find the probability

Question

A company produces steel rods. The lengths of the steel rods are normally distributed with a mean of 175.6-cm and a standard deviation of 1-cm.

Find the probability that the length of a randomly selected steel rod is greater than 174-cm.
P(X > 174-cm) =


Enter your answer as a number accurate to 4 decimal places.

Answer 1

Hello again, Madeline,

Okay, this is the exact same type of problem. You are going to convert your values to a z-score, right? 

Again, the formula for a obtaining a z-score is as follows: 


z = x - xbar
     ----------

          sd 


Now, plug in your known values. 

z = 174 - 175.6
     ---------------

            1 

z = -1.6
      -----

        1 

z = -1.6   Remember the z score table I showed you? Here is the link:  http://www.z-table.com/


You are going to do the same thing again. Make sure you are using the negative value z-score table. 

Then locate the row which is -1.6  Once you locate that value, look for the column at the top which is .00. I have explained why in my previous answer to you. 

The value located at the intersection of -1.6 and .00 is .0548   


Here is the difference between this problem and the last one. You are being asked for the probability that the randomly selected rod is GREATER than the mean of 174, correct? Remember, when you look at the Bell Curve on top of the chart it clearly tells you that the value is for the LEFT side of the Bell Curve when working on this table.  

In your problem you are being asked for the value to the right of the value you just obtained. To determine that, you know the area under the bell curve is 1.000 so you must subtract your obtained value above from this. 

Therefore: 

  1.0000
-  .0548
----------
.9452   (This is the answer to four decimal places). 


To determine the probability in percent all you do is move the decimal two places to the RIGHT because you are being asked for the probability that you randomly pull a rod GREATER THAN your mean. 

Therefore, the probability is 94.52% 


I hope I have helped and that you enjoy your Super Bowl Weekend off from school. Have a great day. If you have any further questions, please do not hesitate to contact me in the comments below. 

Madeline: I am adding an addendum here to ensure you understand the reason we subtracted your obtained z score value from 1.000. The entire area under the normal bell curve is 1.000. The value you obtained, that is the z-score value you originally calculated covers ALL values to the left of that value. Therefore, you MUST subtract that obtained area from 1.000 to arrive at the correct probability since the question is asking the probability you randomly pull a rod greater than the mean. If you have ANY additional questions regarding this, please respond in the comment below. Thanks again.