AP Statistics 3.2.A Mean and SD of Sample Proportions

A large company produces an equal number of brand-name lightbulbs and generic lightbulbs. The director of

quality control sets guidelines that production will be stopped if there is evidence that the proportion of all

lightbulbs that are defective is greater than 0.10. The director also believes that the proportion of brand-name

lightbulbs that are defective is not equal to the proportion of generic lightbulbs that are defective. Therefore, the

director wants to estimate the average of the two proportions.

To estimate the proportion of brand-name lightbulbs that are defective, a simple random sample of

400 brand-name lightbulbs is taken and 44 are found to be defective. Let X represent the number of

brand-name lightbulbs that are defective in a sample of 400, and let $p_{X}$ represent the proportion of all

brand-name lightbulbs that are defective. It is reasonable to assume that X is a binomial random variable.

How many standard errors is the observed value of $\hat{p}_{X}$ from 0.10 ?

To estimate the proportion of generic lightbulbs that are defective, a simple random sample of 400 generic

lightbulbs is taken and 104 are found to be defective. Let Y represent the number of generic lightbulbs that are

defective in a sample of 400. It is reasonable to assume that Y is a binomial random variable and the distribution

of $\hat{p}_{Y}$ is approximately normal, with an approximate standard error of 0.0219. It is also reasonable to assume

that X and Y are independent.

The parameter of interest for the manager of quality control is D, the average proportion of defective lightbulbs

for the brand-name and the generic lightbulbs. D is defined as $D=\frac{p_{X}+p_{Y}}{2}$.