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 http://davidmlane.com/hyperstat/z_table.html
using z-scores, that calculator confirms the results from the table.
the z-score results from that calculator are shown below:
using raw scores, that calculator confirms the results from the z-scores.
the raw score results from that calculator are shown below:
results from the table i used are shown below:
that table can be found at http://www.z-table.com/
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.