The reputations (and hence sales) of many businesses can be severely damaged by

Question

The reputations (and hence sales) of many businesses can be severely damaged by shipments of manufactured items that contain a large percentage of defectives. For example, a manufacturer of alkaline batteries may want to be reasonably certain that fewer than 6% of its batteries are defective. Suppose a random sample of 300 batteries are selected from a very large shipment; each is tested and 5 defective batteries are found. (The manufacturer is analyzing the proportion of DEFECTIVE batteries.) Consider just one experimental unit - that is, one battery. What is the response variable for that one battery? Categorical or quantitative? Verify the three conditions for using the central limit theorem for inference on pi. Conduct a significance test to decide if there is sufficient evidence for the manufacturer to conclude that the fraction defective in the entire shipment is less than 6 percent. Be sure to state your conclusion in plain English in the context of the problem. Choose one of the statements for your conclusion, depending on your P-value: We have evidence to show that the fraction defective in the entire shipment is less than 6 percent (P-value = _) We do not have evidence to show that the fraction defective in the entire shipment is less than 6 percent.

Explanation / Answer

a) The response is categorical i.e. the battery is either defective or not defective. It follows Binomial distribution where the probobability of defective is 0.5 and non defective is 0.5

b)

c)

Test and CI for One Proportion

Test of p = 0.06 vs p < 0.06

95% Upper
Sample X N Sample p Bound Z-Value P-Value
1 5 300 0.016667 0.028824 -3.16 0.001

Using the normal approximation.

Since p value is 0.001 we reject the null hypothesis and conclude that the proportion of defective is less than 6%

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