Beer bottles are filled so that they contain an average of 475 ml of beer in each bottle. suppose that the amount of beer in a bottle is normally distributed with a standard deviation of 8 ml.

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We use the normal distribution density function f(z)  to find the probability of a specific range of values.

Z= ((X-μ)/σ)

μ = 475
σ = 8

a) X< 470 ml

P[X<470] = P[Z<(470-475)/8] = P[Z<(470-475)/8] = P[Z<-0.625] = 0.2659

b) 6 pack with a mean of less than 470ml

Z = ((X-μ)/(σ/√n))

Z = (470-475)/(8/√6) 

P [Z<-1.53] = 0.063

c) 12 pack with a mean of less than 470ml

Z = ((X-μ)/(σ/√n))

Z = (470-475)/(8/√12) 

P [Z<-2.165] = 0.0152