Chapter 6.8, Problem 15E

(a) .

To determine

To Calculate: The probability of the value of a normally distributed random variable being in a certain interval.

Answer to Problem 15E

The probability that a randomly chosen vehicle is travelling at a legal speed is $0.0668$

Explanation of Solution

Given Information:

• The mean of the normally distributed random variable is $\text{μ}=112$ and the standard deviation is $\text{σ}=8$
• The legal speed is $100km/h$

Concept Used:

• The probability distribution function of a normally distributed random variable with mean $\text{μ}$ and standard deviation $\text{σ}$ is given by
$f\left(x\right)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{\frac{-{\left(x-\mu \right)}^2}{2{\sigma }^2}}$
• If $f\left(x\right)$ is a probability density function of a random variable $X$ , then the probability that $X$ lies between $a$ and $b$ , where $a\le b$, is given by
$P\left(a\le X\le b\right)=\underset{a}{\overset{b}{{\displaystyle \int }}}f\left(x\right)dx$
• $e^{-x^2}$ does not have an anti-derivative. So, use a calculator or computer to calculate the definite integrals of such functions.

Calculation: If the legal speed is $100km/h$ , then speed $X$ should be less than $100km/h$ , that is $0\le X\le 100$

S0, put $\text{a}=0$ and $\text{b}=100$ in the above integral for probability to write

  $\text{P}\left(0\le X\le 100\right)=\underset{0}{\overset{100}{{\displaystyle \int }}}\frac{1}{8\sqrt{2\text{π}}}{e}^{-\frac{{\left(x-112\right)}^2}{2\times 8^2}}$

Input the values of lower limit $\text{a}$ , upper limit $\text{b}$ , the mean $\text{μ}$ and the standard deviation $\text{σ}$ in a calculator to get

  $\text{P}\left(0\le X\le 100\right)=0.0668$

Therefore, the probability that a randomly chosen vehicle is travelling at a legal speed is $0.0668$

(b).

To determine

To Calculate: The probability that the value of a random variable, having a normal distribution, is greater than a certain number.

Answer to Problem 15E

  $5.2\%$ of the motorists are targeted by the police for over speeding.

Explanation of Solution

Given Information:

• The mean of the normally distributed random variable is $\mu =112$ and the standard deviation is $\text{σ}=8$
• The probability to be calculated is for a speed of more than $125km/h$ .

Concept Used:

• The probability distribution function of a normally distributed random variable with mean $\text{μ}$ and standard deviation $\text{σ}$ is given by
$f\left(x\right)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{\frac{-{\left(x-\mu \right)}^2}{2{\sigma }^2}}$
• If $f\left(x\right)$ is a probability density function of a random variable $X$ , then the probability that $X$ lies between $a$ and $b$ , where $a\le b$, is given by
$P\left(a\le X\le b\right)=\underset{a}{\overset{b}{{\displaystyle \int }}}f\left(x\right)dx$
• $e^{-x^2}$ does not have an anti-derivative. So, use a calculator or computer to calculate the definite integrals of such functions.

Calculation: Put $\text{a}=125$ and $\text{b}=\infty$ in the above integral for probability to write

  $P\left(125\le X<\infty \right)=\underset{125}{\overset{\infty }{{\displaystyle \int }}}\frac{1}{8\sqrt{2\pi }}{e}^{-\frac{{\left(x-112\right)}^2}{2\times 8^2}}$

Enter the values of lower limit $\text{a}$ , upper limit $\text{b}$ , the mean $\text{μ}$ and the standard deviation $\text{σ}$ in a calculator to get

  $P\left(125\le X\le \infty \right)=0.052=5.2\%$

Note: While using a calculator, avoid the improper integral by replacing $\text{b}=\infty$ with $\text{b}=200$ , that is, a number that is finite but much higher than the mean and standard deviation.

Instead of $200$ , even higher numbers can be entered but the observed difference will be miniscule.

Therefore, $5.2\%$ of the motorists are targeted by the police for over speeding.