Question

The Russell 1000 is a stock market index consisting of the largest U.S. companies. The Dow Jones industrial Average is based on 30 large companies. The data giving the annual percentage returns for each of these stock indexes for 25 years are contained in the Excel Online file below. Construct a spreadsheet to answer the following questions.

Year	DJIA % Return	Russell 1000 % Return
1988	8.82	12.33
1989	26.59	26.44
1990	-3.68	-4.57
1991	16.04	28.88
1992	5.38	1.66
1993	18.58	7.69
1994	6.29	1.76
1995	30.62	37.10
1996	21.49	17.49
1997	19.04	28.68
1998	12.83	29.46
1999	29.15	15.89
2000	-3.01	-6.42
2001	-9.85	-13.16
2002	-15.56	-25.79
2003	27.78	29.69
2004	7.71	10.82
2005	-4.84	8.73
2006	13.34	13.72
2007	8.12	7.04
2008	-31.04	-42.92
2009	20.72	22.47
2010	8.76	9.59
2011	2.80	-3.13
2012	8.40	11.02

a. Which of the following scatter diagrams accurately represents the data set?

#1	Russell 1000 DJIA	#2	Russell 1000 DJIA
#3	Russell 1000 DJIA	#4	Russell 1000 DJIA

_________Scatter diagram #1Scatter diagram #2Scatter diagram #3Scatter diagram #4

b. Compute the sample mean and standard deviation for each index (to 2 decimals).

	sample mean	standard deviation
DJIA:
Russell 1000:

c. Compute the sample correlation coefficient for these data (to 3 decimals).

d. Discuss similarities and differences in these two indexes.

_________There is a strong positive linear association between DJIA and Russell 1000There is a moderate positive linear association between DJIA and Russell 1000There is neither a positive nor a negative linear association between DJIA and Russell 1000There is a moderate negative linear association between DJIA and Russell 1000There is a strong negative linear association between DJIA and Russell 1000

The variance of the Russell 1000 is slightly _________largersmaller than that of the DJIA.

a. Which of the following scatter diagrams accurately represents the data set?

#1	Russell 1000 DJIA	#2	Russell 1000 DJIA
#3	Russell 1000 DJIA	#4	Russell 1000 DJIA

_________Scatter diagram #1Scatter diagram #2Scatter diagram #3Scatter diagram #4

b. Compute the sample mean and standard deviation for each index (to 2 decimals).

	sample mean	standard deviation
DJIA:
Russell 1000:

c. Compute the sample correlation coefficient for these data (to 3 decimals).

d. Discuss similarities and differences in these two indexes.

_________There is a strong positive linear association between DJIA and Russell 1000There is a moderate positive linear association between DJIA and Russell 1000There is neither a positive nor a negative linear association between DJIA and Russell 1000There is a moderate negative linear association between DJIA and Russell 1000There is a strong negative linear association between DJIA and Russell 1000

The variance of the Russell 1000 is slightly _________largersmaller than that of the DJIA.

Answer 1

Given that the Russell 1000 is a stock market index consisting of the largest U.S. companies and that the Dow Jones industrial Average is based on 30 large companies.

Given is a data on the annual percentage returns for each of these stock indexes for 25 years.

Before we go on to solve the problems let us know a bit about scatter plot,sample mean, sample standard deviation, sample correlation coefficient.

Scatter Plot

Scatter plot is a very important type of plot which we usually use for bivariate type of data. Scatter plot gives a fairly good idea about the relation between the two variables.

Sample Mean

The sample mean is given by x̄ which the sum of the observations divided by the number of observations,

Sample Standard Deviation

The sample standard deviation is given by s,

Sample Correlation Coefficient

The sample correlation coefficient is used to measure the degree of linear relationship between two variables x and y and is given by r,

r=-1 [Strong negative relationship]

r=0 [No relationship]

r=1 [Strong positive relationship]

Coming back to our problem,

Our given data is,

let X=DJIA % Return

Y=Russell 1000 % Return

Year	DJIA % Return (x)	Russell 1000 % Return (y)
1988	8.82	12.33
1989	26.59	26.44
1990	-3.68	-4.57
1991	16.04	28.88
1992	5.38	1.66
1993	18.58	7.69
1994	6.29	1.76
1995	30.62	37.1
1996	21.49	17.49
1997	19.04	28.68
1998	12.83	29.46
1999	29.15	15.89
2000	-3.01	-6.42
2001	-9.85	-13.16
2002	-15.56	-25.79
2003	27.78	29.69
2004	7.71	10.82
2005	-4.84	8.73
2006	13.34	13.72
2007	8.12	7.04
2008	-31.04	-42.92
2009	20.72	22.47
2010	8.76	9.59
2011	2.8	-3.13
2012	8.4	11.02

(a) The Scatter Plot is given by,

(b) Here we need to find the sample mean and sample standard deviation for DJIA % Return and Russell 1000 % Return.

