Question

# Consider the following. (Give your answers correct to two decimal places.) (a) Find the standard score...

(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.

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.

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