Question 887125: Could you show me the calculation for the following two problems.
1. The lifetime of light bulbs of a particular type are normally distributed with a mean of 370 hours and a standard deviation of 5 hours. What percentage of bulbs has a lifetimes that lie within 1 standard deviation of the mean on either side? Your answer was 68%, but I'm not able to come up with that same answer and I need to understand the calculation.
2. If the amount of a monthly phone bill is normally distributed with a mean of $50 and a standard deviation of $10. Find the 25th percentile. You gave an answer of 0.674%, but when I do the calculation I come up with a different number. Would you be able to show me the calculation so I can understand this process?
Thanks.
Fran
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! mean = 370
standard deviation = 5
the area of the distribution curve that is within plus or minus 1 standard deviation from the mean can be found in the following manner.
look for a z score of 1 and look for a z score of -1.
you will find that the area to the left of a z score of 1 is equal to .8413
you will find that the area to the left of a z score of -1 is equal to .1587
the area in between is the larger area minus the smaller area which is equal to .6826.
that can be rounded to 68%.
your mean is 370 hours
your standard deviation is 5 hours.
1 standard deviation above 370 would be 370 + 5 = 375
1 standarfd deviation below 370 would be 370 - 5 = 365
68% of your scores will be between 365 and 375.
calculate the z score of 375 as follows:
z score = (x - m) / sd
x is the raw score
m is the mean
sd is the standard deviation.
that becomes (375 - 370) / 5 = 5/5 = 1
calculate the z score of 365 as follows:
z score = (365 - 370)/5 = -5/5 = -1
the z score table i am using can be found here:
http://lilt.ilstu.edu/dasacke/eco148/ztable.htm
you have a mean of 50 and a standard deviation of 10
you want to find the 25th percentile.
this means that 25% of the scores are below that.
you look up in the z score table an area to the left of the z score of .25
in the table i used, that would be a z score between -.67 and -.68
if you used a calculator that calculates it finer than that, then the answer would be -.67447864895
that's a lot more accurate than the table can give you.
with the table a z score of -.67 gives you an area to the left of it of .2514 and a z score of -.68 gives you an area to the left of it of .2483.
if you can't get even close to those numbers than you're doing something wrong.
send me an email and tell me what you are doing and getting so i can determine where you went wrong, if you did.
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