Question

1)

A survey classified a large number of adults according to whether they were diagnosed as needing eyeglasses to correct their reading vision and whether they use eyeglasses when reading. The proportions falling into the four resulting categories are given in the following table. Uses Eyeglasses for Reading Needs glasses Yes No Yes 0.45 0.12 No 0.06 0.37

a) he/she uses reading glasses given that he/she needs reading glasses?

b) he/she doesn't use reading glasses knowing that he/she doesn't need reading glasses?

c) Are the events "Needs reading glasses" and "Uses reading glasses to read" independent? Explain.

Answer 1

(a)

From the given data, the following Table is calculated:

	Uses glasses	Does not use glasses	Total
Needs glasses	0.45	0.12	0.57
Does not need glasses	0.06	0.37	0.43
Total	0.51	0.49	1.00

P(Uses glasses / Needs glasses) = P(Uses glasses AND Needs glasses)/ P(Needs glasses)

                                                   = 0.45/0.57

                                                    = 0.7895

So,

Answer is:

0.7895

(b)

P(Does not use glasses / Does not Need glasses) = P(Does not use glasses AND Does not Need glasses)/ P(Does not Need glasses)

                                                   = 0.37/0.43

                                                    = 0.8605

So,

Answer is:

0.8605

(c)

P(Needs glasses AND Uses glasses) = 0.45

P(Needs glasses) X P(Uses glasses) = 0.57 X 0.51 = 0.2907

Since P(Needs glasses AND Uses glasses) = 0.45 $\neq$ P(Needs glasses) X P(Uses glasses) = 0.2907, the events "Needs reading glasses" and "Uses reading glasses to read" are not independent.