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?
Problem 18E Answer: Step1: We have 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. no. 4 5 6 7 8 9 10 11 12 13 14 15 16 17 21 24 32 Directors Frequency 3 12 13 25 24 42 23 19 16 11 5 4 1 3 1 1 1 We need to find, 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 Step2: a). Consider, No. Directors Frequency Relative Frequency 4 3 0.014705882 5 12 0.058823529 6 13 0.06372549 7 25 0.12254902 8 24 0.117647059 9 42 0.205882353 10 23 0.112745098 11 19 0.093137255 12 16 0.078431373 13 11 0.053921569 14 5 0.024509804 15 4 0.019607843 16 1 0.004901961 17 3 0.014705882 21 1 0.004901961 24 1 0.004901961 32 1 0.004901961 204 1 A histogram of given data is obtained by using Excel and steps are as follows •Enter the given data in Excel sheet with relative frequencies •select relative frequencies column go to insert choose histogram •then press ok. From the above histogram we can say that the given data is bell shaped hence the given data is normally distributed. Step3: b). 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 Except that the three largest values are compacted into the “ 18” category. If this were the originally-presented information, we could not create a histogram, because we would not know the upper boundary for the rectangle corresponding to the “ 18” category. Step4: c). We have n = 204, Consider, No. Directors Frequency 4 3 5 12 6 13 7 25 8 24 9 42 10 23 11 19 12 16 13 11 142 The proportion of these corporations have at most 10 directors is given by = 204 = 0.6960 Therefore, The proportion of these corporations have at most 10 directors is 0.6960. d). We have N = 204 Conside, No. Directors Frequency 16 1 17 3 21 1 24 1 7 The proportion of these corporations have more than 15 directors is given by 7 = 204 = 0.0343 Therefore,The proportion of these corporations have more than 15 directors is 0.0343.
