Please help me to solve this problem.

(d) For small sample sizes, an alternate method of constructing a confidence interval is available. To implement this method, begin by including an additional 4 observations in the sample, 2 of which are marked as successes and 2 of which are marked as failures. Second, calculate a new sample proportion Pnewfor the new sample. Finally, construct a confidence interval using pnew with the formula Pnew + 1.96 pnew(1-pnew) n + 4 For the laptop cases, consider a sample of 5 in which 1 case fails to last through 500 drops. Using the alternate method described, what is the value of pnew ? 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 p 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.AP Statistic Chapter 8 Investigative Task February 6, 2023 The designer of a certain type of new laptop case claims that the average life span of the cases is 500 drops; that is, the case can be dropped at least 500 times before cracking. To investigate the claim, a engineer will select a random sample of new cases and drop each one 500 times. The proportion of cases that fail to last through 500 drops will be recorded. The results will be used to construct a 95% percent confidence interval to estimate the proportion of all such cases that fail to last through 500 drops. (a) Explain in context what it means to be 95% confident. (b) Suppose the engineer runs an investigation with a random sample of 5 new laptop cases, and 1 case out of the 5 fails to last through 500 drops. 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 p is calculated, where p represents the proportion in the sample that display the characteristic. The following table shows the frequency distribution of p in the 1000 trials. Also shown are the upper and lower endpoints of the 95% confidence interval created of p, using the formula p + 1.96 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) p 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 1.020 0.8 28 0.449 1.151 1.0 2 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