**Answer:**

**a) P [ X < 24 mm ] = 0,3015 or P [ X < 24 mm ] = 30,15 %**

**b) P [ X > 32 mm ] = 0,1251 or P [ X > 32 mm ] = 12,51 %**

**c) P [ 25 < X < 30 ] = 0,4964 or P [ 25 < X < 30 ] = 49,64 %**

**d) z(s) = 0,84**

**Step-by-step explanation:**

**Normal Distribution N ( μ₀ ; σ ) is N ( 26,5 ; 4,8 )**

**a) P [ X < 24 mm ] = ( X - μ₀ ) / σ**

**P [ X < 24 mm ] = (24 - 26,5)/ 4,8 = - 0,5208 ≈ - 0,52**

**P [ X < 24 mm ] = - 0,52 **

**And from z-table we find area for z score**

**P [ X < 24 mm ] = 0,3015 or P [ X < 24 mm ] = 30,15 %**

**b)P [ X > 32 mm ] = 1 - P [ X < 32 mm ] **

**P [ X < 32 mm ] = ( 32 - 26,5 ) / 4,8**

**P [ X < 32 mm ] = 5,5/4,8 = 1,1458 ≈ 1,15**

**P [ X < 32 mm ] = 1,15**

**And from z-table we get**

**P [ X < 32 mm ] = 0,8749**

**Then:**

**P [ X > 32 mm ] = 1 - 0,8749**

**P [ X > 32 mm ] = 0,1251 or P [ X > 32 mm ] = 12,51 %**

**c) P [ 25 < X < 30 ] = P [ X < 30 ] - P [ X < 25 ]**

**P [ X < 30 ] = 30 - 26,5 / 4,8 = 0,73**

**From z-table P [ X < 30 ] = 0,7673**

**P [ X < 25 ] = 25 - 26,5 / 4,8 = - 0,3125 ≈ - 0,31**

**From z-table P [ X < 25 ] = 0,2709**

**Then**

** P [ 25 < X < 30 ] = 0,7673 - 0,2709**

** P [ 25 < X < 30 ] = 0,4964 or P [ 25 < X < 30 ] = 49,64 %**

**d) If 20 %**

**z- score for 20% is from z-table**

**z(s) = 0,84**