Q.12 - Elementary Statistics

Question

A quantitative data set has been grouped by using cut-point grouping with equal-width classes of width $8$.

a. If the midpoint of the first class is $10$, what are its lower and upper cut-points?

b. What is the class midpoint of the second class?

c. What are the lower and upper cut-points of the third class?

d. Which class would contain an observation of $22$?

Step-by-Step Solution

Answer

(a)  The lower and upper cut-points are respectively $6$ and $14$.

(b) The midpoint is$18$.

(c) The lower and upper cut-points are therefore $22$ and $33$

(d) The observation of 22 is included in the third class.

Part (a) Step 1: Concept introduction

A grouping system is a type comprising a correlated activity that integrates any number of members of the subset to generate a factor of the range, with a self - concept and an exact opposite for each component.

Part (a) Step 2: Explanation

Cut-point grouping with equal width class of width $8$ was used to organise a quantitative data set.

The difference between the higher and lower cut-points is the common class width.

The lower cut-point is the midpoint that has been reduced by half of the class width, whereas the upper cut-point is the midpoint that has been increased by half of the class width.

As a result, the lower and upper cut-points are respectively $6$ and $14$.

Part (b) Step 1: Concept introduction

A grouping system is a type comprising a correlated activity that integrates any number of members of the subset to generate a factor of the range, with a self - concept and an exact opposite for each component.

Part (b) Step 2: Explanation

The average of the cut-points is the middle.

The former midway is increased by class width to create the new midpoint.

As a result, the midpoint is $18$.

Part (c) Step 1: Concept introduction

A grouping system is a type comprising a correlated activity that integrates any number of members of the subset to generate a factor of the range, with a self - concept and an exact opposite for each component.

Part (c) Step 2: Explanation

The average of the lower and upper bounds is the class mark.

The lower cut-point is the previous class's upper cut-point, whereas the upper cut-point is the lower cut-point multiplied by the class width.

The lower and upper cut-points are therefore $22$ and $30$, respectively.

Part (d) Step 1: Concept introduction

A grouping system is a type comprising a correlated activity that integrates any number of members of the subset to generate a factor of the range, with a self - concept and an exact opposite for each component.

Part (d) Step 2: Explanation

A class's lower class limit is the smallest data value that can be stored in the class.

Because $22$ is the lowest cut-point of the third class, it contains the observation of $22$.

As a result, the observation of $22$ is included in the third class.