Chapter 3, Problem 6RE

a.

To determine

Find the population standard deviation for the prices in May.

Answer to Problem 6RE

• The population standard deviation for the prices in May is 0.791.

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 34.479.

The deviation, deviation’s squares are calculated below:

x	$\left(x-\mu \right)$	${\left(x-\mu \right)}^2$
34.54	0.061	0.004
33.27	–1.209	1.462
34.4	–0.079	0.006
35.67	1.191	1.418
33.16	–1.319	1.74
34.62	0.141	0.02
33.49	–0.989	0.978
34.72	0.241	0.058
34.96	0.481	0.231
32.72	–1.759	3.094
34.35	–0.129	0.017
34.59	0.111	0.012
35	0.521	0.271
34.78	0.301	0.091
33.67	–0.809	0.654
34.98	0.501	0.251
34.84	0.361	0.13
34.99	0.511	0.261
33.72	–0.759	0.576
35	0.521	0.271
35.47	0.991	0.982
35.59	1.111	1.234
Total		${\displaystyle \sum {\left(x-\mu \right)}^2}=13.76$

Substituting ${\displaystyle \sum {\left(x-\mu \right)}^2}=13.76$ and N = 22 in the standard deviation formula,

$\begin{array}{c}\sigma =\sqrt{\frac{{\displaystyle \sum {\left(x-\mu \right)}^2}}{N}}\\ =\sqrt{\frac{13.76}{22}}\\ =0.791\end{array}$

• Hence, the population standard deviation for the prices in May is 0.791.

b.

To determine

Find the population standard deviation for the prices in June.

Answer to Problem 6RE

• The population standard deviation for the prices in June is 0.814.
• .

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 34.052.

The deviation, deviation’s squares are calculated below:

x	$\left(x-\mu \right)$	${\left(x-\mu \right)}^2$
34.9	0.848	0.719
34.15	0.098	0.01
34.08	0.028	8E–04
32.66	–1.392	1.938
35.03	0.978	0.956
34.61	0.558	0.311
33.85	–0.202	0.041
32.99	–1.062	1.128
34.88	0.828	0.686
34.85	0.798	0.637
33.53	–0.522	0.272
33.31	–0.742	0.551
35.02	0.968	0.937
35.08	1.028	1.057
33.03	–1.022	1.044
33.75	–0.302	0.091
34.27	0.218	0.048
34.87	0.818	0.669
32.69	–1.362	1.855
33.49	–0.562	0.316
Total		${\displaystyle \sum {\left(x-\mu \right)}^2}=13.27$

Substituting ${\displaystyle \sum {\left(x-\mu \right)}^2}=13.27$ and N = 20 in the standard deviation formula,

$\begin{array}{c}\sigma =\sqrt{\frac{{\displaystyle \sum {\left(x-\mu \right)}^2}}{N}}\\ =\sqrt{\frac{13.27}{20}}\\ =0.814\end{array}$

• Hence, the population standard deviation for the prices in June is 0.814.

c.

To determine

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

Answer to Problem 6RE

• The spreads for the prices in June 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 0.791.
• From part (b), the population standard deviation for the prices in June is 0.814.
• The standard deviation for the prices in June is slightly more than the population standard deviation for the prices in May.
• Thus, it can be concluded that the spreads for the prices in June is more volatile than May.