Practise deciding whether a sample-proportion distribution is approximately normal by checking expected successes and failures and explaining failed conditions.
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 pX represent the proportion of all
brand-name lightbulbs that are defective. It is reasonable to assume that X is a binomial random variable.
One condition for obtaining an interval estimate for pX is that the distribution of p^X is approximately
normal. Is it reasonable to assume that the condition is met? Justify your answer.
Part (a):
It is reasonable to assume that the distribution is approximately normal. The required condition is that
there are at least 10 successes and 10 failures in the sample. In this case there are 44 defective lightbulbs
and 356 non-defective lightbulbs, thus both exceed the minimum of 10 required.