Question

If \(Z\) is a standard normal variable, use the table of standard normal probabilities to find: (a) \(\operatorname{Pr}(Z<0)\) (b) \(\operatorname{Pr}(-12.54)\).

Short answer

In summary, using the table of standard normal probabilities, we find the following probabilities: (a) \(\operatorname{Pr}(Z<0) = 0.5\) (b) \(\operatorname{Pr}(-12.54) = 0.0056\)

Step-by-step solution

Standard Normal Probability Table

First, make sure you have a table of standard normal probabilities. These tables usually represent the area under the standard normal curve to the left of a given Z value. You can find these tables in your textbook or online (https://en.wikipedia.org/wiki/Standard_normal_table).

Understanding the Table

The table consists of rows representing tenths of the Z-score and columns representing hundredths of the Z-score. For example, to find the area under the curve to the left of Z=1.24, we would go down the rows to the row labeled 1.2, and then go across the columns to the column labeled 0.04, finding the value 0.8925 in the table.

Find the probability for Z

Since Z is a standard normal variable, the mean is 0. The table gives cumulative probabilities, that is, the area under the curve to the left of the given Z score. To find Pr(Z<0), we need to find the area under the curve to the left of 0. Since the standard normal distribution is symmetric about its mean, we know that Pr(Z<0) = Pr(Z>0) = 0.5.

Find the probability for -1

To find the probability Pr(-11)=1-Pr(Z<1)). Now, subtract the two areas: Pr(-1<Z<1) = 0.8413 - 0.1587 = 0.6826.

Find the probability for Z>2.54

To find Pr(Z>2.54), we need to find the area under the curve to the right of Z=2.54. Referring to the probability table, locate the value for Z=2.54, which is 0.9944. This value represents Pr(Z<2.54) or the area under the curve to the left of Z=2.54. Since we want the area to the right, we subtract from 1: Pr(Z>2.54) = 1 - 0.9944 = 0.0056. In summary, the probabilities are: a) Pr(Z<0) = 0.5 b) Pr(-12.54) = 0.0056