Quality control. As part of a quality control process for computer chips, an engineer at a factory randomly samples 212 chips during a week of production to test the current rate of chips with severe defects. She finds that 27 of the chips are defective.
(a) What population is under consideration in the data set?
(b) What parameter is being estimated?
(c) What is the point estimate for the parameter?
(d) What is the name of the statistic can we use to measure the uncertainty of the point estimate?
(e) Compute the value from part (d) for this context.
(f) The historical rate of defects is 10%. Should the engineer be surprised by the observed rate of defects
during the current week?
(g) Suppose the true population value was found to be 10%. If we use this proportion to recompute the value in part (e) using p = 0.1 instead of pˆ, does the resulting value change much?

Given That, As part of a quality control process for computer chips, an engineer at a factory randomly samples 212 chips during a week of production to test the current rate of chips with severe defects. She finds that 27 of the chips are defective.

(a) The sample is from all computer chips manufactured at the factory during the week of production. We might be tempted to generalize the population to represent all weeks, but we should exercise caution here since the rate of defects may change over time.

(b) The fraction of computer chips manufactured at the factory during the week of production that had defects.

(c) Estimate the parameter using the data: phat = 27/212 = 0.127.

(d) Standard error (or SE).

(e) Compute the SE using phat = 0.127 in place of p:

SE ≈ √(phat(1−phat)/n) = 0.023.

(f) The standard error is the standard deviation of phat. A value of 0.10 would be about one standard error away from the observed value, which would not represent a very uncommon deviation. (Usually beyond about 2 standard errors is a good rule of thumb.) The engineer should not be surprised.

(g) Recomputed standard error using p = 0.1: SE = 0.021. This value isn't very different, which is typical when the standard error is computed using relatively similar proportions (and even sometimes when those proportions are quite different!).