Question

Consider the following. (Give your answers correct to two decimal places.)

(a) Find the standard score (z) such that the area below the mean and above z under the normal curve is 0.3961.


(b) Find the standard score (z) such that the area below the mean and above z under the normal curve is 0.4812.


(c) Find the standard score (z) such that the area below the mean and above z under the normal curve is 0.3733.

Answer 1

a) We are to find k here such that:
P( k < Z < 0) = 0.3961 because 0 is the mean for the standard normal variable.

Now, as we know that P(Z < 0) = 0.5, because the distribution is symmetric about mean, therefore
P(Z < k) = 0.5 - P( k < Z < 0) = 0.5 - 0.3961 = 0.1039

From standard normal tables, we have:
P(Z < -1.260) = 0.1039

Therefore -1.26 is the required test statistic value here.

b) Again using the same methodology as above,
P(Z < k) = P(Z < 0.5) - P(k < Z < 0.5) = 0.5 - 0.4812 = 0.0188

From standard normal tables, we have:
P(Z < -2.08) = 0.0188

Therefore -2.08 is the required z value here.

c) Again using the same methodology as above,
P(Z < k) = P(Z < 0.5) - P(k < Z < 0.5) = 0.5 - 0.3733 = 0.1267

From standard normal tables, we have:
P(Z < -1.14) = 0.1267

Therefore -1.14 is the required z value here.