1. (Sec. 6.1) In a random sample of 80 components of a certain type, 12 are found to be defective. (a) Give a point estimate for the proportion of all such components that are not defective. (b) A system is to be constructed by randomly selecting two of these components and connecting them in a series. The series will function only if neither component is defective. Give an estimate for the proportion of all such systems that will function properly.

(a) The point estimate for the proportion of all such components that are not defective is given by the number of non-defective units in the sample divided by the sample size:

[tex]p=\frac{80-12}{80}\\p=0.85[/tex]

The proportion is 0.85.

(b) Assuming that the sample is large enough to accurately provide a point estimate for the whole population, this can be treated as a binomial model with probability of success (non-defective part) p = 0.85. Since both components must be non-defective for the system to work, the probability of two successes in two trials is:

[tex]P(x=2) = 0.85^2\\P(x=2) = 0.7225[/tex]

An estimate of 0.7225 for the proportion of all such systems that will function properly.