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

Find the ${70}^{th}$percentile of the distribution for the time a CD player lasts

a. Sketch the situation. Label and scale the axes. Shade the region corresponding to the lower $70\%$.

b. $P(x<k)=\_\_\_\_\_\_\_\_\_\_$Therefore,$k=\_\_\_\_\_\_\_\_\_$

Short answer

1. The graph for the ${70}^{th}$percentile is shown below

b. The value of ${70}^{th}$percentile is$P(x<k)=0.70$and$k=4.78172$.

Step-by-step solution

Step: Given information (Part a)

Given in the question that

Mean$=4.1$

Standard deviation$=1.3$

solution (Part b)

The graph for the ${70}^{th}$percentile

That is $P(x<k)=0.70$and it has been shown as a shaded region in the below graph

Given information (Part b)

Given in the question that

Mean$=4.1$

Standard deviation$=1.3$

Solution (part b)

The probability can be calculated as

$P(x<k)=0.70$

It can be calculated below

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

$0.70=P\left(Z<\frac{k-4.1}{1.3}\right)$

$\frac{k-4.1}{1.3}=0.5244\left\{\begin{array}{l}\text{From standard}\\ \text{normal table}\end{array}\right\}$

$k-4.1=0.5244\times 1.3$

$k=0.6817+4.1$

$k=4.78172$