Question

Assume a normal distribution with μ = 0 and σ = 1, find the z-score(s) for each statement below:

a) the area is 0.4 to the left of z


b) the area is 0.18 to the right of z


c) the middle area is 0.31 between −z and z
__________________________

The heights of a certain population of corn plants follow a distribution with a mean of 137 cm and a standard deviation 23.19 cm. Find the raw scores for the following z-scores:

a) z = 0.98


b) z = -0.97


c) z = 1.56


c) z = -1.54

Answer 1

Solution :-

a) P(Z < 0.4)

Sketch the curve.

The probability that Z < 0.4 is equal to the blue area under the curve.

Use the standard normal table to find this area

We conclude that:

P(Z < 0.4) = 0.6554

b) P (Z > 0.18)

Sketch the curve.

The probability that Z > 0.18 is equal to the blue area under the curve.

To find the area above 0.18, first we will find the area below 0.18 (the white one).

Use the standard normal table to find the area below 0.18.

We conclude that the white area = 0.5714, so:

Blue area = 1 - white area = 1 - 0.5714 = 0.4286

Last statement can be writed in math notationa as:

P(Z > 0.18) = 1−P(Z < 0.18)

P(Z > 0.18) = 1−0.5714

P(Z > 0.18) = 0.4286

c) P(−0.31< Z < 0.31)

Sketch the curve.

The probability that −0.31<Z<0.31is equal to the blue area under the curve.

To find the probability of P (−0.31<Z<0.31), we use the following formula:

P(−0.31 < Z < 0.31) = P(Z < 0.31) − P(Z < −0.31)

P( Z < 0.31) can be found by using the standard normal table.

We see that P(Z < 0.31) = 0.6217.

P(Z < −0.31) can be found by using the following fomula.

P( Z < −a) = 1−P(Z < a)

After substituting a = 0.31 we have:

P(Z < −0.31) = 1−P(Z < 0.31)

P(Z < 0.31) can be found by using the standard normal table.

We see that P ( Z < 0.31 ) = 0.6217

so,

P(Z < −0.31) = 1−P(Z < 0.31) = 1−0.6217 = 0.3783

At the end we have:

P(-0.31 < Z < 0.31) = 0.2434