Question 1135425
seat designed for pilots weighing between 140 and 181 pounds.


new population of pilots has a mean weight of 150 pounds and a standard deviation of 28.6 pounds.


find probability that a randomly selected pilot will weigh between 140 and 181 pounds.


the following calculator does the job very nicely for you.


it can be found at <a href = "http://davidmlane.com/hyperstat/z_table.html" target = "_blank">http://davidmlane.com/hyperstat/z_table.html</a>


this calculator tells you hat the probability that a randomly selected pilot will weight between 140 and 181 pounds is equal to .4975.


here's what the results of the use of this calculator looks like.


<img src = "http://theo.x10hosting.com/2019/022507.jpg" alt="$$$" >


you can also do it the normal way by finding the z-score and looking up the area in the normal distribution table


doing it that way, you need to find the lower z-score and the upper z-score and then find the area in between.


the z-score formula is z = (x - m) / sd


z is the z-score.
x is the raw score.
m is the mean.
sd is the standard deviation


your mean is 150 and your standard deviation is 28.6.


your lower x-score is 140.
your upper x-score is 181.


the z-score formula is z = (x-m) / sd


the lower z-score would be z = (140 - 150) / 28.6 = -.3496503497.


the upper z-score would be z = (181 - 150) / 28.6 = 1.083916084.


to use the tables, you would round these to -0.35 and 1.08.


in the z-score table, the area to the left of -0.35 equals .36315 and the area for the left of 1.08 = .85993.


the area in between is .85993 minus ..36315 = .49676.


that's pretty close to .4975 using the online calculator, the difference being due to rounding.


here's another normal distributor z-score calculator by university of iowa.


<a href = "" target = "_blank">https://homepage.stat.uiowa.edu/~mbognar/applets/normal.html</a>https://homepage.stat.uiowa.edu/~mbognar/applets/normal.html


this calculator is used similar to how you would use the z-score table, except it allows more details in the z-score and also interpolates to give you a more accurate answer.


the calculator works off the raw score and the mean and the standard deviation, or it works off the z-score.


when in z-score mode, the mean is 0 and the standard deviation is 1.


the first calculator referenced also works off the z-score or off the raw score and the mean and the standard deviation.


here's the results of using the second referenced calculator.


<img src = "http://theo.x10hosting.com/2019/022508.jpg" alt="$$$" >


<img src = "http://theo.x10hosting.com/2019/022509.jpg" alt="$$$" >


the area getween the two z-scores is .8608 minus .3633 = .4975.


that agrees with the resulted from the use of the first referenced calculator.


the z-score table i used can be found at <a href = "https://www.math.arizona.edu/~rsims/ma464/standardnormaltable.pdf" target = "_blank">https://www.math.arizona.edu/~rsims/ma464/standardnormaltable.pdf</a>