Recall that the relative frequency or sample proportion calculated from data can be used to approximate the correspondin

Post by **answerhappygod** » Wed May 11, 2022 6:15 am

: Recall That The Relative Frequency Or Sample Proportion Calculated From Data Can Be Used To Approximate The Correspondin 1 (58.99 KiB) Viewed 26 times

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Recall that the relative frequency or sample proportion calculated from data can be used to approximate the corresponding probability for the entire population (so the probability is equivalent to the population proportion). = Consider the case where X is the binomial random variable with parameters n = the number of trials/sample size and p = the probability of success. Then for a single sample of size n (or, equivalently, a single run of n trials), x would represent the # of "successes." for that sample/run. So, for that single sample/run of n trials, we can approximate the probability of success as the sample proportion p # of "successes" For all such samples/runs of n trials, we can then define the variable P = to be the collection of all possible sample proportions for samples of size n/runs of n trials. 72 72 For large values of n, the binomial random variable, X, is approximately normally distributed with mean 4x = np and standard deviation ox = np(1 - p). Assuming that n is sufficiently large for the normal approximation to apply, do the following.

1. In most cases, the margin of error for a sample statistic/point estimate, ô (which approximates the 0-0 population parameter, e) is the critical value from the distribution of times the standard da crror, 02. Using this format, (1) define the 2-tailed margin of error for as an estimate of p and (2) write out the formula for the 2-tailed (1 - a) x 100% confidence interval for the population proportion p centered at P. Please be sure to show all of your work and to clearly explain your logic throughout. If you reference results from the GW 9 discussion, please be sure to clearly identify the result and the question it is from. It has been shown that replacing p with p in both the margin of error and the endpoints of the confidence interval (which contain the parameter we are estimating, p), results in a confidence interval that will capture the value of the true population parameter, p, approximately (1 - a) x 100% of the time. In your final step for the margin of error and the final step of finding the endpoints for the confidence interval, replace p with p. 2. The article “Limited Yield Estimation for Visual Defect Sources" (IEEE Trans. On Semiconductor Manuf., 1997:17 - 23) reported that in a study of a particular wafer inspection process, 356 dies were examined by an inspection probe and 201 of these passed the probe. Assuming the process is stable, answer the following questions. a. In order for a normal approximation to be appropriate for the binomial random variable X, it needs to be true that both np 5 and n(1-P) 2 5. Clearly define what the variable X represents in this example remembering that X needs to follow a binomial distribution. What are the values of n and p for this example (you may round p to 4 decimal places after giving the exact value) and what do these represent? Show that the criterial are met for X to be approximately normally distributed? Be sure to show all of a your work. b. Construct a 2-sided 95% confidence interval for the proportion of all such dies that would pass an inspection probe out of 356. Please be sure to show all of your work and to interpret this interval in the context of the problem. You may use the nearest z-score rounded to 2 decimal places from the standard normal table - you do not need to interpolate.