Question

section 2.4 spotlight on statistics
objective 1 - compute z - scores and standardized scores
objective 2 - explore the normal distribution
objective 1 - compute z - scores and standardized scores
guided practice activity #12 - z - scores
you get a 63 on a test. is this bad?
to answer this question we need to know how this score compares to the rest of the class.
assuming the mean was 51 we now know you scored above average. but how far above
average?
to answer this question we need to know how spread out the scores are. if all the other scores
are between 49 and 53 then you did extremely well (a+), but if a bunch of people got in the 90s
and a bunch got in the 20s then you just did ok (b - ).
the standard deviation (average distance from the mean) gives us a measure of this spread.
assuming it was 5 points, your 63 is more than twice that distance above the mean.
mean ( overline{x}=51 ) stdev ( s = 5 )
( z=\frac{63 - 51}{5}=\frac{12}{5}=2.4 )
in fact your 63 is 12 points above the mean and the distance of 12 is 2.4 times the average
distance of 5 points. the number + 2.4 is your z - score: it tells me you are 2.4 times the standard
deviation from the mean. someone who got a 39 is 12 points below the mean and would have a
z - score of - 2.4.
definition: the z - score of a data value is the ratio of the data values distance from
the mean to the average distance from the mean:
( z=\frac{x-overline{x}}{s} )

1. z - scores allow you to compare different scales. for example, the admissions office is

comparing two different applicants, one has a math sat score of 640 and the other has a
math act score of 28. which student has a better score? you need to know that for the
sat: ( overline{x}=500 ) and ( s = 100 ), and for the act: ( overline{x}=22 ) and ( s = 5 ).

Explanation:

Step1: Calculate the z - score for the SAT score

The formula for the z - score is \(z=\frac{x - \overline{X}}{s}\). For the SAT score, \(x = 640\), \(\overline{X}=500\), and \(s = 100\).

$$ z_{SAT}=\frac{640 - 500}{100}=\frac{140}{100}=1.4 $$

Step2: Calculate the z - score for the ACT score

For the ACT score, \(x = 28\), \(\overline{X}=22\), and \(s = 5\).

$$ z_{ACT}=\frac{28 - 22}{5}=\frac{6}{5}=1.2 $$

Answer:

The student with the SAT score of 640 has a better score. The z - score for the SAT score (\(z_{SAT}=1.4\)) is higher than the z - score for the ACT score (\(z_{ACT}=1.2\)).