Hens usually begin laying eggs when they are about 6 months old. Young hens tend to lay smaller eggs, often weighing less than the desired minimum weight of 51 grams. Complete parts a) through c) below. a) The average weight of the eggs produced by the young hens is 50.1 grams, and only 25% of their eggs exceed the desired minimum weight. If a Normal model is appropriate, what would the standard deviation of the egg weights be?

Your question has one part only: a) The average weight of the eggs produced by the young hens is 50.1 grams, and only 25% of their eggs exceed the desired minimum weight. If a Normal model is appropriate, what would the standard deviation of the egg weights be?

Solution

You are given the mean, the reference value, and the percent of egss that exceeds that minimum.

In terms of the parameters of a normal distribution that is:

mean = 50.1g (μ)

X = 51 g

Area of the graph above X = 51 g = 25%

Using a standard normal distribution table, you can find the Z-score for which the area under the curve is greater than 25%, i.e. 0.25

The tables with two decimals for the Z-score show probability 0.2514 for Z-score of 0.67 and probabilidad 0.2483 for Z-score = 0.68.

Thus, you must interpolate. Since, (0.2514 + 0.2483)/2 ≈ 0.25, your Z-score is in the middle.

That is, Z-score = (0.67 + 0.68)/2 = 0.675.

Now use the formula for Z-score and solve for the standard deviation (σ):

[tex]Z-score=(x-\mu)/\sigma[/tex]

[tex]0.675=(51g-50.1g)/\sigma[/tex]

[tex]\sigma =0.9g/0.675=1.33g[/tex]