Question

Mopeds (small motorcycles) are very popular in Europe because of their mobility, ease of operation, and low cost. An article2 described a rolling bench test for determining maximum vehicle speed. A normal distribution with mean value 46.8 km/h and standard deviation 1.75 km/h is postulated. Consider randomly selecting a single such moped.

Answer 1

Answer:

a) $P(X<50)=P(\frac{X-\mu}{\sigma}<\frac{50-\mu}{\sigma})=P(Z<\frac{50-46.8}{1.75})=P(z<1.829)$

And we can find this probability using the normal standard table:

$P(z<1.829)=0.966$

b) $P(X>48)=P(\frac{X-\mu}{\sigma}>\frac{48-\mu}{\sigma})=P(Z>\frac{48-46.8}{1.75})=P(Z>0.686)=1-P(z<0.686)$

And we can find this probability using the normal standard table and the complement ruel:

$P(Z>0.686)=1-P(z<0.686)=1-0.754=0.246$

c) $P(46.8-1.5*1.75<X<46.8+1.5*1.75)=P(\frac{44.175-\mu}{\sigma}<\frac{X-\mu}{\sigma}<\frac{49.425-\mu}{\sigma})=P(\frac{44.175-46.8}{2.6}<Z<\frac{49.425-46.8}{2.6})=P(-1.009<z<1.009)$

And we can find this probability with this difference:

$P(-1.009<z<1.009)=P(z<1.009)-P(z<-1.009)$

And in order to find these probabilities we can use the tables for the normal standard distribution, excel or a calculator.  

$P(-1.009<z<1.009)=P(z<1.009)-P(z<-1.009)=0.844-0.156=0.687$

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Assuming the following questions:

Part a: What is the probability that the maximum speed is at most 50 km/h?

Let X the random variable that represent the heights of a population, and for this case we know the distribution for X is given by:

$X \sim N(46.8,1.75)$  

Where $\mu=46.8$ and $\sigma=1.75$

We are interested on this probability

$P(X<50)$

And the best way to solve this problem is using the normal standard distribution and the z score given by:

$z=\frac{x-\mu}{\sigma}$

If we apply this formula to our probability we got this:

$P(X<50)=P(\frac{X-\mu}{\sigma}<\frac{50-\mu}{\sigma})=P(Z<\frac{50-46.8}{1.75})=P(z<1.829)$

And we can find this probability using the normal standard table:

$P(z<1.829)=0.966$

Part b: What is the probability that maximum speed is at least 48 km/h?

$P(X>48)=P(\frac{X-\mu}{\sigma}>\frac{48-\mu}{\sigma})=P(Z>\frac{48-46.8}{1.75})=P(Z>0.686)=1-P(z<0.686)$

And we can find this probability using the normal standard table and the complement ruel:

$P(Z>0.686)=1-P(z<0.686)=1-0.754=0.246$

Part c:What is the probability that maximum speed differs from the mean value by at most 1.5 standard deviations?

$P(46.8-1.5*1.75<X<46.8+1.5*1.75)=P(\frac{44.175-\mu}{\sigma}<\frac{X-\mu}{\sigma}<\frac{49.425-\mu}{\sigma})=P(\frac{44.175-46.8}{2.6}<Z<\frac{49.425-46.8}{2.6})=P(-1.009<z<1.009)$

And we can find this probability with this difference:

$P(-1.009<z<1.009)=P(z<1.009)-P(z<-1.009)$

And in order to find these probabilities we can use the tables for the normal standard distribution, excel or a calculator.  

$P(-1.009<z<1.009)=P(z<1.009)-P(z<-1.009)=0.844-0.156=0.687$