Question

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 temperaturerange of the oven follows a normal distribution with a mean of 450 F and a standard deviation of 25 F. Leslie Larsen, president of the factory, is concerned about thelarge number of defective cores that have been produced in the past several months. If the oven gets hotter than 475 F, the core is defective. What is the probabilitythat the oven will cause a core to be defective? What is the probability that the temperature of the oven will range from 460 F to 470 F.

Answer 1

Almost constant temperatures are maintained by an industrial oven used to cure sand cores for a factory manufacturing engine blocks for small cars.

For the oven, the temperature range follows a normal distribution with a mean and standard deviation.

Thus if random variable X denotes the range of temperature of oven then symbolically,

Or

It is also given that the core will be defective if the oven gets temperature higher than. Thus the probability of core being defective is. Also

The standard normal variate Z, for is,

This implies,

Such normal probabilities are calculated with the help of areas table for standard normal distribution given in Appendix C and since it is known that the normal distribution is symmetric about mean, the table gives the area under the curve to the for the values to left of Z that is for the less than type value of Z.

In the given table, values of Z are listed in the left column with top row showing the second place of decimal of Z .

Here we have to find the value of, therefore find 1.0 in the left column and 0.00 in the top row. Now corresponding to this value, the area listed in table is 0.8413. That is,

Thus probability of the core being defective is.

The probability of having temperature of the oven in range from to is denoted by probability expression. This implies

Thus first obtain the separate probabilities and. Then take their difference to obtain the desired probability.

The standard normal variate Z, for and are calculated as follows:

And

This implies,

And

Now from the table of areas for the standard normal distribution, look for 0.8 in the left-hand column and 0.00 in the top row and find the area listed in the table corresponding to this value as 0.7881. That is,

Again from the table of areas for the standard normal distribution, look for 0.4 in the left-hand column and 0.00 in the top row and find the area listed in the table corresponding to this value as 0.6554. That is,

Therefore the required probability will be,

Hence it is concluded that the probability of having temperature of the oven in the range from to is.