Let Z be the standard normal distribution (having mean 0 and standard deviation 1). For a given probability p, we often need to know the value z such that P(Z <= z) = p, or P(z <= Z) = p, or P(-z <= Z <= z) = p. Such a value z is called a ** z-score**, or critical value. The most commonly used z-scores are 1.645, 1.96. 2.326, and 2.576, where

We can calculate these or any other critical values using the **ZSCORE** program. To execute the program, first enter **1**, **2**, or **3** to specify that you want the critical value from a left probability, or the critical value from a right probability, or the critical values from a middle probability.

If you enter 1, then the program will find the value z such that P(Z <= z) = p. If you enter 2, then the program will find the value z such that P(z <= Z) = p. If you enter 3, then the program will give the values -z and z such that P(-z <= Z <= z) = p. In all cases, the z-scores are rounded to three decimal places.

After specifying either 1, 2, or 3, simply enter the desired probability p.

** Example.** Find the critical value z that satisfies:

(a) P(Z <= z) = 0.15

(b) P(z <= Z) = 0.035

(c) P(-z <= Z <= z) = 0.80

*Solution.* (a) After calling up the **ZSCORE** program, enter **1** to specify a left probability, then enter **.15** for this left probability. We receive a critical value of -1.036. Thus, P(Z <= -1.036) = 0.15.

(b) First enter **2** to specify a right probability, then enter **.035** for this right probability. We receive a critical value of 1.812. Hence, P(1.812 <= Z) = 0.035

(c) First enter **3** to specify a middle probability, then enter **.8** for this middle probability. We receive the critical values -1.282 and 1.282. Hence, P(-1.282 <= Z <= 1.282) = 0.80.

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