Chapter 14.5, Problem 34E

a.

To determine

To find the number of tires would expect to last between $35,000$ and $45,000$ miles

Answer to Problem 34E

  $6830$

Explanation of Solution

Given:

The lifetimes of a certain type of car tire are normally distributed. The mean life time is $40,000$ miles with a standard deviation of $5000$ miles, consider a sample $10,000$ tires.

Calculation:

Find the percent of the data is between $35,000$ and $45,000$

  

The graph shows that $35,000$ − $45,000$ define an interval that can be represented by $\overline{X}\pm ts$

And the values of $35,000$ , $45,000$ can be written as

  $\begin{array}{l}35,000=40,000-5000\\ 45,000=40,000+5000\end{array}$

Thus

  $\begin{array}{l}t\sigma =5000\\ t(5000)=5000\\ t=1\end{array}$

The probability that the score is in between $35,000$ and $45,000$ is $0.683$

Hence the total number of tires that expect to last between $35,000$ and $45,000$ is $0.683(10000)=6830$

b.

To determine

To find the number of tires would expect to last between $35,000$ and $40,000$ miles

Answer to Problem 34E

  $4775$

Explanation of Solution

Given:

The lifetimes of a certain type of car tire are normally distributed. The mean life time is $40,000$ miles with a standard deviation of $5000$ miles, consider a sample $10,000$ tires.

Calculation:

Find the percent of the data is between $35,000$ and $40,000$

  

The graph shows that $35,000$ − $40,000$ define an interval that can be represented by $\overline{X}\pm ts$

And the values of $35,000$ , $40,000$ can be written as

  $\begin{array}{l}35,000=40,000-5000\\ 40,000=40,000+0\end{array}$

Thus

  $\begin{array}{l}40,000-t\sigma =30,000\\ 40,000-t(5000)=30,000\\ t=2\end{array}$

The probability that the score is in between $35,000$ and $40,000$ is $0.4775$

Hence the total number of tires that expect to last between $35,000$ and $40,000$ is $0.4775(10000)=4775$

c.

To determine

To find the number of tires would expect to last less than $40,000$ miles

Answer to Problem 34E

  $5000$

Explanation of Solution

Given:

The lifetimes of a certain type of car tire are normally distributed. The mean life time is $40,000$ miles with a standard deviation of $5000$ miles, consider a sample $10,000$ tires.

Calculation:

Find the percent of the data is less than $40,000$

  

It is half of the total therefore the total number of tires that expect to last less than $40,000$ is $5000$

d.

To determine

To find the number of tires would expect to last more than $50,000$ miles

Answer to Problem 34E

  $225$

Explanation of Solution

Given:

The lifetimes of a certain type of car tire are normally distributed. The mean life time is $40,000$ miles with a standard deviation of $5000$ miles, consider a sample $10,000$ tires.

Calculation:

The probability that the score is in more than $50,000$ is $0.5-0.4775=0.0225$

Hence the total number of tires that expect to last more than $50,000$ milesis $0.0225(10000)=225$

e.

To determine

To find the number of tires would expect to last less than $25,000$ miles

Answer to Problem 34E

  $15$

Explanation of Solution

Given:

The lifetimes of a certain type of car tire are normally distributed. The mean life time is $40,000$ miles with a standard deviation of $5000$ miles, consider a sample $10,000$ tires.

Calculation:

The probability that the score is in less than $25,000$ is $0.5-0.4985=0.0015$

Hence the total number of tires that expect to last less than $25,000$ miles is $0.0015(10000)=15$