Question

A normal population has mean μ-9 and standard deviation σ-5. Round the answers to fou decimal places. Part 1 What proportion of the population is less than 20? The proportion of the population less than 20 is .986 Part 2 out of 2 What is the probability that a randomly chosen value will be greater than 5? The probability that a randomly chosen value will be greater than 5 is

Answer 1

a)
`P(X<20)=P((X-\mu)/(\sigma)<(20-\mu)/(\sigma))=P(Z<(20-9)/(5))=P(z<2.2)`

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

`P(z<2.2)=0.9861`

b)
`P(X<5)=P((X-\mu)/(\sigma)>(5-\mu)/(\sigma))=P(Z>(5-9)/(5))=P(z>-0.8)`

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

`P(z>-0.8)=1-P(z<-0.8)=1-0.2119= 0.7881`

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 variable of interest of a population, and for this case we know the distribution for X is given by:

`X \sim N(9,5)`

Where
`\mu=9` and
`\sigma=5`

Part a

We are interested on this probability

`P(X<20)`

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<20)=P((X-\mu)/(\sigma)<(20-\mu)/(\sigma))=P(Z<(20-9)/(5))=P(z<2.2)`

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

`P(z<2.2)=0.9861`

Part b

We are interested on this probability

`P(X>5)`

If we apply the z score formula to our probability we got this:

`P(X<5)=P((X-\mu)/(\sigma)>(5-\mu)/(\sigma))=P(Z>(5-9)/(5))=P(z>-0.8)`

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

`P(z>-0.8)=1-P(z<-0.8)=1-0.2119= 0.7881`