The weight of the eggs produced by a certain breed of hen is normally distributed with mean 65.9 grams (g) and standard deviation 5.5 g. 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

The probability of the carton weight being between 775g and 825g is 0.71

Step-by-step explanation:

Hello!

The variable of interest is

X: the weight of an egg of a certain breed of hen. (g)

This variable is normally distributed with mean μ= 65.9g and standard deviation σ=5.5 g

If a carton (a random sample of 12 eggs) is taken, you need to calculate the probability of it weighting 775g and 825g, symbolically: P(775≤X[bar]≤825)

For this, you have to use the distribution of the sample mean X[bar]~N(μ;σ²/n)

If the carton weights 775g, then its eggs will have an average weight of 775/12= 64.48g

If the carton weights 825g, then its eggs will have an average weight of 825/12=68.75g

P(64.48≤X[bar]≤68.75)= P(X[bar]≤68.75) - P(X[bar]≤ 64.48)

P(Z≤(68.75-65.5)/(5.5/√12)) - P(Z≤(64.48-65.5)/(5.5/√12))

P(Z≤1.89) - P(Z≤-0.64)= 0.971 - 0.261= 0.71

The probability of the carton weight being between 775g and 825g is 0.71

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