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.

In some studies, large sample sizes are not feasible because of the time and expense involved. However, small sample sizes can create issues with inference procedures. A simulation can be used to investigate what might happen with intervals constructed from small sample sizes. Consider a population in which 30 percent of the population displays a certain characteristic. For each trial of the simulation, 5 observations are selected from the population and the sample proportion pˆ is calculated, where pˆ represents the proportion in the sample that display the characteristic. The following table shows the frequency distribution of pˆ in the 1,000 trials. Also shown are the upper and lower endpoints of a 95 percent confidence created from the value of pˆ, using the formula pˆ±1.96pˆ(1−pˆ)n−−−−−√. For example, the sample proportion of 0.4 occurred 309 times in the 1,000 trials and produced a confidence interval of (−0.029,0.829).

pˆ Frequency Lower Endpoint Upper Endpoint
0 168 0 0
0.2 360 −0.151 0.551
0.4 309 −0.029 0.829
0.6 133 0.171 1.029
0.8 28 0.449 1.151
1.0 2 1 1

(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.

(d) For small sample sizes, an alternate method of constructing a confidence interval is available. To implement this method, first, include an additional 4 observations in the sample, 2 of which are successes and 2 of which are failures. Second, calculate a new sample proportion pˆnew for the new sample. Finally, construct a confidence interval using pˆnew, with the formula pˆnew±1.96pˆnew(1−pˆnew)n+4−−−−−−−− −√.

(i) For the cell phone batteries, consider a sample of 5 in which 1 battery fails to last through 500 charges. Using the alternate method described, what is the value of pˆnew? Show your work.

The following table shows the results of the original simulation with revised 95 percent confidence intervals constructed with the alternate method.

Original pˆ Frequency Revised Lower Endpoint Revised Upper Endpoint
0 168 −0.049 0.494
0.2 360 0.025 0.641
0.4 309 0.120 0.769
0.6 133 0.231 0.880
0.8 28 0.359 0.975
1.0 2 0.506 1.049

(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.