Question

1. The incubation time for a breed of chicks is normally distributed with a mean of 28 days and standard deviation of approximately 1 day. 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 26 to 30 days chicks

(b) in 27 to 29 days chicks

(c) in 28 days or fewer chicks

(d) in 25 to 31 days chicks

Answer 1

P(X < A) = P(Z < (A - mean)/standard deviation)

Mean = 28 days

Standard deviation = 1 day

a) P(hatches in 26 to 30 days) = P(26 < X < 30)

= P(X < 30) - P(X < 26)

= P(Z < (30 - 28)/1) - P(Z < (26 - 28)/1)

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

= 0.9772 - 0.0228

= 0.9544

Number of eggs expected to hatch in this period = 1000x0.9544

= 954 eggs

b) P(hatches in 26 to 30 days) = P(27 < X < 29)

= P(X < 29) - P(X < 27)

= P(Z < (29 - 28)/1) - P(Z < (27 - 28)/1)

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

= 0.8413 - 0.1587

= 0.6826

Number of eggs expected to hatch in this period = 1000x0.6826

= 683 eggs

c) P(in 28 days or fewer) = P(X < 28)

= P(Z < 2)

= 0.9772

Number of eggs expected to hatch in this period = 1000x0.9772

= 977 eggs

d) P(hatches in 25 to 31 days) = P(25 < X < 31)

= P(X < 31) - P(X < 25)

= P(Z < (31 - 28)/1) - P(Z < (25 - 28)/1)

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

= 0.9987 - 0.0013

= 0.9974

Number of eggs expected to hatch in this period = 1000x0.9974

= 997 eggs