Question

in the united states, egg sizes are set by the department of agriculture. a \large\ egg, for example, weighs between 57 and 64 grams. suppose the weights of eggs produced by hens owned by a particular farmer are approximately normally distributed with a mean of 55.8 grams and a standard deviation of 7.5 grams.
(a) what proportion of these eggs weigh less than 50 grams? (round to 4 decimal places.)
(b) what percentage of these eggs would be classified as \large\? % (round to 2 decimal places.)

Explanation:

Step1: Calculate the z - score for part (a)

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\), where \(x = 50\), \(\mu=55.8\), and \(\sigma = 7.5\).

$$z=\frac{50 - 55.8}{7.5}=\frac{-5.8}{7.5}\approx - 0.77$$

Step2: Find the proportion for part (a)

Using the standard normal distribution table or a calculator with a normal - distribution function (\(P(Z<z)\)), for \(z=-0.77\), \(P(X < 50)=P(Z<-0.77)\approx0.2206\)

Step3: Calculate the z - scores for part (b)

For \(x_1 = 57\):

$$z_1=\frac{57 - 55.8}{7.5}=\frac{1.2}{7.5}=0.16$$

For \(x_2 = 64\):

$$z_2=\frac{64 - 55.8}{7.5}=\frac{8.2}{7.5}\approx1.09$$

Step4: Find the proportion for part (b)

\(P(57<X<64)=P(0.16 < Z<1.09)\)
Using the property \(P(a < Z < b)=P(Z < b)-P(Z < a)\)
\(P(Z < 1.09)=0.8621\) and \(P(Z < 0.16)=0.5636\)
\(P(0.16 < Z < 1.09)=0.8621-0.5636 = 0.2985\)

Answer:

(a) \(0.2206\)
(b) \(29.85\%\)