Vehicle speed on a particular bridge in China can be modeled as normally distributed. (a) If 5% of all vehicles travel less than 39.18 m/h and 10% travel more than 73.23 m/h, what are the mean and standard deviation of vehicle speed?

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".

Solution to the problem

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

[tex]X \sim N(\mu,\sigma)[/tex]

For this case we know the following conditions:

[tex] P(X<39.18) = 0.05 [/tex]

[tex]P(X>73.23) = 0.1[/tex]

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

[tex]z=\frac{x-\mu}{\sigma}[/tex]

We look for a one z value that accumulate 0.05 of the area in the left tail and we got: [tex] z_ 1= -1.64[/tex] and we need another z score that accumulates 0.1 of the area on the right tail and we got [tex] z_2 = 1.28[/tex]

And we have the following equations:

[tex] -1.64 = \frac{39.18 -\mu}{\sigma}[/tex] (1)

[tex] 1.28 = \frac{73.23 -\mu}{\sigma}[/tex] (2)

From equation (1) and (2) we can solve for [tex]\mu[/tex] and we got:

[tex] \mu = 39.18 + 1.64 \sigma[/tex] (3)

[tex] \mu = 73.23 - 1.28 \sigma[/tex] (4)

And we can set equal equations (3) and (4) and we got:

[tex] 39.18 +1.64 \sigma = 73.23 -1.28 \sigma[/tex]

And solving for the deviation we got:

[tex] 2.92\sigma = 34.05[/tex]

[tex]\sigma = 11.66[/tex]

And the mean would be:

[tex] \mu = 39.18 +1.64 *11.66 = 58.304[/tex]

joaobezerra joaobezerra · Answer 2 · 2021-11-26T10:39:57+01:00

Using the normal distribution, it is found that:

The mean is of 58.33 m/h.
The standard deviation is of 11.64 m/h.

In a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

It measures how many standard deviations the measure is from the mean.
After finding the z-score, we look at the z-score table and find the p-value associated with this z-score, which is the percentile of X.

In this problem, 39.18 m/h is the 5th percentile, hence, when X = 39.18, Z has a p-value of 0.05, so Z = -1.645.

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]-1.645 = \frac{39.18 - \mu}{\sigma}[/tex]

[tex]39.18 - \mu = -1.645\sigma[/tex]

[tex]\mu = 39.18 + 1.645\sigma[/tex]

Additionally, 73.23 m/h is the 100 - 10 = 90th percentile, hence, when X = 73.23, Z has a p-value of 0.9, so Z = 1.28.

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]1.28 = \frac{73.23 - \mu}{\sigma}[/tex]

[tex]73.23 - \mu = 1.28\sigma[/tex]

[tex]\mu = 73.23 - 1.28\sigma[/tex]

Equaling both equations, we find the standard deviation, hence:

[tex]39.18 + 1.645\sigma = 73.23 - 1.28\sigma[/tex]

[tex]2.925\sigma = 34.05[/tex]

[tex]\sigma = \frac{34.05}{2.925}[/tex]

[tex]\sigma = 11.64[/tex]

Then, we can find the mean:

[tex]\mu = 73.23 - 1.28\sigma = 73.23 - 1.28(11.64) = 58.33[/tex]

A similar problem is given at https://brainly.com/question/24663213