Question

3. Explain all the basic properties of the model of the normal distribution. 4. In a factory, the length of a certain metal bar is normally distributed, and has a mean of 50 cm, and a standard deviation of 0.07 cm. Two bars are selected and the measurements are 50.05cm and 49.99 cm. To calculate the probability that the length falls between 50.05 and 49.99, we need to standardize the normal distribution by calculating the z- scores. What are the z-scores of the lengths of the two bars?

Answer 1

#4]

Let X be the length of certain metal bar, and X be the normally distributed with mean $\mu$ = 50 cm and standard deviation $\sigma$ = 0.07

Now, required probability = Pr( 49.99 < X < 50.05 )

                                        = $Pr(\frac{49.99-50}{0.07}< \frac{X-\mu}{\sigma}<\frac{50.05-50}{0.07} )$

                                        = Pr( -0.1429 < Z < 0.7143 )

Where Z is the standardize normal variates and z-score for 49.99 is -0.1429 and for 50.05 is 0.7143.

Pr( 49.99 < X < 50.05 ) = Pr( X < 50.05) - Pr( X < 49.99)

= Pr( Z < 0.7143) - Pr(Z < -0.1429)        

= 0.7625 - 0.4432

= 0.3193