Question

The incubation time for Rhode Island Red chicks is normally distributed with mean of 22 days and standard deviation of approximately 3 days. Of 1000 eggs are being incubated, how many chicks do we expect will hatch in 19 to 28 days

Answer 1

We should expect 818 chicks to hatch in 19 to 28 days

Explanation:

Normal Probability Distribution

Problems of normal distributions can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the z-score of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

Normally distributed with mean of 22 days and standard deviation of approximately 3 days.

This means that
`\mu = 22, \sigma = 3`

Proportion between 19 and 28 days:

p-value of Z when X = 28 subtracted by the p-value of Z when X = 19.

X = 28

`Z = (X - \mu)/(\sigma)`

`Z = (28 - 22)/(3)`

`Z = 2`

`Z = 2` has a p-value of 0.977.

X = 19

`Z = (X - \mu)/(\sigma)`

`Z = (19 - 22)/(3)`

`Z = -1`

`Z = -1` has a p-value of 0.159.

0.977 - 0.159 = 0.818

Out of 1000:

0.818*1000 = 818

We should expect 818 chicks to hatch in 19 to 28 days