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 ?"Does Optimal Corporate Board Size Exist? An Empirical Analysis" (J. of Applied Finance, 2010: 57-69) ?provided the accompanying data on the number of directors on each board in a random sample of 204 corporations.
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?

Step-by-step explanation:

Hello!

The data set shows the number of broad directors governing corporations.

a) The first step is to calculate the relative frequency of each value of the variable.

The relative frequency h(x)= f(x)/n

f(x)= absolute frequency

n= total of board directors.

(see attachment)

Watching the histogram you can see that the frequency distribution is not symmetric but right-skewed.

Mean X[bar]= ∑x*h(x)= 12.214

Median Position n/2= 204/2= 102, the median is the 102th number of board directors registered; Me= 9

The mode is the value of the variable that shows the greater absolute frequency, in this case, the mode is: Mo= 9

Mo=Me<X[bar]

Out of these three measurements of central tendency, the more affected by extreme/unusual values is the mean. In this case, the observed frequencies corresponding to the values X= 21, X= 24 and X=32 "push" the mean to the right.

b) You have to rearrange the data set so that the last row includes all boards with at least 18 directors, symbolically: X≥18

When working with data sets is not unusual for researchers to join categories of the variable, especially when the observed frequencies in some of them are too low to fulfill the requirements of statistical test or there is, for example, some values so extreme that affect the general distribution of the variable, showing a wider asymmetry that the general values imply or falsely increasing its variability.

If you rearrange the data by joining two or more categories, you have to keep in mind that these new groups of data should be exhaustive (they should be complete and include all possible values of the variable) and complementary (they should not share information).

In this case, we have defined the data group than includes all boards with at least 18 directors X≥18, we could arrange the rest of the data set into its complementary category "all boards with less than 18 directors" X<18.

Resulting

X<18

X≥18

Both groups include all possible observations of the variable and are complementary. Having the data summarized in two groups it makes no sense to graph a histogram, it would not show you any useful information about the variable distribution.

c. What proportion of these corporations have at most 10 directors?

Symbolically P(X≤10)

To calculate this proportion you have to add all relative frequencies corresponding to the boards that have 10 directors or less.

P(X≤10)= h(4)+h(5)+h(6)+h(7)+h(8)+h(9)+h(10)= 0,015 + 0,059 + 0,064 + 0,123 + 0,118 + 0,206 + 0,113 = 0,698

d. What proportion of these corporations have more than 15 directors?

Symbolically: P(X>15)

To calculate this proportion you can add all relative frequencies of the boards that have more than 15 directors (but not include 15 directors) That is:

P(X>15)= h(16)+h(17)+h(21)+h(24)+h(32)= 0,005 + 0,015 + 0,005 + 0,005 + 0,005 = 0,035

I hope you have a SUPER day!

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Step-by-step explanation: