Question

Diane's Tree Trimming Service has trimmed hundreds of trees in the Greater Denver area. The time it takes to trim a tree is normally distributed with a mean of 32 minutes and a standard deviation of 5 minutes (Use this distribution to answer the next several questions). What percent of trees get trimmed between 28.5 and 31.5 minutes? Group of answer choices 21.82% 75.80% 70.22% 29.78%

Answer 1

Answer:
`P(28.5<X<31.5)=P((28.5-\mu)/(\sigma)<(X-\mu)/(\sigma)<(31.5-\mu)/(\sigma))=P((28.5-32)/(5)<Z<(31.5-32)/(5))=P(-0.7<z<-0.1)`And we can find this probability with this difference:

`P(-0.7<z<-0.1)=P(z<-0.1)-P(z<-0.7)`

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

`P(-0.7<z<-0.1)=P(z<-0.1)-P(z<-0.1) =0.4602-0.2420=0.2182`

So then the best answer would be 21.82 %

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".

Solution to the problem

Let X the random variable that represent the time it takes to trim a tree of a population, and for this case we know the distribution for X is given by:

`X \sim N(32,5)`

Where
`\mu=32` and
`\sigma=5`

We are interested on this probability

`P(28.5<X<31.5)`

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(28.5<X<31.5)=P((28.5-\mu)/(\sigma)<(X-\mu)/(\sigma)<(31.5-\mu)/(\sigma))=P((28.5-32)/(5)<Z<(31.5-32)/(5))=P(-0.7<z<-0.1)`And we can find this probability with this difference:

`P(-0.7<z<-0.1)=P(z<-0.1)-P(z<-0.7)`

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

`P(-0.7<z<-0.1)=P(z<-0.1)-P(z<-0.1) =0.4602-0.2420=0.2182`

So then the best answer would be 21.82 %