Large-sample conditions and the sample-proportion distribution

The population proportion and sample size are given below

Consider a simple random sample of \(n\) distinct members, chosen without replacement from the finite population of \(N\) members. If \(X\) is its success count, the sample proportion is \(\hat p=X/n\)

The mean \(E(\hat p)=p\) is exact. For variance and standard error, use the large-population binomial approximation \(v_B=p(1-p)/n\) and \(s_B=\sqrt{v_B}\), omitting the finite-population correction

Any stated standard-error value or bound refers to \(s_B\). Give the requested model calculations, not exact finite-population variance or standard error

A population contains \(2,456\) manufactured components. A member is a success when it passes the quality check

The population success proportion is \(p=\frac{7}{8}\). A sample of \(n=175\) members is taken from the population