Question

A box contains an unknown number of white and black balls. We wish to estimate the proportion ( p ) of white balls in the box. To do so, we draw ( n ) successive balls with replacement. Let ( Z_{n} ) be the proportion of white balls obtained after ( n ) drawings. (a) Use Chebyshev's Inequality to show that, for all ( varepsilon>0 ), [ mathbb{P}left(left|Z_{n}-p ight| geq varepsilon ight) leq frac{1}{4 n varepsilon^{2}} . ] (b) Using the result in part (a), find the smallest value of ( n ) such that, with probability greater than 0.95 , the proportion ( Z_{n} ) in the sample will estimate ( p ) to within an accuracy of 0.1 .

          A box contains an unknown number of white and black balls. We wish to estimate the proportion ( p ) of white balls in the box. To do so, we draw ( n ) successive balls with replacement. Let ( Z_{n} ) be the proportion of white balls obtained after ( n ) drawings.
(a) Use Chebyshev's Inequality to show that, for all ( varepsilon>0 ),
[
mathbb{P}left(left|Z_{n}-p
ight| geq varepsilon
ight) leq frac{1}{4 n varepsilon^{2}} .
]
(b) Using the result in part (a), find the smallest value of ( n ) such that, with probability greater than 0.95 , the proportion ( Z_{n} ) in the sample will estimate ( p ) to within an accuracy of 0.1 .

$A box contains an unknown number of white and black balls. We wish to estimate the proportion ( p ) of white balls in the box. To do so, we draw ( n ) successive balls with replacement. Let ( Zn ) be the proportion of white balls obtained after ( n ) drawings. (a) Use Chebyshev's Inequality to show that, for all ( varepsilon>0 ), [ mathbbPleft(left|Zn-p ight| geq varepsilon ight) leq frac14 n varepsilon^2 . ] (b) Using the result in part (a), find the smallest value of ( n ) such that, with probability greater than 0.95 , the proportion ( Zn ) in the sample will estimate ( p ) to within an accuracy of 0.1 .$

Added by Romeo J.

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