X=DJIA % Return

Y=Russell 1000 % Return

The table of calculations is given below,

Year	DJIA % Return (x)	Russell 1000 % Return (y)	(xi-x̄)^2	(yi-ȳ)^2
1988	8.82	12.33	0.03	11.22
1989	26.59	26.44	310.11	304.85
1990	-3.68	-4.57	160.28	183.6
1991	16.04	28.88	49.84	396.01
1992	5.38	1.66	12.96	53.58
1993	18.58	7.69	92.16	1.66
1994	6.29	1.76	7.24	52.13
1995	30.62	37.1	468.29	790.73
1996	21.49	17.49	156.5	72.42
1997	19.04	28.68	101.2	388.09
1998	12.83	29.46	14.82	419.43
1999	29.15	15.89	406.83	47.75
2000	-3.01	-6.42	143.76	237.16
2001	-9.85	-13.16	354.57	490.18
2002	-15.56	-25.79	602.21	1208.95
2003	27.78	29.69	353.44	428.9
2004	7.71	10.82	1.61	3.39
2005	-4.84	8.73	190.99	0.06
2006	13.34	13.72	19.01	22.47
2007	8.12	7.04	0.74	3.76
2008	-31.04	-42.92	1601.6	2693.61
2009	20.72	22.47	137.83	181.98
2010	8.76	9.59	0.05	0.37
2011	2.8	-3.13	38.19	146.65
2012	8.4	11.02	0.34	4.16
Total	224.48	224.47	5224.6	8143.11

Hence,

	Sample Mean	Sample Standard Deviation
DJIA:	8.98	14.75
Russell 1000:	8.98	18.42

(c) We know that the sample correlation coefficient is given by,

X=DJIA % Return

Y=Russell 1000 % Return

The table of calculations is given below,

Year	DJIA % Return (x)	Russell 1000 % Return (y)	(xi-x̄)^2	(yi-ȳ)^2	(xi-x̄)*(yi-ȳ)
1988	8.82	12.33	0.03	11.22	-0.54
1989	26.59	26.44	310.11	304.85	307.47
1990	-3.68	-4.57	160.28	183.6	171.54
1991	16.04	28.88	49.84	396.01	140.49
1992	5.38	1.66	12.96	53.58	26.35
1993	18.58	7.69	92.16	1.66	-12.38
1994	6.29	1.76	7.24	52.13	19.42
1995	30.62	37.1	468.29	790.73	608.52
1996	21.49	17.49	156.5	72.42	106.46
1997	19.04	28.68	101.2	388.09	198.18
1998	12.83	29.46	14.82	419.43	78.85
1999	29.15	15.89	406.83	47.75	139.37
2000	-3.01	-6.42	143.76	237.16	184.65
2001	-9.85	-13.16	354.57	490.18	416.9
2002	-15.56	-25.79	602.21	1208.95	853.26
2003	27.78	29.69	353.44	428.9	389.35
2004	7.71	10.82	1.61	3.39	-2.34
2005	-4.84	8.73	190.99	0.06	3.46
2006	13.34	13.72	19.01	22.47	20.67
2007	8.12	7.04	0.74	3.76	1.67
2008	-31.04	-42.92	1601.6	2693.61	2077.04
2009	20.72	22.47	137.83	181.98	158.37
2010	8.76	9.59	0.05	0.37	-0.13
2011	2.8	-3.13	38.19	146.65	74.84
2012	8.4	11.02	0.34	4.16	-1.18
Total	224.48	224.47	5224.6	8143.11	5960.29

Hence the sample correlation coefficient is 0.914

(d)

• Since the correlation coefficient is close to one there is a strong positive linear association between DJIA and Russell 1000
• Now variance of DJIA is 217.69 and variance of Russell is 339.30. Hence the variance of the Russell 1000 is slightly larger than that of the DJIA.