Question 1018287: the diameters of pencils produced by a certain machine are normally distributed with a mean of 0.30 inches and a standard deviation of 0.01 inches. In a random sample of 460 pencils, approximately how many would you expect to have a diameter less than 0.293 inches?
Answer by mathmate(429)   (Show Source):
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Question:
the diameters of pencils produced by a certain machine are normally distributed with a mean of 0.30 inches and a standard deviation of 0.01 inches. In a random sample of 460 pencils, approximately how many would you expect to have a diameter less than 0.293 inches?
Solution:
This is a problem solved using the normal distribution, or the equivalent on the calculator.
First we need to find the Z-score, where
Z=(X-μ)/σ
=(0.293-0.30)/0.01
=-0.7
where
X=0.293 (the desired upper limit)
μ=mean,
σ=standard deviation.
Next we look up the normal distribution for Z=-0.7 and find the probability of Z<=-0.07 as 0.24196. Which means
P(X<=0.293)=0.24196
for 460 pencils, we expect 460*0.24196=111.3
So the diameter of 111 pencils are expected to be less than or equal to be 0.293 inches or less.
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