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Magazine Chapter 1: Q18E (page 25)
Every corporation has a governing board of directors. The number of individuals on a board varies from one corporation to another. One of the authors of the article 鈥淒oes Optimal Corporate Board Size Exist? An Empirical Analysis鈥� (J. of Applied Finance, 2010: 57鈥�69) provided the accompanying data on the number of directors on each board in a random sample of 204 corporations.
No. directors: 4 5 6 7 8 9
Frequency: 3 12 13 25 24 42
No. directors: 10 11 12 13 14 15
Frequency: 23 19 16 11 5 4
No. directors: 16 17 21 24 32
Frequency: 1 3 1 1 1
a. Construct a histogram of the data based on relative frequencies and comment on any interesting features.
b. Construct a frequency distribution in which the last row includes all boards with at least 18 directors. If this distribution had appeared in the cited article, would you be able to draw a histogram? Explain.
c. What proportion of these corporations have at most 10 directors?
d. What proportion of these corporations have more than 15 directors?
Short Answer
a.The histogram is represented as,
b.
The required frequency distribution is as follows:
No. directors | Frequency |
4 | 3 |
5 | 12 |
6 | 13 |
7 | 25 |
8 | 24 |
9 | 42 |
10 | 23 |
11 | 19 |
12 | 16 |
13 | 11 |
14 | 5 |
15 | 4 |
16 | 1 |
17 | 3 |
>18 | 3 |
No, both histograms will appear to be similar and will not provide any valuable information about the variable.
c.The proportion of the corporations that have at most 10 directors is 0.698.
d. The proportion of the corporations that have more than 15 directors is 0.03
Step by step solution
Given information
The data on the number of directors on each board in a random sample of 204 corporations is provided.
Construct a histogram and state the features
a.
The relative frequency is computed as,
\({\rm{relative frequency }} = \frac{{frequency}}{{Total\;number\;of\;observations}}\)
The table representing the relative frequency is computed as,
No. directors | Frequency | Relative Frequency |
4 | 3 | 0.015 |
5 | 12 | 0.059 |
6 | 13 | 0.064 |
7 | 25 | 0.123 |
8 | 24 | 0.118 |
9 | 42 | 0.206 |
10 | 23 | 0.113 |
11 | 19 | 0.093 |
12 | 16 | 0.078 |
13 | 11 | 0.054 |
14 | 5 | 0.025 |
15 | 4 | 0.020 |
16 | 1 | 0.005 |
17 | 3 | 0.015 |
21 | 1 | 0.005 |
24 | 1 | 0.005 |
32 | 1 | 0.005 |
Steps to construct a histogram are,
1) Determine the frequency or the relative frequency.
2) Mark the class boundaries on the horizontal axis.
3) Draw a rectangle on the horizontal axis corresponding to the frequency or relative frequency.
The histogram is represented as,
The features that can be observed from the above histogram are,
1)The histogram is unimodal; this implies that there is only one mode; that is 9.
2)The distribution is positively skewed.
3) There are no outliers present in the data.
Given information
The data on the number of directors on each board in a random sample of 204 corporations is provided.
Construct a frequency distribution
b.
A frequency distribution in which the last row includes all boards with at least 18 directors is represented as,
No. directors | Frequency |
4 | 3 |
5 | 12 |
6 | 13 |
7 | 25 |
8 | 24 |
9 | 42 |
10 | 23 |
11 | 19 |
12 | 16 |
13 | 11 |
14 | 5 |
15 | 4 |
16 | 1 |
17 | 3 |
>18 | 3 |
Comment on whether histogram can be drawn using above frequency distribution
Since the observed frequency is too low i.e.,>18 has frequency of 3 which is not so extreme to affect the general distribution of the given variable.
The histogram can be constructed as:
As both histograms will appear almost similar (both are right-skewed) which is of no use (not provide any useful information).
Therefore, it makes no sense to draw a histogram based on above frequency distribution.
Given information
The data on the number of directors on each board in a random sample of 204 corporations is provided.
Calculate the proportion
Let X represents the number of directors.
Referring to the relative frequencies computed in part a,
c.
The proportion of the corporations that have at most 10 directors is computed as,
\(\begin{aligned}P\left( {X \le 10} \right) &= P\left( {X = 4} \right) + P\left( {X = 5} \right) + P\left( {X = 6} \right) + ... + P\left( {X = 10} \right)\\ &= 0.015 + 0.059 + 0.064 + ... + 0.113\\ &= 0.698\end{aligned}\)
Therefore, the proportion of the corporations that have at most 10 directors is 0.698.
Given information
The data on the number of directors on each board in a random sample of 204 corporations is provided.
Calculate the proportion
Let X represents the number of directors.
Referring to the relative frequencies computed in part a.
d.
The proportion of the corporations that have more than 15 directors is computed as,
\(\begin{aligned}P\left( {X > 15} \right) &= P\left( {X = 16} \right) + P\left( {X = 17} \right) + P\left( {X = 21} \right) + P\left( {X = 24} \right) + P\left( {X = 32} \right)\\ &= 0.005 + 0.015 + 0.005 + 0.005 + 0.005\\ &= 0.035\end{aligned}\)
Therefore, the proportion of the corporations that have more than 15 directorsis 0.035.
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