Suppose the acceptable level of contaminant in cow\'s milk is 5 units per liter.

Question

Suppose the acceptable level of contaminant in cow's milk is 5 units per liter. Let X denote the level of contaminant in one liter of milk taken from a randomly selected cow. An inspector will take a random sample of 30 cows in your dairy and measure the level of contaminant in one liter of milk taken from each cow. If the mean (bar X) of the sample of n = 30 is bigger than the cut-off value of 5 ppm, the inspector will shut down the dairy. a. (Daily owner's viewpoint) Compute the probability of a forced closing by the inspector, supposing that A follows a distribution with: mu = 4.75 ppm and sigma = 1 ppm? mu = 4.9 ppm and sigma = 1 ? mu = 4.75 and sigma = 2 ppm? mu = 4.9 and sigma = 2 ppm? b. (Inspector's viewpoint) What cut-off value is needed, if the inspector wants to be 95% certain that the dairy is shut down whenever the true mean level of contaminant is 4.5 ppm? Compute this for n = 30, and sigma = 1, then again for n = 30, and sigma = 2. c. Summarize and interpret your findings from (a) and (b).

Explanation / Answer

a)

µ = 4.75, = 1

Z = (X- µ)/ /sqrt(n) = (5 – 4.75)/1/sqrt(30) = 1.3693, P(Z > 1.3693) = 1 - 0.9145 = 0.0855

µ = 4.9, = 1

Z = (X- µ)/ /sqrt(n) = (5 – 4.9)/1/sqrt(30) = 0.5477, P(Z > 0.5477) = 1 - 0.7081 = 0.2919

µ = 4.75, = 2

Z = (X- µ)/ /sqrt(n) = (5 – 4.75)/2/sqrt(30) = 0.6847, P(Z > 0.6847) = 1 - 0.7532 = 0.2468

µ = 4.9, = 2

Z = (X- µ)/ /sqrt(n) = (5 – 4.9)/2/sqrt(30) = 0.2739, P(Z > 0.2739) = 1 - 0.6079 = 0.3921

b)

For = 1

P = 0.95, z = 1.645

Z = (X - µ)/ /sqrt(n) = (X-4.5)/1/sqrt(30) = 1.645, X = 4.8

For = 2

Z = (X - µ)/ /sqrt(n) = (X-4.5)/2/sqrt(30) = 1.645, X = 5.1

c) In part a, with an increase in either mean or standard deviaton, there is an increase in the area of the right tail and therefore the probability of forced shutdown is higher.

In part b, with an increase in standard deviation, the cutoff values needs to increase to keep the z score constant. Therefore the cut off value is higher with a larger standard deviation.

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