解决方案-第14章统计-练习14.4

问题1.以下分配给出了如果有工厂，则每天有50名工人的日收入。将上面的分布转换为小于类型的累积频率分布，并绘制其形状。

Daily income in Rupees	100-120	120-140	140-160	160-180	180-200
Number of workers	12	14	8	6	10

解决方案：

According to the question, we convert the given distribution to a less than type cumulative frequency distribution, 

Now according to the table we plot the points that are corresponding to the ordered pairs (120, 12), (140, 26), (160, 34), (180, 40), and (200, 50) on a graph paper. Here x-axis represents the upper limit and y-axis represent the frequency. The curve obtained from the graph is known as less than type ogive curve. 

问题2：在对35名班级学生进行身体检查时，他们的体重记录如下：

Daily income	Frequency	Cumulative Frequency
Less than 120	12	12
Less than 140	14	26
Less than 160	8	34
Less than 180	6	40
Less than 200	10	50

为给定数据绘制小于类型的ogive。因此，从图表中获取体重的中位数，并使用公式验证结果。

解决方案：

According to the give table, we get the points (38, 0), (40, 3), (42, 5), (44, 9), (46, 14), (48, 28), (50, 32) and (52, 35). Now using these points we draw an ogive, where the x-axis represents the upper limit and y-axis represents the frequency. The curve obtained is known as less than type ogive.

Now, locate the point 17.5 on the y-axis and draw a line parallel to the x-axis cutting the curve at a point. From this point, now we draw a perpendicular line to the x-axis and the intersection point which is perpendicular to x-axis is the median of the given data. After, locating point now we create a table to find the mode:

Class interval	Number of students(Frequency)	Cumulative Frequency
Less than 38	0	0
Less than 40	3 – 0 = 3	3
Less than 42	5 – 3 = 2	8
Less than 44	9 – 5 = 4	9
Less than 46	14 – 9 = 5	14
Less than 48	28 – 14 = 14	28
Less than 50	32 – 28 = 4	32
Less than 52	35 – 22 = 3	35

The class 46 – 48 has the maximum frequency, hence, this is the modal class

l = 46, h = 2, f₁ = 14, f₀ = 5 and f₂ = 4

Now we find the mode:

Mode = $l+\left(\frac{f_1-f_0}{2f_1-f_0-f_2}\right)\times h$

On substituting the values in the given formula, we get

= 46 + 0.95 = 46.95

Hence, the mode is verified.

问题3.下表列出了一个村庄的100个农场的每公顷小麦的产量。

Weight in kg	Number of students
Less than 38	0
Less than 40	3
Less than 42	5
Less than 44	9
Less than 46	14
Less than 48	28
Less than 50	32
Less than 52	35

将分布更改为多于类型的分布，并绘制其外观。

解决方案：

According to the question, we change the distribution to a more than type distribution.

Production Yield (kg/ha)	Number of farms
More than or equal to 50	100
More than or equal to 55	100 – 2 = 98
More than or equal to 60	98 – 8 = 90
More than or equal to 65	90 – 12 = 78
More than or equal to 70	78 – 24 = 54
More than or equal to 75	54 – 38 = 16

Now, according to the table we draw the ogive by plotting the points. Here, the a-axis represents the upper limit and y-axis represents the cumulative frequency. And the points are(50, 100), (55, 98), (60, 90), (65, 78), (70, 54) and (75, 16) on this graph paper. The graph obtained is known as more than type ogive curve.