An electrical component for one of the main products your company produces is specified to have an electrical impedance of 2.15 ohms.However,the manufacturing process is not perfect and there is some variation on the actual impedance of these components.A recent statistical study indicated that in fact,the impedances are normally distributed with a mean impedance of 2.15 ohms and a standard deviation of 0.05 ohms.When the impedance exceeds 2.25 ohms,the product malfunctions so that component must be rejected and replaced by one whose impedance is within the acceptable limit.If you need 1350 usable components,how many components should you order?
A) 1608
B) 1422
C) 1421
D) 1385
E) 1384

Question

Quizplus · Accepted Answer

To ensure that we have enough usable components, we need to order enough so that the probability of receiving at least 1350 usable components is high.

Let X be the number of usable components out of a sample of n components. We need to find the smallest value of n such that P(X ≥ 1350) is close to 1.

To find this value, we can use the normal approximation to the binomial distribution, since n is large and the probability of success (getting a usable component) is close to 0.5 (since the distribution is symmetric and the cutoff is at the mean).

The mean of X is np = n(0.5) = 0.5n. The standard deviation of X is sqrt(np(1-p)) = sqrt(0.5n(1-0.5)) = sqrt(0.25n) = 0.5sqrt(n).

We want to find the smallest value of n such that:

P(X ≥ 1350) = P((X - 0.5n)/sqrt(0.25n) ≥ (1350 - 0.5n)/sqrt(0.25n))

Using the standard normal distribution, we can find the z-score corresponding to (1350 - 0.5n)/sqrt(0.25n).

z = (1350 - 0.5n)/sqrt(0.25n) - 2.15)/(0.05)

We want to find the smallest value of n such that:

P(Z ≥ z) = 1 - P(Z < z) ≥ 0.99

From standard normal tables or using a calculator, we can find that z = 2.326.

Substituting in the expression for z, we have:

(1350 - 0.5n)/sqrt(0.25n) - 2.15)/(0.05) = 2.326

Solving for n (e.g. by trial and error or using a calculator), we get n ≈ 1385.

Therefore, we should order at least 1385 components to ensure that we have a high probability of obtaining at least 1350 usable components.

The closest answer choice to this is D) 1385.

An Electrical Component for One of the Main Products Your