Mathematics, 17.12.2019 00:31 genyjoannerubiera

1. suppose the failure time of a certain mechanical component is normally distributed with mean value of 10 years and standard deviation of 2 years. suppose a mechanical component of above type is randomly selected. what is the probability that the failure time of the component is (a) between 8 years to 12 years. (b) more than 13 years.

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Answer from: leiort0930

a) P(8

b) P(X13)=1-0.933=0.0668

Step-by-step explanation:

1) 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".

2) Part a

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:

$X \sim N(10,2)$

Where $\mu=10$ and $\sigma=2$

We are interested on this probability

P(8

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(8

And we can find this probability on this way:

P(-1

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

P(-1

3) Part b

We are interested on this probability

P(X13)

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:

$P(X13)=P(\frac{X-\mu}{\sigma}\frac{13-\mu}{\sigma})=P(Z\frac{13-10}{2})=P(Z1.5)$

And we can use the complement rule to find this probability:

$P(Z1.5)=1-P(Z\leq 1.5)$

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

P(Z1.5)=1-P(Z

Answer from: Quest

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step-by-step explanation:

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Answer from: Quest

the solution of the equation is 2.23 ⇒ answer b

step-by-step explanation:

* lets revise the meaning of exponential function

- the form of the exponential function is y = ab^x, where a ≠ 0, b > 0 ,

b ≠ 1, and x is any real number

- it has a constant base b

- it has a variable exponent x

- to solve an exponential equation, take the log or ln of both sides,

and solve for the variable

* lets solve the problem

∵ $7^{x}=77$

- the base is 7 and the exponent is x

- insert ㏑ in both sides

∴ $ln(7^{x})=ln(77)$

- use the rule $ln(a^{n})= nln(a)$

∴ x ㏑(7) = ㏑(77)

- to find x divide both sides by ㏑(7)

∴ $x=\frac{ln(77)}{ln(7)}=2.23$

* the solution of the equation is 2.23 to the nearest 2 decimal places

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