Question 1178774
Let's solve this problem step-by-step:

**Given Information:**

* X ~ N(190, 21²) (X is normally distributed with mean μ = 190 minutes and standard deviation σ = 21 minutes)

**a. Less than 160 minutes?**

1.  **Calculate the z-score:**
    * z = (X - μ) / σ
    * z = (160 - 190) / 21
    * z = -30 / 21
    * z ≈ -1.4286

2.  **Find the probability:**
    * P(X < 160) = P(z < -1.4286)
    * Using a z-table or calculator, P(z < -1.43) ≈ 0.0764

**b. Greater than 210 minutes?**

1.  **Calculate the z-score:**
    * z = (X - μ) / σ
    * z = (210 - 190) / 21
    * z = 20 / 21
    * z ≈ 0.9524

2.  **Find the probability:**
    * P(X > 210) = P(z > 0.9524)
    * Using a z-table or calculator, P(z > 0.95) ≈ 0.1711

**c. 215 to 245 minutes?**

1.  **Calculate the z-scores:**
    * z₁ = (215 - 190) / 21
    * z₁ = 25 / 21
    * z₁ ≈ 1.1905
    * z₂ = (245 - 190) / 21
    * z₂ = 55 / 21
    * z₂ ≈ 2.6190

2.  **Find the probability:**
    * P(215 < X < 245) = P(1.1905 < z < 2.6190)
    * Using a z-table or calculator:
        * P(z < 2.6190) ≈ 0.9956
        * P(z < 1.1905) ≈ 0.8830
        * P(1.1905 < z < 2.6190) = P(z < 2.6190) - P(z < 1.1905) ≈ 0.9956 - 0.8830 ≈ 0.1126