Question

Assume that z-scores are normally distributed with a mean of 0 and a standard deviation of 1.

If P(0<z<a)=0.4554, find a.

(Round to two decimal places.)

Answer 1

Solution:

Given:  z-scores are normally distributed with a mean of 0 and a standard deviation of 1.

That is: z ~ Standard Normal distribution ( Mean = 0 , Standard Deviation = 1 )

We have to find value of 'a' such that:

P( 0 < z < a ) = 0.4554

Thus we use following steps:

P( 0 < z < a ) = P( z < a ) - P( z < 0 )

0.4554 = P( z < a ) - P( z < 0 )

To get P( z < 0) , look in z table for z = 0.0 and 0.00 and find area.

P( z < 0.00 ) = 0.5000

Thus we get:

0.4554 = P( z < a ) - P( z < 0 )

0.4554 = P( z < a ) - 0.5000

( Adding 0.5000 on both sides , we get)

0.4554 + 0.5000 = P( z < a ) - 0.5000 + 0.5000

0.9554 = P(z < a)

That is:

P( z < a) = 0.9554

Now look in z table for area = 0.9554 or its closest area and find corresponding z value.

Area 0.9554 corresponds to 1.7 and 0.00

thus z value = 1.70

That means: P( z < 1.70 ) = 0.9554

That is: P( z < a )= P( z < 1.70) =0.9554

Thus required value of z = 1.70.