6.4 Find the Value of z

Suppose SAT Critical Reading scores are normally distributed with a mean of 503 and a standard deviation of 109. A university plans to offer tutoring jobs to students whose scores are in the top 10%. What is the minimum score required for the job offer?

Step 1: find Z Use: "negative infinity to -z" look for the probability closest to 0.10 (10%) Looking at the row and column titles, you will see that z=1.28 (1.2+0.08) when you add them together Step 2: plug the info into the formula Z=X-µ÷σ 1.28=X-503÷109 109*1.28=X-503 139.52=X-503 139.52+503=X X=642.52 ≈643

find the value of z which divides the normal distribution so that half the area is on one side and half is on the other

The standard normal curve has a mean value of 0 and a standard deviation of 1. Since the standard normal curve is symmetrical about its mean, the area under the standard normal curve is divided into two equal halves by the mean. Thus the value of z which divides the normal curve into two equal halves is given by z=mean=0-1.

Find the value of z such that 0.9464 of the area lies between −z and z.

Use: "area under the normal curve (Z) from 0-z" divide 0.9464 by 2 (0.9464÷2=.4732) look for the probability closest to 0.4732. Looking at the row and column titles, you will see that z=1.93 (0.03+1.9) when you add them together

Find the value of z such that the area to the left of −z plus the area to the right of z is 0.2846.

Use: "negative infinity to -z" divide 0.2846 by 2 (0.2846÷2=.1423) look for the probability closest to 0.1423. Looking at the row and column titles, you will see that z=1.07 (0.07+1.0) when you add them together

Area in the Tails

Use: "negative infinity to -z" divide the value given in half scan through the body of the table until you find the divided value

What z-value has an area of 0.352 to its right?

Use: "negative infinity to -z" look for the probability closest to 0.3520. Looking at the row and column titles, you will see that z=.38 (0.3+0.08) when you add them together

What z-value has an area of 0.4 to its left?

Use: "negative infinity to -z" look for the probability closest to 0.4. Looking at the row and column titles, you will see that z=−0.25 (0.2+0.05) when you add them together

Area Between −z and z

to find the value of z such that a given area lies between −z and z find the z-value for a given area between the 2 z values Use: "area under the normal curve (Z) from 0-z" divide the value given in half scan through the body of the table until you find the divided value

Area to the Left of z

to find the value of z that corresponds to a tail probability of some value find the z-value for a given area to the left of z. Use: "negative infinity to -z" scan through the body of the table until you find the given area

Area to the Right of z

to find the value of z that corresponds to a tail probability of some value find the z-value for a given area to the right of z. Use: "negative infinity to -z" scan through the body of the table until you find the given area