Question

In order to get promoted to the next grade, a student is required to score above 55 in the final test. Assume the scores to be normally distributed with a mean of 60 and a standard deviation of 5.5. Calculate the percentage of students who will be promoted to the next grade.

Answer 1

`P(X>55)=P((X-\mu)/(\sigma)>(55-\mu)/(\sigma))=P(Z>(55-60)/(5.5))=P(z>-0.909)`

And we can find this probability with the complement rule:

`P(z>-0.909)=1-P(z<-0.909)`

And we can use excel or the normal standard table and we got:

`P(z>-0.909)=1-P(z<-0.909)=1-0.182= 0.818`

So then we expect about 81.8% of students that will be promoted

Explanation:

Let X the random variable that represent the scores of promotion of a population, and for this case we know the distribution for X is given by:

`X \sim N(60,5.5)`

Where
`\mu=60` and
`\sigma=5.5`

We want to find the following probability:

`P(X>55)`

And we can use the z score formula given by:

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

Using the z score formula we got:

`P(X>55)=P((X-\mu)/(\sigma)>(55-\mu)/(\sigma))=P(Z>(55-60)/(5.5))=P(z>-0.909)`

And we can find this probability with the complement rule:

`P(z>-0.909)=1-P(z<-0.909)`

And we can use excel or the normal standard table and we got:

`P(z>-0.909)=1-P(z<-0.909)=1-0.182= 0.818`

So then we expect about 81.8% of students that will be promoted