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...

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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**