Question

For each of the following z-score locations in a normal distribution, determine whether the tail is on the left side or the right side of the distribution and find/give the proportion that is located in the tail (There will be two answers for each question and drawn a distrubtion.

a. z = +1.75
b. z = +0.80
c. z = –0.70

Answer 1

a) For this case we have
`z =1.75`, and this value is higher than 0, so then would be on the right tail. And we can find the probability in the tail like this:

`P(z>1.75) =1-P(z<1.75) =1-0.960= 0.04`

`P(z<1.75)= 0.960`

b) For this case we have
`z =0.8`, and this value is higher than 0, so then would be on the right tail. And we can find the probability in the tail like this:

`P(z>0.8) =1-P(z<0.8) =1-0.788= 0.212`

`P(z<0.8)= 0.788`

c) For this case we have
`z =-0.70`, and this value is lower than 0, so then would be on the left tail. And we can find the probability in the tail like this:

`P(z<-0.7)= 0.242`

`P(z>-0.7)=1- P(Z<-0.7) =1-0.242= 0.758`

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

Part a

For this case we have
`z =1.75`, and this value is higher than 0, so then would be on the right tail. And we can find the probability in the tail like this:

`P(z>1.75) =1-P(z<1.75) =1-0.960= 0.04`

`P(z<1.75)= 0.960`

Part b

For this case we have
`z =0.8`, and this value is higher than 0, so then would be on the right tail. And we can find the probability in the tail like this:

`P(z>0.8) =1-P(z<0.8) =1-0.788= 0.212`

`P(z<0.8)= 0.788`

Part c

For this case we have
`z =-078`, and this value is lower than 0, so then would be on the left tail. And we can find the probability in the tail like this:

`P(z<-0.7)= 0.242`

`P(z>-0.7)=1- P(Z<-0.7) =1-0.242= 0.758`

The results are on the figure attached for this case.