An industrial oven used to cure sand cores for a factory man

An industrial oven used to cure sand cores for a factory manufacturing engine blocks for small cars is able to maintain fairly constant temperatures.The temperature range of the oven follows a normal distribution with a mean 430 F and a standard deviation of 25 F. Leslie Larsen, president of the factory.,is corcerned about the large number of defective cores that have been produced in the past several months. If the oven gets hotter than 451 F the core is defective .What is the probability that the oven will cause a core to be defective?What is the probability that the temperature of the oven will range from 440 F to 458 F?

Solution

Mean ( u ) =430
Standard Deviation ( sd )=25
Normal Distribution = Z= X- u / sd ~ N(0,1)                  
a)              
P(X > 451) = (451-430)/25
= 21/25 = 0.84
= P ( Z >0.84) From Standard Normal Table
= 0.2005                  
b)
To find P(a < = Z < = b) = F(b) - F(a)
P(X < 440) = (440-430)/25
= 10/25 = 0.4
= P ( Z <0.4) From Standard Normal Table
= 0.65542
P(X < 458) = (458-430)/25
= 28/25 = 1.12
= P ( Z <1.12) From Standard Normal Table
= 0.86864
P(440 < X < 458) = 0.86864-0.65542 = 0.2132