Question

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 PXrepresent the proportion of all brand-name lightbulbs that are defective. It is reasonable to assume that X is a binomial random variable.

Requried:
One condition for obtaining an interval estimate for Px is that the distribution of Px is approximately normal. Is it reasonable to assume that the condition is met?

Answer 1

X is a binomial random variable

Then the condition is met

Explanation:

Sample size n₁ = 400

X represents the number of brand-name lightbulbs)

P(X) = 44/400

P(X) = 0,11

X is a binomial random variable it only could be either no defective or defective ( only two conditions or values).

To make use of the condition of approximation of binomial distribution to a normal distribution it is required that the products:

p*n = 0,11*400 = 44 and q = 1 - 0,11 q = 0,89

q*n = 0,89 * 400 = 356

both p*n q*n are greater than 5.

Then the condition is met