Question

4. Researchers have found volatile organic chemical markers in the exhaled breath of cancer sufferers but lab analysis is not yet sensitive enough for clinical diagnosis. As a result, work has been done using dogs for the purpose. Assuming, conservatively, that it is not clear what the success rate of a trained dog will be, (a) calculate how many different breath specimens would be required in order to estimate the long-run proportion of correct diagnoses for a particular dog with a margin of error, E, of 5% and with 95% confidence? A German study of the same kind was later found to have been published in which the dog’s success rate was 71%. Knowing this, (b) calculate the number of samples to be taken to achieve the same precision (i.e. the same margin of error) and the same level of confidence.

Answer 1

a. For this case p=0.5, E=0.05 and z value is 1.96 as P(-1.96<z<1.96)

So using formula of E we will find n

$E=z*\sqrt{\frac{pq}{n}}$

So $n=(\frac{z}{E})^2*p*q=(\frac{1.96}{0.05})^2*0.5*0.5=384.16=385$

b. Now for p=0.71

$n=(\frac{z}{E})^2*p*q=(\frac{1.96}{0.05})^2*0.71*0.29=316.39=317$ $n=(\frac{z}{E})^2*p*q = (\frac{1.96}{0.05})^2*0.71*0.29=316.39=317$