The strength of a steel beam is measured in the size of deflection (in micromete
ID: 3131921 • Letter: T
Question
The strength of a steel beam is measured in the size of deflection (in micrometers, umm) that results when subjecting the beam to a force of 10 000 pounds. The strength of steel beams is believed to be normally distributed. Suppose a materials engineer selects a random sample of 5 beams, tests them, and finds their deflections as follows (in mum):72 78 68 73 75 Find a 95% confidence interval for the true mean strength of beams of this kind (as measured in mum of deflection). How many more beams would need to be tested in order to construct a 95% confidence interval which would estimate the true mean strength with a maximum error of plusminus 2 mum in deflection?Explanation / Answer
a)
Note that
Lower Bound = X - t(alpha/2) * s / sqrt(n)
Upper Bound = X + t(alpha/2) * s / sqrt(n)
where
alpha/2 = (1 - confidence level)/2 = 0.01
X = sample mean = 73.2
t(alpha/2) = critical t for the confidence interval = 3.746947388
s = sample standard deviation = 3.701351105
n = sample size = 5
df = n - 1 = 4
Thus,
Lower bound = 66.99769846
Upper bound = 79.40230154
Thus, the confidence interval is
( 66.99769846 , 79.40230154 ) [ANSWER]
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b)
Note that
n = z(alpha/2)^2 s^2 / E^2
where
alpha/2 = (1 - confidence level)/2 = 0.025
Using a table/technology,
z(alpha/2) = 1.959963985
Also,
s = sample standard deviation = 3.701351105
E = margin of error = 2
Thus,
n = 13.15699646
Rounding up,
n = 14
Hence, as we alredy have 5 data poins, we need 14 - 5 = 9 more. [ANSWER: 9]
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