Question

The incubation time for Rhode Island Red chicks is normally distributed with a mean of 23 days and standard deviation of approximately 2 days. Look at the figure below and answer the following questions. If 1000 eggs are being incubated, how many chicks do we expect will hatch in the following time periods? (Note: In this problem, let us agree to think of a single day or a succession of days as a continuous interval of time. Assume all eggs eventually hatch.) (a) in 19 to 27 days chicks (b) in 21 to 25 days chicks (c) in 23 days or fewer chicks (d) in 17 to 29 days chicks

Answer 1

Given,

= 23, = 2

We convert this to standard normal as

P( X < x) = P( Z < x - / )

a)

P( 19 < X < 27) = P( X < 27 ) - P( X < 19)

= P( Z < 27 - 23 / 2) - P( Z < 19 - 23 / 2)

= P( Z < 2 ) - P( Z < -2)

= P( Z < 2) - ( 1 - P( Z < 2) )

= 0.9772 - ( 1 - 0.9772)

= 0.9544

Among 1000, number of chicks we will expect to hatch = 0.9544 * 1000 = 954

b)

P( 21 < X < 25) = P( X < 25 ) - P( X < 21)

= P( Z < 25 - 23 / 2) - P( Z < 21 - 23 / 2)

= P( Z < 1) - P( Z < -1)

= P( Z < 1) - ( 1 - P( Z < 1) )

= 0.8413 - ( 1 - 0.8413)

= 0.6827

Among 1000, number of chicks we will expect to hatch = 0.6827 * 1000 = 683

c)

P( X <= 23) = P( Z <= 23 - 23 / 2)

= p( Z <= 0)

= 0.50

Amoung 1000, number of chicks we will expect to hatch = 0.5 * 1000 = 500

d)

P( 17 < X < 29) = P( X < 29 ) - P( X < 17)

= P( Z < 29 - 23 / 2) - P( Z < 17 - 23 / 2)

= P( Z < 3) - P( Z < -3)

= P( Z < 3) - ( 1 - P( Z < 3) )

= 0.9987 - ( 1 - 0.9987 )

= 0.9974

Amoung 1000, number of chicks we will expect to hatch = 0.9974 * 1000 = 997