Q. 11 - Elementary Statistics

Question

A quantitative data set has been grouped by using cutpoint grouping with equal-width classes.

a) If the lower and upper cutpoints of the first class are 5 and 15, respectively, what is the common class width?

b) What is the midpoint of the second class?

c) What are the lower and upper cutpoints of the third class?
d)Which class would contain an observation of $32.4$?

Step-by-Step Solution

Answer

Ans:  

(a)  The class width is 10. 

(b)  The midpoint of the second class is $20$.

(c)  The lower and upper cutpoints of the third class are $25 and 35$.

(d)  The third class cutpoints are $25 and 35$. Thus, the third class cutpoints include observation 32.4. 

Step 1. Given information.

given,

     A quantitative data set has been grouped by using cutpoint grouping with equal-width classes. 

Step 2. A set of quantitative data is collected using collections of points of equal width.

a) If the lower and upper cutpoints of the first class $5 and 15$, we find the width of the standard class is shown below:

The minimum limit for the first class is 5.

The maximum limit for the first phase is 15.

   $$\begin{gathered} Class width = Upper limit - Lowerlimit \\ = 15-5 =10 \end{gathered}$$

Therefore, the class width is 10.

Step 3. (b) Now, find the second class midpoint given below:

The lower cutpoint for the first class is $5$.

The upper cutpoint for the first class is $15$.

In Cutpoint integration the lower cut area of the second phase is equal to the uppercut area of the first phase.

So the Lower cutpoint of the second phase is $15$

The more cutting point of the second phase is,

   $$\begin{gathered} = Lower cutpoint of the secon class + Class width \\ =15+10 =25 \end{gathered}$$

By definition, the midpoint is given by,

   $$\begin{gathered} Midpoint=\frac{Lower cutpoint +Upper cutpoint }{2} \\ =\frac{15+25}{2} = 20 \end{gathered}$$

Therefore, the midpoint of the second class is $20$.

Step 4. Find the lower and upper cutpoints for the third class are given below:

The Lower cutpoint of the second class is $15$.

The upper cutpoint for the second phase is $25$.

In Cutpoint collection the lower cut area of the third phase is equal to the uppercut area of the second phase.

So the minimum point for the third stage is $25$

The top cutpoint for the third stage is,

    $$\begin{gathered} = Lower limit of the third class + Class width \\ =25+10 =35 \end{gathered}$$

Thus, the lower and upper cutpoints of the third class are $25 and 35$.

Step 5. (d) The class would contain an observation of 32 . 4 ,

The third class cutpoints are $25 and 35$
 Thus, the third class cutpoints include observation 32.4.