# Normal Probability distribution to find true length of an object

• Jul 17th 2010, 12:39 PM
cruxkitty
Normal Probability distribution to find true length of an object
The question is as follows:

The normal distribution is commonly used to model the variability expected when making measurements. In this context, a measured quantity x is assumed to have a normal distribution whose mean is assumed to be the true value of the object being measured. The precision of the measuring instrument determines the standard deviation of the distribution.

a) If the measurements of the length of an object have a normal probability distribution with a standard deviation 1mm what is the probability that a single measurement will lie within 2mm of the true length of the object?

b) Suppose the measuring instrument in part a is replaced witha a more precise measuring instrument having a standard deviation of .5mm What is the probability that a measurement from the new instrument lies within 2mm of the true length of the object?

- What I think part a is asking is for me to find the probability of P(x-mu/sigma<x<x-mu/sigma) but I am confused because it also makes me think that it is asking P(-1<z<3) and I am just not sure how to even approach solving this problem. I think that when it asks for the probability of 2mm of the true length that it is asking for the area between two sets of z values. Any help , examples, or explanations would be appreciated.
• Jul 18th 2010, 01:31 AM
CaptainBlack
Quote:

Originally Posted by cruxkitty
The question is as follows:

The normal distribution is commonly used to model the variability expected when making measurements. In this context, a measured quantity x is assumed to have a normal distribution whose mean is assumed to be the true value of the object being measured. The precision of the measuring instrument determines the standard deviation of the distribution.

a) If the measurements of the length of an object have a normal probability distribution with a standard deviation 1mm what is the probability that a single measurement will lie within 2mm of the true length of the object?

b) Suppose the measuring instrument in part a is replaced witha a more precise measuring instrument having a standard deviation of .5mm What is the probability that a measurement from the new instrument lies within 2mm of the true length of the object?

- What I think part a is asking is for me to find the probability of P(x-mu/sigma<x<x-mu/sigma) but I am confused because it also makes me think that it is asking P(-1<z<3) and I am just not sure how to even approach solving this problem. I think that when it asks for the probability of 2mm of the true length that it is asking for the area between two sets of z values. Any help , examples, or explanations would be appreciated.

It is asking what is the probability of the absolute value of a RV with zero mean (the error in the measured length) and SD 1mm being less than 2mm. Which is the same thing in this case as asking the probability that a standard normal RV lies between +/-2 SD.

In the secod case this translates to the same as asking the probability that a standard normal RV lies between +/-4 SD

CB