Question at position 47 Items produced by a manufacturing process are supposed to weigh 90 grams. The manufacturing process is such, however, that there is variability in the items produced and they do not all weigh exactly 90 grams. The distribution of weights can be approximated by a Normal distribution with a mean of 90 grams and a standard deviation of 1 gram. About what percentage of the items will either weigh less than 87 or more than 93 grams?6%0.3%0.15%94%99.7%

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BlackTom AI · Accepted Answer

We’re told the weights are Normal with mean 90 g and standard deviation 1 g, and we want the probability that a weight is either below 87 g or above 93 g.
First, convert the cutoffs to z-scores: (87 − 90)/1 = −3 and (93 − 90)/1 = +3. So we’re looking for P(X < 87 or X > 93) = P(|Z| > 3) where Z ~ N(0,1).
For a standard normal, P(|Z| > 3) is the tail probability beyond ±3 sigma. The one-sided tail P(Z > 3) is about 0.00135, so the two tails combined are approximately 0.0027, i.e., 0.27%.
Now evaluate each option:
- 6%: This is far too large; it would correspond to a much wider interval than ±3 sigma (roughly ±1.55 to capture 94% in the center), not ±3 sigma. Misconception: treating 3-sigma as frequent as 6% outside.
- 0.3%: This matches the calculation 0.27% which is very close to 0.3% when rounded to one significant figure or a single decimal place. Reasonable approximation of the two-tail area beyond ±3.
- 0.15%: This is roughly half of the true two-tail probability beyond ±3 sigma, suggesting the student may have halved the tail or used an incorrect z-value.
- 94%: This would imply only 6% outside, which corresponds to a much smaller cutoff (about ±1.5 sigma) and is inconsistent with ±3 sigma.
- 99.7%: This is the central portion within ±3 sigma, not the outside region. If 99.7% are inside, only 0.3% are outside, but the phrasing here would be about the inside probability, not the outside.
In summary, the correct interpretation of the ±3 sigma threshold yields a two-tailed outside probability of roughly 0.27%, which is best represented by the option closest to that value.

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