The manufacturer of a certain type of new cell phone battery claims that the average life span of the batteries is 500 charges; that is, the battery can be charged at least 500 times before failing. To investigate the claim, a consumer group will select a random sample of cell phones with the new battery and use the phones through 500 charges of the battery. The proportion of batteries that fail to last through 500 charges will be recorded. The results will be used to construct a 95 percent confidence interval to estimate the proportion of all such batteries that fail to last through 500 charges. (a) Explain in context what it means to be 95 percent confident.
b) Suppose the consumer group conducts its investigation with a random sample of 5 cell phones with the new battery, and 1 battery out of the 5 fails to last through 500 charges. Verify all conditions for inference for a 95 percent confidence interval for a population proportion. Indicate whether any condition has not been met. Do not construct the interval.
(c) Based on the simulation, what proportion of the 95 percent confidence intervals capture the population proportion of 0.3 ? Explain how you determined your answer.
(ii) Based on the results of the simulation, is the alternate method better than the traditional method described in part (b) to construct a 95 percent confidence interval with a small sample size? Explain your reasoning.