Question

The life of Sunshine CD players is normally distributed with a mean of $4.1$years and a standard deviation of $1.3$years. A CD player is guaranteed for three years. We are interested in the length of time a CD player lasts. Find the probability that a CD player will break down during the guarantee period.

a. Sketch the situation. Label and scale the axes. Shade the region corresponding to the probability.

b. $P(0<x<\_\_\_\_\_\_\_\_\_\_\_\_)=\_\_\_\_\_\_\_\_\_\_\_$(Use zero for the minimum value of x.)

Short answer

1. The graph for the $P(0<x<3)$is

b. The value of the$P(0<x<3)$is$0.1979$.

Step-by-step solution

Given information (Part a)

Given in the question that

Mean$=4.1$

Standard deviation$=1.3$

Solution (Part a)

The graph for the $P(0<x<3)$with the shaded region is given below

Given information (Part b)

Given in the question that

Mean$=4.1$

Standard deviation$=1.3$

Solution (Part b)

Here we need to calculate the probability is below,

$P(0<x<3)=P(x<3)-P(x<0)$

$=P\left(\frac{x-\mu }{\sigma }<\frac{3-\mu }{\sigma }\right)-P\left(\frac{x-\mu }{\sigma }<\frac{0-\mu }{\sigma }\right)$

$=P\left(Z<\frac{3-4.1}{1.3}\right)-P\left(Z<\frac{0-4.1}{1.3}\right)$

$=P(Z<-0.8462)-P(Z<-3.1538)$

$=0.1987-0.0008\left\{\begin{array}{l}\text{From standard}\\ \text{normal table}\end{array}\right\}$

$=0.1979$