Question

The incubation time for Rhode Island Red chicks is normally distributed with a mean of 21 days and standard deviation of approximately 1.5 days. If 1000 eggs are being incubated, how many chicks do we expect will hatch (what proportion of 1,000

1. in 18 to 24 days (18 > x < 24)?
2. in 19.5 to 22.5 days (19.5 > x < 22.5)?
3. in 21 days or fewer (X < 21)?
4. in 15 to 20 days (15 > x < 20)?
5. In more that 24 days (x> 24)?
6. on 25^th day (i.e., x=25)?

Answer 1

Solution :

Given that,

mean = = 21

standard deviation = =1.5

a ) P (18 > x < 24 )

P ( x -  / ) < (18 - 21 / 1.5)

P ( - 3 / 1.5 )

P ( z < - 2 )

Using z table

= 0.0228

P (18 > x < 24 )

P ( x -  / ) < (24- 21 / 1.5)

P ( -3 / 1.5 )

P ( z < 2 )

= 0.9544

= 0.0228 + 0.9772

Probability = 1

b ) P (19.5 > x < 22.5 )

P ( x -  / ) < (19.5 - 21 / 1.5)

P ( - 1.5 / 1.5 )

P ( z < - 1 )

Using z table

= 0.1587

P (18 > x < 24 )

P ( x -  / ) < (22.5 - 21 / 1.5)

P ( 1.5 / 1.5 )

P ( z < 1 )

= 0.8413

= 0.1587+ 0.8413

Probability = 1

c ) (X < 21)

P ( x -  / ) < (21 - 21 / 1.5)

P ( 0 / 1.5 )

P ( z < 0)

= 0.5000

Probability = 0.5000

d ) P (15 > x < 20 )

P ( x -  / ) < (15 - 21 / 1.5)

P ( - 6 / 1.5 )

P ( z < - 4 )

Using z table

= 0.0000

P (18 > x < 24 )

P ( x -  / ) < (20 - 21 / 1.5)

P ( -1 / 1.5 )

P ( z < - 0.67 )

= 0.8413

= 0.0000 + 0.7486

Probability = 0.7486

e )  P ( x > 24 )

= 1 - P ( x > 24 )

= 1 - P ( x -  / ) < (24 - 21 / 1.5)

= 1 -P ( 3 / 1.5 )

= 1 -P ( z < 2 )

= 1 - 0.9544

= 0.0456

Probability = 0.0456

f ) x = 25

using z score formula

z = x -  /

= 25 - 21 / 1.5

=3 / 1.5

= 2

z = 2

using z table

= 0.9544

Probability = 0.9544