What Are Shadow Stocks?

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The following table shows a data set containing information for 25 of the shadow stocks tracked by the American Association of Individual Investors. Shadow stocks are common stocks of smaller companies that are not closely followed by Wall Street analysts. 

How many variables are in the data set? 

The variables are Exchange, Market Capitalization, Price/ Earnings Ratio and Gross Profit Margin. These are variables as the values of these variables is not constant and change from time to time. 

Which of the variables are categorical and which are quantitative? 

A quantitative variable is the one which can be quantitatively measured. i.e. it is a numerical value. 

A categorical variable is the one that can take one value from a limited number of fixed values. 

Exchange is a Categorical Variable. 

Price/Earnings Ratio is a Quantitative Variable. 

Gross Profit Margin (%) is a Quantitative Variable. 

For the Exchange variable, show the frequency and the percent frequency for AMEX, NYSE, and OTC. Construct a bar graph similar to Figure 1.4 (in the textbook) for the Exchange variable. 

Out of the 25 stocks, 

AMEX is the exchange for 5 stocks. So percent frequency is 5/25 = 0.2 = 20%. 

NYSE is the exchange for 3 stocks. So percent frequency is 3/25 = 0.12 = 12%. 

OTC is the exchange for 17 stocks. So percent frequency is 17/25 = 0.68 = 68%. 

Show the frequency distribution for the Gross Profit Margin using he five intervals: 0 – 14.9, 15 – 29.9, 30 – 44.9, 45 – 59.9, and 60 – 74.9. Construct a histogram similar to Figure 1.5 (in the textbook). 

Gross Profit Margin	Tally	Frequency
0–14.9	||	2
15–29.9	|||||     |	6
30–44.9	|||||     |||	7
45–59.9	|||||     |	6
60–74.9	|||	3

What is the average price/earnings ratio? 

Average price/earnings ratio  

 16 

DATA SET FOR 25 SHADOW STOCKS 

Company	Exchange	Ticker Symbol	Market Cap ($ millions)	Price/ Earnings Ratio	Gross Profit Margin (%)
DeWolfe Companies	AMEX	DWL	36.4	8.4	36.7
North Coast Energy	OTC	NCEB	52.5	6.2	59.3
Hansen Natural Corp.	OTC	HANS	41.1	14.6	44.8
MarineMax, Inc.	NYSE	HZO	111.5	7.2	23.8
Nanometrics Inc.	OTC	NANO	228.6	38.0	53.3
Team Staff, Inc.	OTC	TSTF	92.1	33.5	4.1
Environmental Tectonics	AMEX	ETC	51.1	35.8	35.9
Measurement Specialties	AMEX	MSS	101.8	26.8	37.6
SEMCO Energy, Inc.	NYSE	SEN	193.4	18.7	23.6
Party City Corporation	OTC	PCTY	97.2	15.9	36.4
Embrex, Inc.	OTC	EMBX	136.5	18.9	59.5
Tech/Ops Sercon, Inc.	AMEX	TO	23.2	20.7	35.7
ARCADIS NV	OTC	ARCAF	173.4	8.8	9.6
Qiao Xing Universal Tele.	OTC	XING	64.3	22.1	30.8
Energy West Inc.	OTC	EWST	29.1	9.7	16.3
Barnwell Industries, Inc.	AMEX	BRN	27.3	7.4	73.4
Innodata Corporation	OTC	INOD	66.1	11.0	29.6
Medical Action Industries	OTC	MDCI	137.1	26.9	30.6
Instrumentarium Corp.	OTC	INMRY	240.9	3.6	52.1
Petroleum Development	OTC	PETD	95.9	6.1	19.4
Drexler Technology Corp.	OTC	DRXR	233.6	45.6	53.6
Gerber Childrenswear, Inc.	NYSE	GCW	126.9	7.9	25.8
Gaiam, Inc,	OTC	GAIA	295.5	68.2	60.7
Artesian Resources Corp.	OTC	ARTNA	62.8	20.5	45.5
York Water Company	OTC	YORW	92.2	22.9	74.2

People often wait till middle age to worry about having a healthy heart. However, recent studies have shown that earlier monitoring of risk factors such as blood pressure can be very beneficial ( The Wall Street Journal, January 10, 2012). Having higher than normal blood pressure, a condition known as hypertension, is a major risk factor for heart disease. Suppose a large sample of individuals of various ages and gender was selected and that each individual’s blood pressure was measured to determine if they have hypertension. For the sample data, the following table shows the percentage of individuals with hypertension. 

