Question 1191724: A light bulb manufacturer finds that 5% of his bulbs last more than 500 hours. An improvement in the process meant that the mean lifetime was increased by 50 hours . In a new test, 20% of bulbs now lasted longer than 500 hours.
Find the mean and standard deviation of the original process
Found 2 solutions by Theo, ikleyn:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! i think i have it.
at least it looks good.
here's what the results look like.
in the first display, the probability of getting more than 500 is .05.
in the second diaply, the probability of getting more than 500 is .20 (rounded up to 2 decimal places.
in problems such as this, there is always the question about using rounded numbers or unrounded numbers.
since there are never any guidelines from the instructors that i can see, then it's always a question as to what amount of rounding is expected.
i'll show you what i did.
you are welcome to redo the numbers with the rounding required by your instructor.
the method should be the same.
here's what i did.
the z-score formula is z = (x - m) / s
z is the -score
x is the raw score
m is the mean
s is the standard deviation.
the z-score that has 5% of the area under the normal distribution curve to the right of it is equal to 1.645.
the z-score that has 20% of the area under the normal distribution curve to the right of it is equal to .842.
these are both rounded to 3 decimal places.
the 5% z-score formula becomes 1.645 = (500 - m) / s
the 20% z-score formula becomes .842 = (500 - (m + 50)) / s
simplify these formulas to get:
1.645 = (500 - m) / s
.842 = (450 - m) / s
multiply both sides of these formulas by s to get:
1.645 * s = 500 - m
.842 * s = 450 - m
add m to both sides of these formulas to get:
1.645 * s + m = 500
.842 * s + m = 450
subtract the second equation from the first to get:
.803 * s = 50
solve for s to get:
s = 50 / .803 = 62.26650062.
solve for m in both equations to get:
m = 500 - 1.645 * s
m = 450 - .842 * s
when s = 62.26650062, you get:
m = 500 - 1.645 * s = 397.5716065
m = 450 - .842 * s = 397.5716065.
you have:
s = 62.26650062
m = 397.5716065
m + 50 = 447.5716065
use these values to solve for the respective z-scores.
z = (500 - m) / s becomes z = 1.645
z = (500 - (m + 50)) / s becomes z = .842.
solve for the area to the right of these z-scores.
area to the right of 1.645 = .04998..... which rounds to .05.
area to the right of .842 = .19989..... which rounds to .20.
since i used rounded numbers, i didn't get the answer right on.
if i used numbers that weren't rounded, i would have been right on.
you can use rounded or unrounded numbers as required by your instructor to repeat this procedure, if that is necessary.
if not, then you're good to go.
for my calculations, i used the ti-84 plus calculator.
let me know if you have any questions.
theo
Answer by ikleyn(52780)  (Show Source):
You can put this solution on YOUR website! .
Looking at this problem, I'd say that its formulation is incorrect (or incomplete).
Would it be correct and complete, it must say that the standard deviation after improvement remained the same as before it.
Since the problem is silent about it (leaving it for the reader to guess), this problem is more like a puzzle than a Math problem.
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