Question

A large company produces an equal number of brand-mame lightbulbs and generic lightbulbs. The director of quality control sets guidelines that lightbulbs that are defect lightbulbs that are defective is not equal to the director wants to estimate the average of the two proportions. Production will be stopped if there is evidence that the proportion of all ve is greater than 0. 10. The director also believes that the proportion of brand-name proportion of generic lightbulbs that are defective. Therefore, the o estmate the proportion of brand-name lightbulbs that are defective, a simple random sample of 400 brand-name lightbulbs 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.

(a) One condition for obtaining an interval estimate for px is that the distribution of px is approximately number of is taken and 44 are found to be defective. Let X represent the normal Is it reasonable to assume that the condition is met? Justify your answer.

(b) The suandard error of hr is approumately O 0156 Show how the value of the standard error is calculated

Answer 1

Using the Central Limit Theorem, we have that:

a) Since there are at least 10 successes and 10 failures, the condition is met.

b) Using the formula $s = \sqrt{\frac{p(1 - p)}{n}}$, with n = 400 and p = 0.11, the standard error is of 0.0156.

What does the Central Limit Theorem state?

It states that for a proportion p in a sample of size n, the sampling distribution of sample proportion is approximately normal with mean $\mu = p$ and standard deviation $s = \sqrt{\frac{p(1 - p)}{n}}$, as long as $np \geq 10$ and $n(1 - p) \geq 10$.

Item a:

44 are found to be defective, hence:

• np = 44.

• n(1 - p) = 356.

Which means that the conditions are met.

Item b:

We have that the parameters are:

$n = 400, p = \frac{44}{400} = 0.11$

Hence, the standard error is given by:

$s = \sqrt{\frac{0.11(0.89)}{400}} = 0.0156$

More can be learned about the Central Limit Theorem at https://brainly.com/question/24663213