Question 1122524
mean = 16 ounces.
standard deviation = .5 ounces.
weights are normally distributed.
what percent will weight:
a.  less than 15.5 ounces.
b.  between 15.8 and 16.2 ounces.


there are calculators that make this easy, but i will show you how to do it manually using just the normal distribution table.


you need to find the z-scores before using the z-score tables.


the formula for the z-score is:


z = (x-m)/s


z is the z-score
x is the raw score you are comparing against the mean.
m is the mean
s is the standard deviation.


in your problem, ....


m = 26
s = .5


for part a, your raw score is 15.5.


formula becomes z = (15.5 - 16) / .5
result is z-score = -1.


look up the z-score of -1.0 in the z-score table to find that the area to the left of that z-score is .1587.


that means that 15.87% of the boxes will weight less than 15.5 ounces.


for part b, you will be looking for the z-scores associated with a raw score of 15.8 and a raw score of 16.2.


formula for 15.8 becomes z = (15.8 - 16) / .5.
result is z-score = -.4


formula for 16.2 becomes z = (16.2 - 16) / .5.
result is z-score = .4


you will look up the area to the left of a z-score of -.4 and to the left of a z-score of .4.


those areas will be .3446 to the left of -.4 and .6554 to the left of .4


subtract the smaller area from the larger area to get .3108 which tells you that 31.08% of the scores will be between 15.8 and 16.2 ounces.


there is an online calculator that i use to check my results.


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


using z-scores, that calculator confirms the results from the table.


the z-score results from that calculator are shown below:


<img src = "http://theo.x10hosting.com/2018/090701.jpg" alt="$$$" >


<img src = "http://theo.x10hosting.com/2018/090702.jpg" alt="$$$" >


using raw scores, that calculator confirms the results from the z-scores.


the raw score results from that calculator are shown below:


<img src = "http://theo.x10hosting.com/2018/090703.jpg" alt="$$$" >


<img src = "http://theo.x10hosting.com/2018/090704.jpg" alt="$$$" >


results from the table i used are shown below:


<img src = "http://theo.x10hosting.com/2018/090705.jpg" alt="$$$" >


<img src = "http://theo.x10hosting.com/2018/090706.jpg" alt="$$$" >


that table can be found at <a href = "http://www.z-table.com/" target = "_blank">http://www.z-table.com/</a>


when you are using the table, the first column is the integer part the the tenth part of the z-score.


the second column through the 10th column provide the hundredth portion of the z-score.


a z-score of -.4 is equal to -4.0 plus .00 column which is the area in the second column of that row.


a z-score of -.42 would be equal to a -4.0 plus .03 .02 column which would be the area in the 4th column of that row.


in numbers, -.4 gets you .3446 which is in the second column of that row and -.42 gets you .3372 which is in the fourth column of that row.


-.42 is equal to -.4 + .02 = -.42


the .01, .02, .03, ... etc columns are shown at the top of the table.


you can see that clearer in the second of the two displays of the z-score table.