# Mathematical Statistics and Data Analysis - Solutions

### Chapter 7, Survey Sampling

#### Solution 39

The probability of choosing atleast one defected item in the sample, $\, p \,$ = 1 - Probability of choosing no defected item in the sample.

We have total $\, N \,$ items and $\, k \,$ of them are defected. Thus the probability of choosing a sample of size $\, n \,$ with no defected item = $\, \frac { {N-k} \choose n} {N \choose n} \,$.

Thus $\, p = 1 - \frac { {N-k} \choose n} {N \choose n} \,$. Now we can plot this function with $\, n \,$ and $\, p \,$ as variables to get the desired sample size. Skipping the sketch part :)

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