Question

Sound City sells the ClearTone-400 satellite car radio. For this radio, historical sales records over the last 100 weeks show 5 weeks with no radios sold, 27 weeks with one radio sold, 27 weeks with two radios sold, 27 weeks with three radios sold, 9 weeks with four radios sold, and 5 weeks with five radios sold. Calculate μx, σx2, and σx, of x, the number of ClearTone-400 radios sold at Sound City during a week using the estimated probability distribution. (Round your answers to 2 decimal places.)

µ_{x ?}

σx^2 ?

σ_{x ?}

Answer 1

Answer: Sound City sells the ClearTone-400 satellite car radio. For this radio, historical sales records over the last 100 weeks show 5 weeks with no radios sold, 27 weeks with one radio sold, 27 weeks with two radios sold, 27 weeks with three radios sold, 9 weeks with four radios sold, and 5 weeks with five radios sold.

Solution:

Radio (x)	week	P(x)	x*P(x)	x^2*P(x)
0	5	5/100 = 0.05	0	0.00
1	27	27/100 = 0.27	0.27	0.27
2	27	27/100 = 0.27	0.54	1.08
3	27	27/100 = 0.27	0.81	2.43
4	9	9/100 = 0.09	0.36	1.44
5	5	5/100 = 0.05	0.25	1.25
Total	100	1.00	2.23	6.47

Mean, μx​​ = ∑x * P(x) = 2.23

Therefore, mean = 2.23

Variance, σ​​​​​​x^2 = ∑x^2 * P(x) - (∑x*P(x))^2

σ​​​​​​​​​​​​​​​​​​x^2 = 6.47 - (2.23)^2 = 1.4971

σ​​​​​​x^2 = 1.50

Therefore, variance = 1.50

Std.Dev, σ​​​​​​x = √variance = √1.50 = 1.2247

Therefore, standard deviation = 1.22