Question 1122524: The mean weight of a box of cereal filled by a machine is 16.0 ounces, with a standard deviation of 0.5 ounce. If the weights of all the boxes filled by the machine are normally distributed, what percent of the boxes will weigh the following amounts?
a. less than 15.5 ounces
b.between 15.8 and 16.2 ounces
I would like to see the steps if possible
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! 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 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.
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