Example 4.7 from Statistics, 4.2 Mean or Expected Value and Standard Deviation

Mean or Expected Value and Standard Deviation: example 4.7

Toss a fair, six-sided die twice. Let X = the number of faces that show an even number. Construct a table like Table 4.12 and calculate the mean μ and standard deviation σ of X.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution

Tossing one fair six-sided die twice has the same sample space as tossing two fair six-sided dice. The sample space has 36 outcomes.

(1, 1)	(1, 2)	(1, 3)	(1, 4)	(1, 5)	(1, 6)
(2, 1)	(2, 2)	(2, 3)	(2, 4)	(2, 5)	(2, 6)
(3, 1)	(3, 2)	(3, 3)	(3, 4)	(3, 5)	(3, 6)
(4, 1)	(4, 2)	(4, 3)	(4, 4)	(4, 5)	(4, 6)
(5, 1)	(5, 2)	(5, 3)	(5, 4)	(5, 5)	(5, 6)
(6, 1)	(6, 2)	(6, 3)	(6, 4)	(6, 5)	(6, 6)

Table 4.13

Use the sample space to complete the following table.

x	P(x)	xP(x)	(x – μ)² $\cdot$ P(x)
0	$\frac{9}{36}$	0	(0 – 1)² ⋅ $\frac{9}{36}$ = $\frac{9}{36}$
1	$\frac{18}{36}$	$\frac{18}{36}$	(1 – 1)² ⋅ $\frac{18}{36}$ = 0
2	$\frac{9}{36}$	$\frac{18}{36}$	(2 – 1)² ⋅ $\frac{9}{36}$ = $\frac{9}{36}$

Table 4.14 Calculating μ and σ.

Add the values in the third column to find the expected value: μ = $\frac{36}{36}$ = 1. Use this value to complete the fourth column.

Add the values in the fourth column and take the square root of the sum: σ = $\sqrt{\frac{18}{36}}$ ≈ .7071.

How did it go?