# The weight of the eggs produced by a certain breed of hen is normally distributed with mean 64.2 grams (g) and standard deviation 6.5g. If cartons of such eggs can be considered to be SRSs of...

The weight of the eggs produced by a certain breed of hen is normally distributed with mean 64.2 grams (g) and standard deviation 6.5g. If cartons of such eggs can be considered to be SRSs of size 12 from the population of all eggs, what is the probability that the weight of a carton falls between 775 g and 825 g? (Round your answer to four decimal places.)

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Since the sample is a simple random sample (SRS), then the box of twelve eggs should be a representative sample of the population as a whole. Given that the weight of eggs is Normally distributed with mean 64.2g and sd 6.5g we would expect the weight of the twelve eggs to be Normally distributed with mean m and sd s, where

m = 12*64.2 = 770.4g

s = sqrt(12*6.5^2) = 2*sqrt(3)*6.5 = 22.52

Standardizing to standard Normal (subtract the mean and divide by the standard deviation) and using lookup tables, the probability of the 12 eggs weighing more than 775g is

`1-Phi((775-770.4)/22.517) = 1-Phi(0.2043) = 0.4191`

where `Phi` is the cdf of the standard Normal distribution. The probability of the 12 eggs weighing more than 825g is

`1-Phi((825-770.4)/22.517) = 1-Phi(2.4249) = 0.0077`

**Therefore the probability of the weight of the 12 eggs falling in the range 775-825g is 0.4191 - 0.0077 = 0.4114 to 3 dp**