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.)

Answer 1

Answer:

μ =2.23

$\sigma_x^2= 1.49$

$\sigma_x=1.22$

Step-by-step explanation:

Radio(x)                            weeks                           P(x)

0                                         5                                 5/100 = 0.05

1                                           27                               27/100=0.27

2                                          27                                 27/100 = 0.27

3                                         27                                 27/100 = 0.27  

4                                           9                                    9/100=0.09

5                                             5                                    5/100=0.05

We know that

Mean ,μ

$\mu =\sum x.P(x)$

μ = 0 x 0.05 + 1 x 0.27 + 2 x 0.27 + 3 x 0.27 + 4 x 0 .09 + 5 x 0.05

μ =2.23

$\sigma_x^2= \sum x^2\times P(x)-(\sum x.P(x))^2$

$\sigma_x^2= 0^2\times 0.05+ 1^2\times 0.27+ 2^2\times 0.27+ 3^2\times 0.27+ 4^2\times 0.09+5^2\times 0.05-2.23^2$

$\sigma_x^2= 1.49$

$\sigma_x= \sqrt{\sigma_x^2}$

$\sigma_x=1.22$

Answer 2

The mean, variance and standard deviation of the distribution are 2.23, 1.50 and 1.22 respectively.

Total number of weeks = 100

Weeks with no radio sold, x = 0 :

• P(x) = 5/100 = 0.05

Weeks with one radio sold, x = 1 :

• P(x) = 27/100 = 0.27

Weeks with two radio sold, x = 2 :

• P(x) = 27/100 = 0.27

Weeks with three radio sold, x = 3 :

• P(x) = 27/100 = 0.27

Weeks with four radio sold, x = 4 :

• P(x) = 9/100 = 0.09

Weeks with five radio sold, x = 5 :

• P(x) = 5/100 = 0.05

The probability distribution :

X ____ 0 ____ 1 ____ 2 ____ 3 _____ 4 __ 5

P(X) _ 0.05 _ 0.27 _ 0.27 _ 0.27 __ 0.09 _ 0.05

The mean :

Mean, μ = Σ P(X)X

μ = (0×0.05) + (1×0.27 + (2×0.27) + (3×0.27) + (4×0.09) + (5×0.05) = 2.23

The Variance, σ² :

σ² = Σ( X² × p(x)) - μ²

σ² = [(0²×0.05) + (1²×0.27 + (2²×0.27) + (3²×0.27) + (4²×0.09) + (5²×0.05) - 2.23²]

σ² = 6.47 - 4.9729 = 1.4971

σ² = 1.50

The standard deviation, σ :

σ = √σ²

σ = √1.4971

σ = 1.22

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