Chapter 3, Problem 6RE

a)

To determine

The population standard deviation of the prices in May.

Answer to Problem 6RE

Population standard deviation of the prices in May = 0.79

Explanation of Solution

May:

First need to find mean.

Formula:

  $\overline{x}=\frac{{\displaystyle \sum _{i=1}^n{x}_i}}{n}$

Formula:

  ${\sigma }^{}=\sqrt{\frac{{\displaystyle \sum {(x_i-\overline{x})}^2}}{n}}$

Calculation:

Using data,

Mean = $\overline{x}$ = 34.48

And n = 22

Creating a table for finding population standard deviation.

$x$	$x-\overline{x}$	${(x-\overline{x})}^2$
34.54	0.06	0.00
34.62	0.14	0.02
34.35	-0.13	0.02
33.67	-0.81	0.66
33.72	-0.76	0.58
33.27	-1.21	1.46
33.49	-0.99	0.98
34.59	0.11	0.01
34.98	0.5	0.25
35	0.52	0.27
34.4	-0.08	0.01
34.72	0.24	0.06
35	0.52	0.27
34.84	0.36	0.13
35.47	0.99	0.98
35.67	1.19	1.42
34.96	0.48	0.23
34.78	0.3	0.09
34.99	0.51	0.26
35.59	1.11	1.23
33.16	-1.32	1.74
32.72	-1.76	3.10
Sum		13.76

From table,

  ${\displaystyle \sum {(x_i-\overline{x})}^2}=13.76$

Put in a formula,

  $\sigma =\sqrt{\frac{13.76}{22}}=0.79$

Population standard deviation = 0.79

b)

To determine

The population standard deviation of June.

Answer to Problem 6RE

Population standard deviation for June is 0.81.

Explanation of Solution

June:

First need to find mean.

Formula:

  $\overline{x}=\frac{{\displaystyle \sum _{i=1}^n{x}_i}}{n}$

Calculation:

Using data,

Mean = $\overline{x}$ = 34.05

Creating a table for finding population standard deviation.

$x$	$x-\overline{x}$	${(x-\overline{x})}^2$
34.9	0.85	0.72
35.03	0.98	0.96
34.88	0.83	0.69
35.02	0.97	0.94
34.27	0.22	0.05
34.15	0.1	0.01
34.61	0.56	0.31
34.85	0.8	0.64
35.08	1.03	1.06
34.87	0.82	0.67
34.08	0.03	0.00
33.85	-0.2	0.04
33.53	-0.52	0.27
33.03	-1.02	1.04
32.69	-1.36	1.85
32.66	-1.39	1.93
32.99	-1.06	1.12
33.31	-0.74	0.55
33.75	-0.3	0.09
33.49	-0.56	0.31
Sum		13.27

From table,

  ${\displaystyle \sum {(x_i-\overline{x})}^2}=13.27$

Put in a formula,

  $\sigma =\sqrt{\frac{13.27}{20}}=0.81$

Population standard deviation = 0.81

c)

To determine

To check: The volatility of the prices of May and June.

Answer to Problem 6RE

Variation (volatility) of Jun is slightly higher than May.

Explanation of Solution

To check variation in prices for May and June, first need to check coefficient of variance for both months. The lower the value of the coefficient of variation (CV), the more precise the estimate.The higher the CV, the higher would be the variation in prices.

Formula:

  $CV=\frac{\sigma }{\overline{x}}\times 100$

Calculation:

May:

  $\begin{array}{l}CV=\frac{0.79}{34.48}\times 100\\ CV=2.29\%\end{array}$

June:

  $\begin{array}{l}CV=\frac{34.05}{34.05}\times 100\\ CV=2.39\%\end{array}$

Here the CV of June is slightly higher than May. This shows the variation in prices in June which is slightly higher as compared to the prices of May.