Mopeds (small motorcycles) are very popular in Europe because of their mobility, ease of operation, and low cost. An article2 described a ro

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.

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Raelynn 4 weeks 2021-11-10T01:12:45+00:00 1 Answer 0 views 0

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    2021-11-10T01:13:49+00:00

    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

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45:7+7-4:2-5:5*4+35:2 =? ( )