Chapter 3.2, Problem 37E

a.

To determine

Find the population standard deviation for the prices in May.

Answer to Problem 37E

• The population standard deviation for the prices in May is 18.52.

Explanation of Solution

Calculation:

The closing prices of google stock in May and June are given.

Standard deviation:

Let $x_1,x_2\mathrm{...}{x}_N$ be the population observations of size N. the population mean is denoted by $\mu$. Hence, the population standard deviation can be expressed as,

$\begin{array}{c}\sigma =\sqrt{\frac{{\displaystyle \sum {\left(x-\mu \right)}^2}}{N}}\\ =\sqrt{\frac{{\displaystyle \sum x^2}-N{\mu }^2}{N}}\end{array}$

Population standard deviation:

Finding mean:

Software procedure:

Step-by-step procedure to find mean using the MINITAB software:

• Choose Stat > Basic Statistics > Display Descriptive Statistics.
• In Variables enter the columns May.
• Choose option statistics, and select N total, Mean.
• Click OK.

Output using the MINITAB software is given below:

• 
• From the MINITAB output the mean is 872.51.

The deviation, deviation’s squares are calculated below:

x	$\left(x-\mu \right)$	${\left(x-\mu \right)}^2$
880.37	7.86	61.7796
877.07	4.56	20.7936
873.65	1.14	1.2996
866.2	–6.31	39.8161
869.79	–2.72	7.3984
829.61	–42.9	1840.41
880.93	8.42	70.8964
884.74	12.23	149.5729
900.68	28.17	793.5489
900.62	28.11	790.1721
886.25	13.74	188.7876
820.43	–52.08	2,712.326
875.04	2.53	6.4009
877	4.49	20.1601
871.98	–0.53	0.2809
879.81	7.3	53.29
890.22	17.71	313.6441
879.73	7.22	52.1284
864.64	–7.87	61.9369
859.7	–12.81	164.0961
859.1	–13.41	179.8281
867.6	–4.91	24.1081
Total		${\displaystyle \sum {\left(x-\mu \right)}^2}=7,552.675$

Substituting ${\displaystyle \sum {\left(x-\mu \right)}^2}=7,552.675$ and N = 22 in the standard deviation formula,

$\begin{array}{c}\sigma =\sqrt{\frac{{\displaystyle \sum {\left(x-\mu \right)}^2}}{N}}\\ =\sqrt{\frac{7,552.675}{22}}\\ =18.52\end{array}$

• Hence, the population standard deviation for the prices in May is 18.52.

b.

To determine

Find the population standard deviation for the prices in June.

Answer to Problem 37E

• The population standard deviation for the prices in June is 18.48.

Explanation of Solution

Calculation:

Population standard deviation:

Finding mean:

Software procedure:

Step-by-step procedure to find mean using the MINITAB software:

• Choose Stat > Basic Statistics > Display Descriptive Statistics.
• In Variables enter the columns June.
• Choose option statistics, and select N total, Mean.
• Click OK.

Output using the MINITAB software is given below:

• 
• From the MINITAB output the mean is 881.8.

The deviation, deviation’s squares are calculated below:

x	$\left(x-\mu \right)$	${\left(x-\mu \right)}^2$
871.22	–10.58	111.9364
870.76	–11.04	121.8816
868.31	–13.49	181.9801
881.27	–0.53	0.2809
873.32	–8.48	71.9104
882.79	0.99	0.9801
889.42	7.62	58.0644
906.97	25.17	633.5289
908.53	26.73	714.4929
909.18	27.38	749.6644
903.87	22.07	487.0849
915.89	34.09	1162.128
887.1	5.3	28.09
877.53	–4.27	18.2329
880.23	–1.57	2.4649
871.48	–10.32	106.5024
873.63	–8.17	66.7489
857.23	–24.57	603.6849
861.55	–20.25	410.0625
845.72	–36.08	1,301.766
Total		${\displaystyle \sum {\left(x-\mu \right)}^2}=6,831.486$

Substituting ${\displaystyle \sum {\left(x-\mu \right)}^2}=6,831.486$ and N = 20 in the standard deviation formula,

$\begin{array}{c}\sigma =\sqrt{\frac{{\displaystyle \sum {\left(x-\mu \right)}^2}}{N}}\\ =\sqrt{\frac{6,831.486}{20}}\\ =18.48\end{array}$

• Hence, the population standard deviation for the prices in June is 18.48.

c.

To determine

Find the month when the price of Google stock more volatile.

Answer to Problem 37E

• The spreads for the prices in May has increased in 2013.

Explanation of Solution

• The financial analysts use the term “volatility” for referring the price variation of assets as stocks.
• From part (a), the population standard deviation for the prices in May is 18.52.
• From part (b), the population standard deviation for the prices in June is 18.48.
• The standard deviation for the prices in May is slightly more than the population standard deviation for the prices in June.
• Thus, it can be concluded that the spreads for the prices in May is more volatile than June.