Develop a side-by-side bar chart with age on the horizontal axis, the percentage of individuals with hypertension on the vertical axis, and side-by-side bars based on gender. 

What does the display you developed in part (a), indicate about hypertension and age? 

As the age increases hypertension also increases. 

Comment on differences in gender. 

The percentages of male having age between 20 and 64 with hypertension is greater than the percentages of females having age between 20 and 64 with hypertension. But the percentages of females having age 65 and above with hypertension is greater than the percentage of males having age 65 and above with hypertension. 

Age	Male	Female
20 - 34	11.0%	9.0%
35 – 44	24.0%	19.0%
45 – 54	39.0%	37.0%
55 – 64	57.0%	56.0%
65 – 74	62.0%	64.0%
75 +	73.3%	79.0%

The department of transportation’s study on driving speed and miles per gallon for midsize automobiles resulted in the data below. 

Plot the speed and miles per gallon using a scatter plot with speed on the horizontal axis. 

Compute the sample mean and standard deviation for each variable. 

Speed (Miles per Hours) 

Sample Mean 

Standard deviation 

Miles per Gallon 

Sample Mean 

Standard deviation 

Compute the sample correlation coefficient. Is there a positive or negative correlation between these two variables? Comment. 

Speed (Miles per Hour)	30	50	40	55	30	25	60	25	50	55
Miles per Gallon	28	25	25	23	30	32	21	35	26	25

Let x= Speed (Miles per Hours) 

y= Miles per Gallon 

There is a strong negative linear correlation between the two variables. A negative correlation implies that as the average speed increases the fuel efficiency decreases. 

The U.S. Census Bureau serves as the leading source of quantitative data about the nation’s people and economy. The following cross-tabulation shows the number of households (1,000s) and the household income by the highest of level of education for the head of the household (U.S. Census Bureau website, 2013). Only households in which the head has a high school diploma or more are included. 

Develop a joint probability table 

	Under $25,000	$25,000 to $49,999	$50,000 to $99,999	$100,000 and Over
High School Graduate	0.1505	0.1519	0.1438	0.0530
Bachelor’s Degree	0.0378	0.0634	0.1168	0.1191
Master’s Degree	0.0104	0.0184	0.0460	0.0624
Doctoral Degree	0.0012	0.0024	0.0064	0.0164

What is the probability of the head of one of these households having a master’s degree or more education? 

What is the probability of a household headed by someone with a high school diploma earning $100,000 or more? 

What is the probability of one of these households having an income below $25,000? 

What is the probability of a household headed by someone with a bachelor’s degree earning less than $25,000? 

Is household income independent of educational level? 

Highest Level of Education	Household Income
	Under $25,000	$25,000 to $49,999	$50,000 to $99,999	$100,000 and Over	TOTAL
High School Graduate	9880  6554.20  1687.61	9970  7737.93  643.86	9441  10258.66  65.17	3482  8222.21  2732.79	32,773
Bachelor’s Degree	2484  4425.93  852.05	4164  5225.28  215.55	7666  6927.48  78.73	7817  5552.30  923.73	22,131
Master’s Degree	685  1800.49  691.10	1205  2725.67  398.76	3019  2818.13  14.32	4094  2258.70  1491.26	9003
Doctoral Degree	79  347.38  207.34	160  410.12  152.54	422  543.72  27.25	1076  435.78  940.55	1737
TOTAL	13,128	15,499	20,548	16,469	65,644

Since  

P-value = 0.0000 

So household income is not independent of educational level. 

Highest Level of Education	Household Income
	Under $25,000	$25,000 to $49,999	$50,000 to $99,999	$100,000 and Over	TOTAL
High School Graduate	9880	9970	9441	3482	32,773
Bachelor’s Degree	2484	4164	7666	7817	22,131
Master’s Degree	685	1205	3019	4094	9003
Doctoral Degree	79	160	422	1076	1737
TOTAL	13,128	15,499	20,548	16,469	65,644

Reference 

Anderson, D. R., Sweeney, D. J., Williams, T. A., Camm, J. D., & Cochran, J. J. (2016).  Statistics for business & economics . Nelson Education.