Question

A box contains 2 defective light bulbs and 8 working light bulbs. Four bulbs are drawn at random, without replacement. Find the standard error for the percentage of defective bulbs drawn.

Answer 1

x = no. of defective bulbs

P(x) :

P(0) = (select 4 working bulbs out of 8)/(select 4 bulbs out of 10)

= 8C4 / 10C4

= 0.3333

P(1) = (select 3 working bulbs out of 8)* (select 1 defective bulbs out of 2)/(select 4 bulbs out of 10)

= (8C3 * 2C1) / 10C4

= 0.5333

P(2) = (select 2 working bulbs out of 8)* (select 2 defective bulbs out of 2)/(select 4 bulbs out of 10)

= 8C2 * 2C2 / 10C4

= 0.1333

E(percentage) = sum of percenatge*P(x)

= sum of (x/4)*P(x) * 100%

= 0% * 0.3333 + 25% * 0.5333 + 50%*0.1333

= 19.9975 %

E(percentage ^ 2) = (0/100)^2 * 0.3333 + (25/100)^2 * 0.5333 + (50/100)^2*0.1333

= 0.0667

= 6.67 %

var(percentage) = E(percentage^2) - E(percentnage)^2

= (6.67/100) - (19.9975/100)^2

= 2.6710 %

SE = [var/n]^0.5

= [(2.6710/100)/4]^0.5

= 8.1716 %

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