Q. 19 - Elementary Statistics

Question

Inauguration Ages. From the Information Please Almanac, we obtained the ages at the inauguration of the first 44 presidents of the United States (from George Washington to Barack H. Obama).

a. Identify the classes for grouping these data, using limit grouping with classes of equal width 5 and a first-class of 40-44.

b. Identify the class marks of the classes found in part (a).

c. Construct frequency and relative frequency distributions of the inauguration ages based on your classes obtained in part (a).

d. Draw a frequency histogram for the inauguration ages based on your grouping in part (a).

e. Classify the shape of the distribution of inauguration ages for the first 44 presidents of the United States both generally and specifically.

Step-by-Step Solution

Answer

Ans:  

(a)  

Limit grouping classes	Frequency
40-44	2
45-49	7
50-54	13
55-59	12
60-64	7
65-69	3
Total	44

(b)  

Limit grouping classes	Frequency	Classmarks
40-44	2	42
45-49	7	47
50-54	13	52
55-59	12	57
60-64	7	62
65-69	3	67
Total	44

(c)  

Limit grouping classes	Frequency	Relative Frequency
40-44	2	0.045
45-49	7	0.159
50-54	13	0.295
55-59	12	0.273
60-64	7	0.159
65-69	3	0.0678
Total	44	0.999

(d) 

(e)  According to the Histogram in part (d), it is clear that there is only one peak, and therefore it is a  unimodal distribution. And the histogram is symmetric.  

Step 1. Given information.

given,

Step 2. (a) Find the frequency distribution using limit grouping with a first class of 40 - 44 and a class width of 5 .

Given that the first section is $40-44$ and the width is 5. Thus, the second phase is $45-49$, and so on. Also, the highest view is 69. Therefore, the final category is $65-69.$ Therefore, the frequency distribution using the boundary division is given below:

Limit grouping classes	Frequency
40-44	2
45-49	7
50-54	13
55-59	12
60-64	7
65-69	3
Total	44

Step 3. (b) Find the class marks for the classes obtained in part (a).

The class marks are found by using the following formula,

    $Class Marks=\frac{Upper class limit - Lower class limit }{2}$

Therefore, the class marks for the classes obtained in part (a) are shown below:

   

Limit grouping classes	Frequency	Classmarks
40-44	2	42
45-49	7	47
50-54	13	52
55-59	12	57
60-64	7	62
65-69	3	67
Total	44

Step 4. (c) Find the relative-frequency distribution using limit grouping with a first class of 40 - 44 and a class width of 5 .

The relative frequency is found using the formula,
$Relative frequency =\frac{Frequency}{Number of observations}$

Therefore, the relative frequency distribution is given below:

  

Limit grouping classes	Frequency	Relative Frequency
40-44	2	0.045
45-49	7	0.159
50-54	13	0.295
55-59	12	0.273
60-64	7	0.159
65-69	3	0.0678
Total	44	0.999

Step 5. (d) Now, use Minitab to draw a histogram for given data based on the result in part (a).

The lower limits of the limited collection are used to construct a histogram.

MINITAB process-

Step 1) Select Graph> Histogram

Step 2) Select Easy, then click OK.

Step 3) For graph variables, enter the corresponding column of AGE.

Step 4) Click OK.

Step 6. Minitab Output,

According to Minitab output, it is clear that the height of the bar represents the frequency found in part (a).

Step 7. (e) Explanation,

 According to the Histogram in part (d), it is clear that there is only one peak, and therefore it is a  unimodal distribution. And the histogram is symmetric.