find probability

Question

A manufacturer knows that their items have a normally distributed length, with a mean of 18.8 inches, and standard deviation of 2.3 inches.

If one item is chosen at random, what is the probability that it is less than 15.8 inches long? Round your answer to 4 decimal places.

Answer 1

Hello Madeline,


This is a typical z-score transformation problem: 

The formula for a z-score is: 

z = x - xbar
     -----------

            sd 


x is your value given 
xbar is your mean
sd is your standard deviation 


Therefore: 

z = 15.8 - 18.8
     --------------
             2.3 

z = -3
     ----
     2.3 

z = -1.30 


Now, you need to look up the associated value on the z-score table. I will provide you a link to do so. http://www.z-table.com/ 

If you will notice the first column on the first table (which is for negative z scores) is your z value. Go down that column until you come to 1.3. Once there look immediately to your right at the column, .00

PLEASE NOTE: If you would have had a value of let's just say 1.35 you would have gone to the column .05 which adds .05 to the z score of 1.3. However, your z score is 1.3 which means you are interested in the next column which is .00  All you are doing is adding the .00 to the original z score you arrived at. 

When you look at the intersecting value for the row of 1.3 and .00 you find the following value, right? 

.0968  This is your answer to four decimal places. But you want a probability which is determined by you moving your decimal two places to the RIGHT. 

That means the probability that one item chosen at random is less than 15.8 inches 9.7%  You round up due to the thousandths being an eight. 



Finally, as a tutor I would not be doing my job if I did not explain this answer for you a bit more. Notice the values (probabilities) you arrive at are for those to the LEFT of the arrived value. You can see this on the bell curve provided on the chart. This is due to your z-score being a negative value.  

I hope this assists and if you have ANY questions, please feel free to ask in a follow-up comment so I can answer you. I hope you have a great weekend.