Question

Joses ACT score had a z-score of .75.what was his ACT score?

mean score=21.1 sd=5.2

Answer 1

`X \sim N(21.1,5.2)`

Where
`\mu=21.1` and
`\sigma=5.2`

And the z score is given by:

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

And since z = 0.75 we can replace like this:

`0.75 = (x- 21.1)/(5.2)`

And if we solve for x we got:

`x = 21.1 +0.75*5.2 = 25`

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

`X \sim N(21.1,5.2)`

Where
`\mu=21.1` and
`\sigma=5.2`

And the z score is given by:

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

And since z = 0.75 we can replace like this:

`0.75 = (x- 21.1)/(5.2)`

And if we solve for x we got:

`x = 21.1 +0.75*5.2 = 25`