SOLUTION: A machine that produces a major part for an airplane engine is monitored closely. In the past, 10% of the parts produced would be defective. With a .95 probability, the sample size

Question 1190683: A machine that produces a major part for an airplane engine is monitored closely. In the past, 10% of the parts produced would be defective. With a .95 probability, the sample size that needs to be taken if the desired margin of error is .04 or less is
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
i think it works like this.

p = .10
q = 1 - p = .9
s = sqrt(p*q/n)

p is the probability of success.
q is the probability of failure.
s is the standard error.

moe is the margin of error.

at 95% confidence interval, the critical z-score will be plus or minus 1.96.

the 95% confidence interval is assumed to be two tailed, meaning that .025 of the area under the normal distribution curve will be in the tail on the left of the confidence interval and .025 of the area under the normal distribution curve will be in the tail on the right of the confidence interval, with .95 of the area under the normal distribution curve in the middle.

s = sqrt(p * q / n) = sqrt(.1 * .9 / n) = sqrt(.09 / n)
desired moe = plus or minus .04.
critical z = plus or minus 1.96.
z-score formula is z = (x - m) / s
with moe of plus or minus .04, then look at the positive moe and the formula becomes:
1.96 = .04 / s
multiply both sides by s and divide both sides by 1.96 and the formula becomes:
s = .04 / 1.96
since s = sqrt(.09/n), the formula becomes:
sqrt(.09/n) = .04/1.96
square both sides of the equation to get:
.09/n = (.04/1.96)^2
multiply both sides by n and divide both sides by (.04/1.96)^2 to get:
.09 / (.04/1.96)^2 = n
solve for n to get:
n = 216.09

with a sample size of 216.09, the value of s becomes sqrt(.09/216.09) = .0204081633.
z = (x - m) / s is the z-score formula.
solve for (x - m) to get:
(x - m) = z * s
when z = 1.96, this becomes (x - m) = 1.96 * s = 1.96 * .0204081633 = .04.

if the sample size were 216.09, then you would get an moe of exactly .04.
since the sample size needs to be an integer, then we go to the next higher integer to get a sample size of 217.
when the sample size is 217, s = sqrt(.09/217) = .020365327.

at 95% confidence interval, .....

-1.96 = (x - m) / s and 1.96 = (x - m) / s

solve these equations for (x - m) to get:

(x - m) = -1.96 * s for the lower confidence limit.
(x - m) = 1.96 * s for the higher confidence limit.

since s = .020365327, these formulas become:

(x - m) = -.0399160409 for the lower limit.
(x - m) = .0399160409 for the upper limit.

both margins of error are less than .04.
this is what is desired.
it is achieved when the sample size is 217 or greater.

let me know if you have any questions.
theo

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