# 5-2. Exercises

## 1. Exercises

**Ex 1.** Use Figure 12.2 "Cumulative Normal Probability" to find the probability indicated.

1. $$P(Z < −1.72)$$&#x20;
2. $$P(Z < 2.05)$$&#x20;
3. $$P(Z < 0)$$&#x20;
4. $$P(Z > −2.11)$$&#x20;
5. $$P(Z > 1.63)$$&#x20;
6. $$aP(Z > 2.36)$$&#x20;

**Ex 2.** Use Figure 12.2 "Cumulative Normal Probability" to find the probability indicated.

1. $$P(Z < −1.17)$$&#x20;
2. $$P(Z < −0.05)$$&#x20;
3. $$P(Z < 0.66)$$&#x20;
4. $$P(Z > −2.43)$$&#x20;
5. $$P(Z > −1.00)$$&#x20;
6. $$P(Z > 2.19)$$&#x20;

**Ex 3.** Use Figure 12.2 "Cumulative Normal Probability" to find the probability indicated.

1. $$P(−2.15 < Z < −1.09)$$&#x20;
2. $$P(−0.93 < Z < 0.55)$$&#x20;
3. $$P(0.68 < Z < 2.11)$$&#x20;

**Ex 4.** Use Figure 12.2 "Cumulative Normal Probability" to find the probability indicated.

1. $$P(−1.99 < Z < −1.03)$$&#x20;
2. $$P(−0.87 < Z < 1.58)$$&#x20;
3. $$P(0.33 < Z < 0.96)$$&#x20;

**Ex 5.** Use Figure 12.2 "Cumulative Normal Probability" to find the probability indicated.

1. $$P(−4.22 < Z < −1.39)$$&#x20;
2. $$P(−1.37 < Z < 5.11)$$&#x20;
3. $$P(Z < −4.31)$$&#x20;
4. $$P(Z < 5.02)$$&#x20;

**Ex 6.** Use Figure 12.2 "Cumulative Normal Probability" to find the probability indicated.

1. $$P(Z > −5.31)$$&#x20;
2. $$P(−4.08 < Z < 0.58)$$&#x20;
3. $$P(Z < −6.16)$$&#x20;
4. $$P(−0.51 < Z < 5.63)$$&#x20;

**Ex 7.** Use Figure 12.2 "Cumulative Normal Probability" to find the first probability listed. Find the second probability without referring to the table, but using the symmetry of the standard normal density curve instead. Sketch the density curve with relevant regions shaded to illustrate the computation.

1. $$P(Z < −1.08), P(Z > 1.08)$$&#x20;
2. $$P(Z < −0.36), P(Z > 0.36)$$&#x20;
3. $$P(Z < 1.25), P(Z > −1.25)$$&#x20;
4. $$P(Z < 2.03), P(Z > −2.03)$$&#x20;

**Ex 8.** Use Figure 12.2 "Cumulative Normal Probability" to find the first probability listed. Find the second probability without referring to the table, but using the symmetry of the standard normal density curve instead. Sketch the density curve with relevant regions shaded to illustrate the computation.

1. $$P(Z < −2.11), P(Z > 2.11)$$&#x20;
2. $$P(Z < −0.88), P(Z > 0.88)$$&#x20;
3. $$P(Z < 2.44), P(Z > −2.44)$$&#x20;
4. $$P(Z < 3.07), P(Z > −3.07)$$&#x20;

**Ex 9.** The probability that a standard normal random variable *Z* takes a value in the union of intervals $$(−∞, −a] ∪ \[a, ∞),$$ which arises in applications, will be denoted $$P(Z ≤ −a or Z ≥ a)$$ . Use Figure 12.2 "Cumulative Normal Probability" to find the following probabilities of this type. Sketch the density curve with relevant regions shaded to illustrate the computation. Because of the symmetry of the standard normal density curve you need to use Figure 12.2 "Cumulative Normal Probability" only one time for each part.

1. $$P(Z < −1.29 or Z > 1.29)$$&#x20;
2. $$P(Z < −2.33 or Z > 2.33)$$&#x20;
3. $$P(Z < −1.96 or Z > 1.96)$$&#x20;
4. $$P(Z < −3.09 or Z > 3.09)$$&#x20;

**Ex 10.** The probability that a standard normal random variable *Z* takes a value in the union of intervals $$(−∞, −a] ∪ \[a, ∞)$$ , which arises in applications, will be denoted $$P(Z ≤ −a or Z ≥ a)$$ . Use Figure 12.2 "Cumulative Normal Probability" to find the following probabilities of this type. Sketch the density curve with relevant regions shaded to illustrate the computation. Because of the symmetry of the standard normal density curve you need to use Figure 12.2 "Cumulative Normal Probability" only one time for each part.

1. $$P(Z < −2.58 or Z > 2.58)$$&#x20;
2. $$P(Z < −2.81 or Z > 2.81)$$&#x20;
3. $$P(Z < −1.65 or Z > 1.65)$$&#x20;
4. $$P(Z < −2.43 or Z > 2.43)$$&#x20;


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