Question

It is known that the life of a particular auto transmission follows a normal distribution with mean 72,000 miles and standard deviation of 12,000 miles. a. A manufacturer warranties the transmission up to 40,000 miles. What percent of the transmissions will fail before the end of the warranty period? Would it be unusual for a transmission to expire before the warranty period? Explain. b. What percent of the transmissions will last longer than 65,000 miles? c. What percent of the transmissions last longer than 100,000 miles? Is this unusual? d. A transmission in the top 10% has been running for how many miles?

Answer 1

a)
`P(X<40000)=P((X-\mu)/(\sigma)<(40000-\mu)/(\sigma))=P(Z<(40000-72000)/(12000))=P(z<-2.67)`

`P(z<-2.67)=0.0038`

b)
`P(X>65000)=P((X-\mu)/(\sigma)>(65000-\mu)/(\sigma))=P(Z>(65000-72000)/(12000))=P(z>-0.583)`

`P(z>-0.583)=1-P(Z<-0.583)=1-0.280=0.720`

c)
`P(X>100000)=P((X-\mu)/(\sigma)>(100000-\mu)/(\sigma))=P(Z>(100000-72000)/(12000))=P(z>2.33)`

`P(z>2.33)=1-P(Z<2.33)=1-0.990=0.01`

Sicne this probability just represent 1% of the data we can consider this value as unusual.

d)
`z=1.28<(a-72000)/(12000)`

And if we solve for a we got

`a=72000 +1.28*12000=87360`

So the value of height that separates the bottom 90% of data from the top 10% is 87360.

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".

Part a

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

`X \sim N(72000,12000)`

Where
`\mu=72000` and
`\sigma=12000`

We are interested on this probability

`P(X<40000)`

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

`z=(x-\mu)/(\sigma)`

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

`P(X<40000)=P((X-\mu)/(\sigma)<(40000-\mu)/(\sigma))=P(Z<(40000-72000)/(12000))=P(z<-2.67)`

And we can find this probability using excel or the normal standard table and we got:

`P(z<-2.67)=0.0038`

Part b

`P(X>65000)`

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

`z=(x-\mu)/(\sigma)`

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

`P(X>65000)=P((X-\mu)/(\sigma)>(65000-\mu)/(\sigma))=P(Z>(65000-72000)/(12000))=P(z>-0.583)`

And we can find this probability using the complement rule and excel or the normal standard table and we got:

`P(z>-0.583)=1-P(Z<-0.583)=1-0.280=0.720`

Part c

`P(X>100000)`

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

`z=(x-\mu)/(\sigma)`

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

`P(X>100000)=P((X-\mu)/(\sigma)>(100000-\mu)/(\sigma))=P(Z>(100000-72000)/(12000))=P(z>2.33)`

And we can find this probability using the complement rule and excel or the normal standard table and we got:

`P(z>2.33)=1-P(Z<2.33)=1-0.990=0.01`

Sicne this probability just represent 1% of the data we can consider this value as unusual.

Part d

For this part we want to find a value a, such that we satisfy this condition:

`P(X>a)=0.1` (a)

`P(X<a)=0.9` (b)

Both conditions are equivalent on this case. We can use the z score again in order to find the value a.

As we can see on the figure attached the z value that satisfy the condition with 0.9 of the area on the left and 0.1 of the area on the right it's z=1.28. On this case P(Z<1.28)=0.9 and P(z>1.28)=0.1

If we use condition (b) from previous we have this:

`P(X<a)=P((X-\mu)/(\sigma)<(a-\mu)/(\sigma))=0.9`

`P(z<(a-\mu)/(\sigma))=0.9`

But we know which value of z satisfy the previous equation so then we can do this:

`z=1.28<(a-72000)/(12000)`

And if we solve for a we got

`a=72000 +1.28*12000=87360`

So the value of height that separates the bottom 90% of data from the top 10% is 87360.