Question

In preparation for the upcoming school year, a teacher looks at raw test scores on the statewide standardized test for the students in her class. Instead of looking at the scores relative to the norms in the state, the teacher wants to understand the scores relative to the students who will be in the class. To do so, she decides to convert the test scores into z-scores relative to the mean and standard deviation of the students in the class. The mean test score in her upcoming class is 49, and the standard deviation is 20.7. The teacher wants to identify those students who may need extra challenges. As a first cut, she decides to look at students who have z-scores above z = 2.00 Identify the test score corresponding to a z-score of above z=2.00. Round to the nearest whole number.

Answer 1

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

And we can solve for the value of X like this:

`X = \mu + z*\sigma`

And since we know that z=2 we can replace and we got:

`X = 49 +2*20.7= 90.4 \approx 90`

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

`X \sim N(49,20.7)`

Where
`\mu=49` and
`\sigma=20.7`

And for this case the z score is given by:

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

And we can solve for the value of X like this:

`X = \mu + z*\sigma`

And since we know that z=2 we can replace and we got:

`X = 49 +2*20.7= 90.4 \approx 90`